Digital electronics

Basics of Electronics

Electronics
Relationship between Voltage, Current Resistance

All materials are made up from atoms, and all atoms consist of protons, neutrons and electrons. Protons, have a
positive electrical charge. Neutrons have no electrical charge while Electrons, have a negative electrical charge.
Atoms are bound together by powerful forces of attraction existing between the atoms nucleus and the electrons
in its outer shell. When these protons, neutrons and electrons are together within the atom they are happy and
stable. However, if we separate them they exert a potential of attraction called a potential difference. If we
create a circuit or conductor for the electrons to drift back to the protons the flow of electrons is called a current.
The electrons do not flow freely through the circuit, the restriction to this flow is called resistance. Then all
basic electrical or electronic circuit consists of three separate but very much related quantities, Voltage, ( v ),
Current, ( i ) and Resistance, ( Ω ).

Voltage
Voltage, ( V ) is the potential energy of an electrical supply stored in the form of an electrical charge. Voltage
can be thought of as the force that pushes electrons through a conductor and the greater the voltage the greater is
its ability to "push" the electrons through a given circuit. As energy has the ability to do work this potential
energy can be described as the work required in joules to move electrons in the form of an electrical current
around a circuit from one point or node to another. The difference in voltage between any two nodes in a circuit
is known as the Potential Difference, p.d. sometimes called Voltage Drop.

The Potential difference between two points is measured in Volts with the circuit symbol V, or lowercase "v",
although Energy, E lowercase "e" is sometimes used. Then the greater the voltage, the greater is the pressure (or
pushing force) and the greater is the capacity to do work.

A constant voltage source is called a DC Voltage with a voltage that varies periodically with time is called an
AC voltage. Voltage is measured in volts, with one volt being defined as the electrical pressure required to force
an electrical current of one ampere through a resistance of one Ohm. Voltages are generally expressed in Volts
with prefixes used to denote sub-multiples of the voltage such as microvolts ( μV = 10-6 V ), millivolts ( mV =
10-3 V ) or kilovolts ( kV = 103 V ). Voltage can be either positive or negative.

Batteries or power supplies are mostly used to produce a steady D.C. (direct current) voltage source such as 5v,
12v, 24v etc in electronic circuits and systems. While A.C. (alternating current) voltage sources are available for
domestic house and industrial power and lighting as well as power transmission. The mains voltage supply in
the United Kingdom is currently 230 volts a.c. and 110 volts a.c. in the USA. General electronic circuits operate
on low voltage DC battery supplies of between 1.5V and 24V d.c. The circuit symbol for a constant voltage
source usually given as a battery symbol with a positive, + and negative, - sign indicating the direction of the
polarity. The circuit symbol for an alternating voltage source is a circle with a sine wave inside.

Voltage Symbols
A simple relationship can be made between a tank of water and a voltage supply. The higher the water tank
above the outlet the greater the pressure of the water as more energy is released, the higher the voltage the
greater the potential energy as more electrons are released. Voltage is always measured as the difference
between any two points in a circuit and the voltage between these two points is generally referred to as the
"Voltage drop". Any voltage source whether DC or AC likes an open or semi-open circuit condition but hates
any short circuit condition as this can destroy it.

Electrical Current
Electrical Current, ( I ) is the movement or flow of electrical charge and is measured in Amperes, symbol i, for
intensity). It is the continuous and uniform flow (called a drift) of electrons (the negative particles of an atom)
around a circuit that are being "pushed" by the voltage source. In reality, electrons flow from the negative (-ve)
terminal to the positive (+ve) terminal of the supply and for ease of circuit understanding conventional current
flow assumes that the current flows from the positive to the negative terminal. Generally in circuit diagrams the
flow of current through the circuit usually has an arrow associated with the symbol, I, or lowercase i to indicate
the actual direction of the current flow. However, this arrow usually indicates the direction of conventional
current flow and not necessarily the direction of the actual flow.

In electronic circuits, a current source is a circuit element that provides a specified amount of current for
example, 1A, 5A 10 Amps etc, with the circuit symbol for a constant current source given as a circle with an
arrow inside indicating its direction. Current is measured in Amps and an amp or ampere is defined as the
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number of electrons or charge (Q in Coulombs) passing a certain point in the circuit in one second, (t in
Seconds). Current is generally expressed in Amps with prefixes used to denote micro amps ( μA = 10-6A ) or
milli amps ( mA = 10-3A ). Note that electrical current can be either positive in value or negative in value
depending upon its direction of flow.

Current that flows in a single direction is called Direct Current, or D.C. and current that alternates back and
forth through the circuit is known as Alternating Current, or A.C.. Whether AC or DC current only flows
through a circuit when a voltage source is connected to it with its "flow" being limited to both the resistance of
the circuit and the voltage source pushing it. Also, as AC currents (and voltages) are periodic and vary with
time the "effective" or "RMS", (Root Mean Squared) value given as Irms produces the same average power loss
equivalent to a DC current Iaverage . Current sources are the opposite to voltage sources in that they like short
or closed circuit conditions but hate open circuit conditions as no current will flow.

Resistance
The Resistance, ( R ) of a circuit is its ability to resist or prevent the flow of current (electron flow) through
itself making it necessary to apply a greater voltage to the electrical circuit to cause the current to flow again.
Resistance is measured in Ohms, Greek symbol ( Ω, Omega ) with prefixes used to denote Kilo-ohms ( kΩ =
103Ω ) and Mega-ohms ( MΩ = 106Ω ). Note that Resistance cannot be negative in value only positive.

Resistance can be linear in nature or non-linear in nature. Linear resistance obeys Ohm's Law and controls or
limits the amount of current flowing within a circuit in proportion to the voltage supply connected to it and
therefore the transfer of power to the load. Non-linear resistance, does not obey Ohm's Law but has a voltage
drop across it that is proportional to some power of the current. Resistance is pure and is not affected by
frequency with the AC impedance of a resistance being equal to its DC resistance and as a result can not be
negative. resistance is always positive. Also, resistance is an attenuator which has the ability to change the
characteristics of a circuit by the effect of load resistance or by temperature which changes its resistivity.
For very low values of resistance, for example milli-ohms, ( mΩ´s ) it is sometimes more easier to use the
reciprocal of resistance ( 1/R ) rather than resistance ( R ) itself. The reciprocal of resistance is called
Conductance, symbol ( G ) and represents the ability of a conductor or device to conduct electricity. In other
words the ease by which current flows. High values of conductance implies a good conductor such as copper
while low values of conductance implies a bad conductor such as wood. The standard unit of measurement
given for conductance is the Siemen, symbol (S).
Quantity Symbol Unit of Measure Abbreviation

Voltage V or E Volt V
Current I Amp A
Resistance R Ohms Ω

Ohms Law
The relationship between Voltage, Current and Resistance in any DC electrical circuit was firstly discovered by
the German physicist Georg Ohm, (1787 - 1854). Georg Ohm found that, at a constant temperature, the
electrical current flowing through a fixed linear resistance is directly proportional to the voltage applied across
it, and also inversely proportional to the resistance. This relationship between the Voltage, Current and
Resistance forms the bases of Ohms Law and is shown below.

Ohms Law Relationship

By knowing any two values of the Voltage, Current or Resistance quantities we can use Ohms Law to find the
third missing value. Ohms Law is used extensively in electronics formulas and calculations so it is "very
important to understand and accurately remember these formulas".
To find the Voltage, ( V )
[V=IxR] V (volts) = I (amps) x R (Ω)

To find the Current, ( I )
[I=V÷R] I (amps) = V (volts) ÷ R (Ω)

To find the Resistance, ( R )
[R=V÷I] R (Ω) = V (volts) ÷ I (amps)

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It is sometimes easier to remember Ohms law relationship by using pictures. Here the three quantities of V, I
and R have been superimposed into a triangle (affectionately called the Ohms Law Triangle) giving voltage at
the top with current and resistance at the bottom. This arrangement represents the actual position of each
quantity in the Ohms law formulas.

Ohms Law Triangle

and transposing the above equation gives us the following combinations of the same equation:

Then by using Ohms Law we can see that a voltage of 1V applied to a resistor of 1Ω will cause a current of 1A
to flow and the greater the resistance, the less current will flow for any applied voltage. Any Electrical device or
component that obeys "Ohms Law" that is, the current flowing through it is proportional to the voltage across it
(I α V), such as resistors or cables, are said to be "Ohmic" in nature, and devices that do not, such as transistors
or diodes, are said to be "Non-ohmic" devices.

Power in Electrical Circuits

Electrical Power, (P) in a circuit is the amount of energy that is absorbed or produced within the circuit. A
source of energy such as a voltage will produce or deliver power while the connected load absorbs it. The
quantity symbol for power is P and is the product of voltage multiplied by the current with the unit of
measurement being the Watt (W) with prefixes used to denote milliwatts (mW = 10-3W) or kilowatts (kW =
103W). By using Ohm's law and substituting for V, I and R the formula for electrical power can be found as:

To find the Power (P)

[P=VxI] P (watts) = V (volts) x I (amps) Also,
[ P = V ÷ R ] P (watts) = V2 (volts) ÷ R (Ω) Also,
2

[ P = I2 x R ] P (watts) = I2 (amps) x R (Ω)
Again, the three quantities have been superimposed into a triangle this time called the Power Triangle with
power at the top and current and voltage at the bottom. Again, this arrangement represents the actual position of
each quantity in the Ohms law power formulas.

The Power Triangle

One other point about Power, if the calculated power is positive in value for any formula the component absorbs
the power, but if the calculated power is negative in value the component produces power, in other words it is a
source of electrical energy. Also, we now know that the unit of power is the WATT but some electrical devices
such as electric motors have a power rating in Horsepower or hp. The relationship between horsepower and
watts is given as: 1hp = 746W.

Ohms Law Pie Chart

We can now take all the equations from above for finding Voltage, Current, Resistance and Power and
condense them into a simple Ohms Law pie chart for use in DC circuits and calculations.

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Ohms Law Pie Chart

Example No1

For the circuit shown below find the Voltage (V), the Current (I), the Resistance (R) and the Power (P).

Voltage [ V = I x R ] = 2 x 12Ω = 24V
Current [ I = V ÷ R ] = 24 ÷ 12Ω = 2A
Resistance [ R = V ÷ I ] = 24 ÷ 2 = 12 Ω
Power [ P = V x I ] = 24 x 2 = 48W

Power within an electrical circuit is only present when BOTH voltage and current are present for example, In
an Open-circuit condition, Voltage is present but there is no current flow I = 0 (zero), therefore V x 0 is 0 so the
power dissipated within the circuit must also be 0. Likewise, if we have a Short-circuit condition, current flow
is present but there is no voltage V = 0, therefore 0 x I = 0 so again the power dissipated within the circuit is 0.

As electrical power is the product of V x I, the power dissipated in a circuit is the same whether the circuit
contains high voltage and low current or low voltage and high current flow. Generally, power is dissipated in
the form of Heat (heaters), Mechanical Work such as motors, etc Energy in the form of radiated (Lamps) or
as stored energy (Batteries).

Energy in Electrical Circuits

Electrical Energy that is either absorbed or produced is the product of the electrical power measured in Watts
and the time in Seconds with the unit of energy given as Watt-seconds or Joules.

Although electrical energy is measured in Joules it can become a very large value when used to calculate the
energy consumed by a component. For example, a single 100 W light bulb connected for one hour will consume
a total of 100 watts x 3600 sec = 360,000 Joules. So prefixes such as kilojoules (kJ = 103J) or megajoules (MJ
= 106J) are used instead. If the electrical power is measured in "kilowatts" and the time is given in hours then
the unit of energy is in kilowatt-hours or kWh which is commonly called a "Unit of Electricity" and is what
consumers purchase from their electricity suppliers.

Electrical Units of Measure

The standard SI units used for the measurement of voltage, current and resistance are the Volt [ V ], Ampere
[ A ] and Ohms [ Ω ] respectively. Sometimes in electrical or electronic circuits and systems it is necessary to
use multiples or sub-multiples (fractions) of these standard units when the quantities being measured are very
large or very small. The following table gives a list of some of the standard units used in electrical formulas and
component values.

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Standard Electrical Units

Parameter Symbol Measuring Unit Description

Voltage Volt V or E Unit of Electrical Potential V = I × R
Current Ampere I or i Unit of Electrical Current I = V ÷ R
Resistance Ohm R or Ω Unit of DC Resistance R = V ÷ I
Conductance Siemen G or ℧ Reciprocal of Resistance G = 1 ÷ R
Capacitance Farad C Unit of Capacitance C = Q ÷ V
Charge Coulomb Q Unit of Electrical Charge Q = C × V
Inductance Henry L or H Unit of Inductance VL = -L(di/dt)
Power Watts W Unit of Power P = V × I or I2 × R
Impedance Ohm Z Unit of AC Resistance Z2 = R2 + X2
Frequency Hertz Hz Unit of Frequency ƒ = 1 ÷ T
Multiples and Sub-multiples

There is a huge range of values encountered in electrical and electronic engineering between a maximum value
and a minimum value of a standard electrical unit. For example, resistance can be lower than 0.01Ω's or higher
than 1,000,000Ω's. By using multiples and submultiple's of the standard unit we can avoid having to write too
many zero's to define the position of the decimal point. The table below gives their names and abbreviations.

Prefix Symbol Multiplier Power of Ten
Terra T 1,000,000,000,000 1012
Giga G 1,000,000,000 109
Mega M 1,000,000 106
kilo k 1,000 103
none none 1 100
centi c 1/100 10-2
milli m 1/1,000 10-3
micro µ 1/1,000,000 10-6
nano n 1/1,000,000,000 10-9
Pico P 1/1,000,000,000,000 10-12
So to display the units or multiples of units for Resistance, Current or Voltage we would use as an example:

1kV = 1 kilo-volt - which is equal to 1,000 Volts.
1mA = 1 milli-amp - which is equal to one thousandths (1/1000) of an Ampere.
47kΩ = 47 kilo-ohms - which is equal to 47 thousand Ohms.
100uF = 100 micro-farads - which is equal to 100 millionths (1/1,000,000) of a Farad.
1kW = 1 kilo-watt - which is equal to 1,000 Watts.
1MHz = 1 mega-hertz - which is equal to one million Hertz.
To convert from one prefix to another it is necessary to either multiply or divide by the difference between the
two values. For example, convert 1MHz into kHz.

Well we know from above that 1MHz is equal to one million (1,000,000) hertz and that 1kHz is equal to one
thousand (1,000) hertz, so one 1MHz is one thousand times bigger than 1kHz. Then to convert Mega-hertz into
Kilo-hertz we need to multiply mega-hertz by one thousand, as 1MHz is equal to 1000 kHz. Likewise, if we
needed to convert kilo-hertz into mega-hertz we would need to divide by one thousand. A much simpler and
quicker method would be to move the decimal point either left or right depending upon whether you need to
multiply or divide.

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As well as the "Standard" electrical units of measure shown above, other units are also used in electrical
engineering to denote other values and quantities such as:

• Wh − The Watt-Hour, The amount of electrical energy consumed in the circuit by a load of one watt
drawing power for one hour, eg a Light Bulb. It is commonly used in the form of kWh (Kilowatt-
hour) which is 1,000 watt-hours or MWh (Megawatt-hour) which is 1,000,000 watt-hours.

• dB − The Decibel, The decibel is a one tenth unit of the Bel (symbol B) and is used to represent gain
either in voltage, current or power. It is a logarithmic unit expressed in dB and is commonly used
to represent the ratio of input to output in amplifier, audio circuits or loudspeaker systems.
For example, the dB ratio of an input voltage (Vin) to an output voltage (Vout) is expressed as
20log10 (Vout/Vin). The value in dB can be either positive (20dB) representing gain or negative (-
20dB) representing loss with unity, ie input = output expressed as 0dB.

• θ− Phase Angle, The Phase Angle is the difference in degrees between the voltage waveform and the
current waveform having the same periodic time. It is a time difference or time shift and
depending upon the circuit element can have a "leading" or "lagging" value. The phase angle of a
waveform is measured in degrees or radians.

• ω− Angular Frequency, Another unit which is mainly used in a.c. circuits to represent the Phasor
Relationship between two or more waveforms is called Angular Frequency, symbol ω. This is a
rotational unit of angular frequency 2πƒ with units in radians per second, rads/s. The complete
revolution of one cycle is 360 degrees or 2π, therefore, half a revolution is given as 180 degrees or
π rad.
• τ− Time Constant, The Time Constant of an impedance circuit or linear first-order system is the time
it takes for the output to reach 63.7% of its maximum or minimum output value when subjected to
a Step Response input. It is a measure of reaction time.

Kirchoffs Circuit Law
We saw in the Resistors tutorial that a single equivalent resistance, ( RT ) can be found when two or more
resistors are connected together in either series, parallel or combinations of both, and that these circuits obey
Ohm's Law. However, sometimes in complex circuits such as bridge or T networks, we can not simply use
Ohm's Law alone to find the voltages or currents circulating within the circuit. For these types of calculations
we need certain rules which allow us to obtain the circuit equations and for this we can use Kirchoffs Circuit
Law.

In 1845, a German physicist, Gustav Kirchoff developed a pair or set of rules or laws which deal with the
conservation of current and energy within electrical circuits. These two rules are commonly known as: Kirchoffs
Circuit Laws with one of Kirchoffs laws dealing with the current flowing around a closed circuit, Kirchoffs
Current Law, (KCL) while the other law deals with the voltage sources present in a closed circuit, Kirchoffs
Voltage Law, (KVL).

Kirchoffs First Law - The Current Law, (KCL)
Kirchoffs Current Law or KCL, states that the "total current or charge entering a junction or node is exactly
equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within
the node". In other words the algebraic sum of ALL the currents entering and leaving a node must be equal to
zero, I(exiting) + I(entering) = 0. This idea by Kirchoff is commonly known as the Conservation of Charge.

Kirchoffs Current Law

Here, the 3 currents entering the node, I1, I2, I3 are all positive in value and the 2 currents leaving the node, I4
and I5 are negative in value. Then this means we can also rewrite the equation as;

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I1 + I2 + I3 - I4 - I5 = 0

The term Node in an electrical circuit generally refers to a connection or junction of two or more current
carrying paths or elements such as cables and components. Also for current to flow either in or out of a node a
closed circuit path must exist. We can use Kirchoff's current law when analysing parallel circuits.

Kirchoffs Second Law - The Voltage Law, (KVL)

Kirchoffs Voltage Law or KVL, states that "in any closed loop network, the total voltage around the loop is
equal to the sum of all the voltage drops within the same loop" which is also equal to zero. In other words the
algebraic sum of all voltages within the loop must be equal to zero. This idea by Kirchoff is known as the
Conservation of Energy.

Kirchoffs Voltage Law

Starting at any point in the loop continue in the same direction noting the direction of all the voltage drops,
either positive or negative, and returning back to the same starting point. It is important to maintain the same
direction either clockwise or anti-clockwise or the final voltage sum will not be equal to zero. We can use
Kirchoff's voltage law when analyzing series circuits.

When analysing either DC circuits or AC circuits using Kirchoffs Circuit Laws a number of definitions and
terminologies are used to describe the parts of the circuit being analyzed such as: node, paths, branches, loops
and meshes. These terms are used frequently in circuit analysis so it is important to understand them.
Circuit - a circuit is a closed loop conducting path in which an electrical current flows.
Path - a line of connecting elements or sources with no elements or sources included more than once.
Node - a node is a junction, connection or terminal within a circuit were two or more circuit elements
are connected or joined together giving a connection point between two or more branches. A node is
indicated by a dot.
Branch - a branch is a single or group of components such as resistors or a source which are connected
between two nodes.
Loop - a loop is a simple closed path in a circuit in which no circuit element or node is encountered
more than once.
Mesh - a mesh is a single open loop that does not have a closed path. No components are inside a mesh.
Components are connected in series if they carry the same current.
Components are connected in parallel if the same voltage is across them.

Example No1
Find the current flowing in the 40Ω Resistor, R3

The circuit has 3 branches, 2 nodes (A and B) and 2 independent loops.

Using Kirchoffs Current Law, KCL the equations are given as;
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At node A : I1 + I2 = I3
At node B : I3 = I1 + I2
Using Kirchoffs Voltage Law, KVL the equations are given as;
Loop 1 is given as : 10 = R1 x I1 + R3 x I3 = 10I1 + 40I3
Loop 2 is given as : 20 = R2 x I2 + R3 x I3 = 20I2 + 40I3
Loop 3 is given as : 10 - 20 = 10I1 - 20I2
As I3 is the sum of I1 + I2 we can rewrite the equations as;
Eq. No 1 : 10 = 10I1 + 40(I1 + I2) = 50I1 + 40I2
Eq. No 2 : 20 = 20I2 + 40(I1 + I2) = 40I1 + 60I2
We now have two "Simultaneous Equations" that can be reduced to give us the value of both I1 and I2

Substitution of I1 in terms of I2 gives us the value of I1 as -0.143 Amps
Substitution of I2 in terms of I1 gives us the value of I2 as +0.429 Amps
As : I3 = I1 + I2
The current flowing in resistor R3 is given as : -0.143 + 0.429 = 0.286 Amps
and the voltage across the resistor R3 is given as : 0.286 x 40 = 11.44 volts
The negative sign for I1 means that the direction of current flow initially chosen was wrong, but never the less
still valid. In fact, the 20v battery is charging the 10v battery.

Application of Kirchoffs Circuit Laws

These two laws enable the Currents and Voltages in a circuit to be found, ie, the circuit is said to be "Analysed",
and the basic procedure for using Kirchoff's Circuit Laws is as follows:

1. Assume all voltages and resistances are given. ( If not label them V1, V2,... R1, R2, etc. )
2. Label each branch with a branch current. ( I1, I2, I3 etc. )
3. Find Kirchoff's first law equations for each node.
4. Find Kirchoff's second law equations for each of the independent loops of the circuit.
5. Use Linear simultaneous equations as required to find the unknown currents.

As well as using Kirchoffs Circuit Law to calculate the various voltages and currents circulating around a
linear circuit, we can also use loop analysis to calculate the currents in each independent loop which helps to
reduce the amount of mathematics required by using just Kirchoff's laws. In the next tutorial about DC Theory
we will look at Mesh Current Analysis to do just that.

Circuit Analysis

In the previous tutorial we saw that complex circuits such as bridge or T-networks can be solved using
Kirchoff's Circuit Laws. While Kirchoff´s Laws give us the basic method for analysing any complex electrical
circuit, there are different ways of improving upon this method by using Mesh Current Analysis or Nodal
Voltage Analysis that results in a lessening of the math's involved and when large networks are involved this
reduction in maths can be a big advantage.

For example, consider the circuit from the previous section.

Mesh Analysis Circuit

One simple method of reducing the amount of math's involved is to analyse the circuit using Kirchoff's Current
Law equations to determine the currents, I1 and I2 flowing in the two resistors. Then there is no need to calculate
the current I3 as its just the sum of I1 and I2. So Kirchhoff’s second voltage law simply becomes:
Equation No 1 : 10 = 50I1 + 40I2
Equation No 2 : 20 = 40I1 + 60I2
therefore, one line of math's calculation have been saved.

Mesh Current Analysis
A more easier method of solving the above circuit is by using Mesh Current Analysis or Loop Analysis which
is also sometimes called Maxwell´s Circulating Currents method. Instead of labeling the branch currents we

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need to label each "closed loop" with a circulating current. As a general rule of thumb, only label inside loops in
a clockwise direction with circulating currents as the aim is to cover all the elements of the circuit at least once.
Any required branch current may be found from the appropriate loop or mesh currents as before using
Kirchoff´s method.

For example: : i1 = I1 , i2 = -I2 and I3 = I1 - I2

We now write Kirchoff's voltage law equation in the same way as before to solve them but the advantage of this
method is that it ensures that the information obtained from the circuit equations is the minimum required to
solve the circuit as the information is more general and can easily be put into a matrix form.


These equations can be solved quite quickly by using a single mesh impedance matrix Z. Each element ON the
principal diagonal will be "positive" and is the total impedance of each mesh. Where as, each element OFF the
principal diagonal will either be "zero" or "negative" and represents the circuit element connecting all the
appropriate meshes. This then gives us a matrix of:

Where:
[ V ] gives the total battery voltage for loop 1 and then loop 2.
[ I ] states the names of the loop currents which we are trying to find.
[ R ] is called the resistance matrix.
and this gives I1 as -0.143 Amps and I2 as -0.429 Amps
As : I3 = I1 - I2
The current I3 is therefore given as: -0.143 - (-0.429) = 0.286 Amps
Which is the same value of 0.286 amps, we found using Kirchoff´s circuit law in the previous tutorial.

Mesh Current Analysis Summary.

This "look-see" method of circuit analysis is probably the best of all the circuit analysis methods with the basic
procedure for solving Mesh Current Analysis equations is as follows:

1. Label all the internal loops with circulating currents. (I1, I2, ...IL etc)
2. Write the [ L x 1 ] column matrix [ V ] giving the sum of all voltage sources in each loop.
3. Write the [ L x L ] matrix, [ R ] for all the resistances in the circuit as follows;
o R11 = the total resistance in the first loop.
o Rnn = the total resistance in the Nth loop.
o RJK = the resistance which directly joins loop J to Loop K.
4. Write the matrix or vector equation [V] = [R] x [I] where [I] is the list of currents to be found.
As well as using Mesh Current Analysis, we can also use node analysis to calculate the voltages around the
loops, again reducing the amount of mathematics required using just Kirchoff's laws.
Nodal Voltage Analysis
As well as using Mesh Analysis to solve the currents flowing around complex circuits it is also possible to use
nodal analysis methods too. Nodal Voltage Analysis complements the previous mesh analysis in that it is
equally powerful and based on the same concepts of matrix analysis. As its name implies, Nodal Voltage
Analysis uses the "Nodal" equations of Kirchoff's first law to find the voltage potentials around the circuit. By
adding together all these nodal voltages the net result will be equal to zero. Then, if there are "N" nodes in the

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circuit there will be "N-1" independent nodal equations and these alone are sufficient to describe and hence
solve the circuit.

At each node point write down Kirchoff's first law equation, that is: "the currents entering a node are exactly
equal in value to the currents leaving the node" then express each current in terms of the voltage across the
branch. For "N" nodes, one node will be used as the reference node and all the other voltages will be referenced
or measured with respect to this common node.

Nodal Voltage Analysis Circuit

In the above circuit, node D is chosen as the reference node and the other three nodes are assumed to have
voltages, Va, Vb and Vc with respect to node D. For example;

As Va = 10v and Vc = 20v , Vb can be easily found by:

Again is the same value of 0.286 amps, we found using Kirchoff's Circuit Law in the previous tutorial.

From both Mesh and Nodal Analysis methods we have looked at so far, this is the simplest method of solving
this particular circuit. Generally, nodal voltage analysis is more appropriate when there are a larger number of
current sources around. The network is then defined as: [ I ] = [ Y ] [ V ] where [ I ] are the driving current
sources, [ V ] are the nodal voltages to be found and [ Y ] is the admittance matrix of the network which
operates on [ V ] to give [ I ].

Nodal Voltage Analysis Summary.

The basic procedure for solving Nodal Analysis equations is as follows:

1. Write down the current vectors, assuming currents into a node are positive. ie, a (N x 1) matrices
for "N" independent nodes.
2. Write the admittance matrix [Y] of the network where:
o Y11 = the total admittance of the first node.
o Y22 = the total admittance of the second node.
o RJK = the total admittance joining node J to node K.
3. For a network with "N" independent nodes, [Y] will be an (N x N) matrix and that Ynn will be
positive and Yjk will be negative or zero value.
4. The voltage vector will be (N x L) and will list the "N" voltages to be found.

Thevenins Theorem

In the previous 3 tutorials we have looked at solving complex electrical circuits using Kirchoff's Circuit Laws,
Mesh Analysis and finally Nodal Analysis but there are many more "Circuit Analysis Theorems" available to
calculate the currents and voltages at any point in a circuit. In this tutorial we will look at one of the more
common circuit analysis theorems (next to Kirchoff´s) that has been developed, Thevenins Theorem.

Thevenins Theorem states that "Any linear circuit containing several voltages and resistances can be replaced
by just a Single Voltage in series with a Single Resistor". In other words, it is possible to simplify any "Linear"
circuit, no matter how complex, to an equivalent circuit with just a single voltage source in series with a

K. Adisesha Page 10


resistance connected to a load as shown below. Thevenins Theorem is especially useful in analyzing power or
battery systems and other interconnected circuits where it will have an effect on the adjoining part of the circuit.

Thevenins equivalent circuit.

As far as the load resistor RL is concerned, any "one-port" network consisting of resistive circuit elements and
energy sources can be replaced by one single equivalent resistance Rs and equivalent voltage Vs, where Rs is
the source resistance value looking back into the circuit and Vs is the open circuit voltage at the terminals.


Firstly, we have to remove the centre 40Ω resistor and short out (not physically as this would be dangerous) all
the emf´s connected to the circuit, or open circuit any current sources. The value of resistor Rs is found by
calculating the total resistance at the terminals A and B with all the emf´s removed, and the value of the voltage
required Vs is the total voltage across terminals A and B with an open circuit and no load resistor Rs connected.
Then, we get the following circuit.

Find the Equivalent Resistance (Rs)

Find the Equivalent Voltage (Vs)

We now need to reconnect the two voltages back into the circuit, and as VS = VAB the current flowing around
the loop is calculated as:

so the voltage drop across the 20Ω resistor can be calculated as:
VAB = 20 - (20Ω x 0.33amps) = 13.33 volts.

Then the Thevenins Equivalent circuit is shown below with the 40Ω resistor connected.

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and from this the current flowing in the circuit is given as:

Which again, is the same value of 0.286 amps, we found using Kirchoff´s circuit law in the previous tutorial.

Thevenins theorem can be used as a circuit analysis method and is particularly useful if the load is to take a
series of different values. It is not as powerful as Mesh or Nodal analysis in larger networks because the use of
Mesh or Nodal analysis is usually necessary in any Thevenin exercise, so it might as well be used from the start.
However, Thevenins equivalent circuits of Transistors, Voltage Sources such as batteries etc, are very useful
in circuit design.

Thevenins Theorem Summary

The basic procedure for solving a circuit using Thevenins Theorem is as follows:

1. Remove the load resistor RL or component concerned.
2. Find RS by shorting all voltage sources or by open circuiting all the current sources.
3. Find VS by the usual circuit analysis methods.
4. Find the current flowing through the load resistor RL.

Norton Theorem

In some ways Norton's Theorem can be thought of as the opposite to "Thevenins Theorem", in that Thevenin
reduces his circuit down to a single resistance in series with a single voltage. Norton on the other hand reduces
his circuit down to a single resistance in parallel with a constant current source. Nortons Theorem states that
"Any linear circuit containing several energy sources and resistances can be replaced by a single Constant
Current generator in parallel with a Single Resistor". As far as the load resistance, RL is concerned this single
resistance, RS is the value of the resistance looking back into the network with all the current sources open
circuited and IS is the short circuit current at the output terminals as shown below.

Nortons equivalent circuit.

The value of this "constant current" is one which would flow if the two output terminals where shorted together
while the source resistance would be measured looking back into the terminals, (the same as Thevenin).

For example, consider our now familiar circuit from the previous section.

To find the Nortons equivalent of the above circuit we firstly have to remove the centre 40Ω load resistor and
short out the terminals A and B to give us the following circuit.

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When the terminals A and B are shorted together the two resistors are connected in parallel across their two
respective voltage sources and the currents flowing through each resistor as well as the total short circuit current
can now be calculated as:

with A-B Shorted Out

If we short-out the two voltage sources and open circuit terminals A and B, the two resistors are now effectively
connected together in parallel. The value of the internal resistor Rs is found by calculating the total resistance at
the terminals A and B giving us the following circuit.

Find the Equivalent Resistance (Rs)

Having found both the short circuit current, Is and equivalent internal resistance, Rs this then gives us the
following Nortons equivalent circuit.

Nortons equivalent circuit.

Ok, so far so good, but we now have to solve with the original 40Ω load resistor connected across terminals A
and B as shown below.

Again, the two resistors are connected in parallel across the terminals A and B which gives us a total resistance
of:

The voltage across the terminals A and B with the load resistor connected is given as:

Then the current flowing in the 40Ω load resistor can be found as:

K. Adisesha Page 13


which again, is the same value of 0.286 amps, we found using Kirchoff´s circuit law in the previous tutorials.

Nortons Theorem Summary
The basic procedure for solving a circuit using Nortons Theorem is as follows:

1. Remove the load resistor RL or component concerned.
2. Find RS by shorting all voltage sources or by open circuiting all the current sources.
3. Find IS by placing a shorting link on the output terminals A and B.
4. Find the current flowing through the load resistor RL.

Maximum Power Transfer

We have seen in the previous tutorials that any complex circuit or network can be replaced by a single energy
source in series with a single internal source resistance, RS. Generally, this source resistance or even impedance
if inductors or capacitors are involved is of a fixed value in Ohm´s. However, when we connect a load
resistance, RL across the output terminals of the power source, the impedance of the load will vary from an
open-circuit state to a short-circuit state resulting in the power being absorbed by the load becoming dependent
on the impedance of the actual power source. Then for the load resistance to absorb the maximum power
possible it has to be "Matched" to the impedance of the power source and this forms the basis of Maximum
Power Transfer.
The Maximum Power Transfer Theorem is another useful analysis method to ensure that the maximum
amount of power will be dissipated in the load resistance when the value of the load resistance is exactly equal
to the resistance of the power source. The relationship between the load impedance and the internal impedance
of the energy source will give the power in the load. Consider the circuit below.

Thevenins Equivalent Circuit.

In our Thevenin equivalent circuit above, the maximum power transfer theorem states that "the maximum
amount of power will be dissipated in the load resistance if it is equal in value to the Thevenin or Norton source
resistance of the network supplying the power" in other words, the load resistance resulting in greatest power
dissipation must be equal in value to the equivalent Thevenin source resistance, then RL = RS but if the load
resistance is lower or higher in value than the Thevenin source resistance of the network, its dissipated power
will be less than maximum. For example, find the value of the load resistance, RL that will give the maximum
power transfer in the following circuit.

Example No1.

Where:
RS = 25Ω
RL is variable between 0 - 100Ω
VS = 100v

Then by using the following Ohm's Law equations:

We can now complete the following table to determine the current and power in the circuit for different values
of load resistance.

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Table of Current against Power

RL I P RL I P
0 0 0 25 2.0 100
5 3.3 55 30 1.8 97
10 2.8 78 40 1.5 94
15 2.5 93 60 1.2 83
20 2.2 97 100 0.8 64
Using the data from the table above, we can plot a graph of load resistance, RL against power, P for different
values of load resistance. Also notice that power is zero for an open-circuit (zero current condition) and also for
a short-circuit (zero voltage condition).

Graph of Power against Load Resistance

From the above table and graph we can see that the Maximum Power Transfer occurs in the load when the
load resistance, RL is equal in value to the source resistance, RS so then: RS = RL = 25Ω. This is called a
"matched condition" and as a general rule, maximum power is transferred from an active device such as a power
supply or battery to an external device occurs when the impedance of the external device matches that of the
source. Improper impedance matching can lead to excessive power use and dissipation.

Transformer Impedance Matching

One very useful application of impedance matching to provide maximum power transfer is in the output stages
of amplifier circuits, where the speakers impedance is matched to the amplifier output impedance to obtain
maximum sound power output. This is achieved by using a matching transformer to couple the load to the
amplifiers output as shown below.

Transformer Coupling

The maximum power transfer can be obtained even if the output impedance is not the same as the load
impedance. This can be done using a suitable "turns ratio" on the transformer with the corresponding ratio of
load impedance, ZLOAD to output impedance, ZOUT matches that of the ratio of the transformers primary turns to
secondary turns as a resistance on one side of the transformer becomes a different value on the other. If the load
impedance, ZLOAD is purely resistive and the source impedance is purely resistive, ZOUT then the equation for
finding the maximum power transfer is given as:

Where: NP is the number of primary turns and NS the number of secondary turns on the transformer. Then by
varying the value of the transformers turns ratio the output impedance can be "matched" to the source
impedance to achieve maximum power transfer. For example,

Example No2.

If an 8Ω loudspeaker is to be connected to an amplifier with an output impedance of 1000Ω, calculate the turns
ratio of the matching transformer required to provide maximum power transfer of the audio signal. Assume the
amplifier source impedance is Z1, the load impedance is Z2 with the turns ratio given as N.

K. Adisesha Page 15


Generally, small transformers used in low power audio amplifiers are usually regarded as ideal so any losses
can be ignored.

Star Delta Transformation

We can now solve simple series, parallel or bridge type resistive networks using Kirchoff´s Circuit Laws,
mesh current analysis or nodal voltage analysis techniques but in a balanced 3-phase circuit we can use different
mathematical techniques to simplify the analysis of the circuit and thereby reduce the amount of math's
involved which in itself is a good thing. Standard 3-phase circuits or networks take on two major forms with
names that represent the way in which the resistances are connected, a Star connected network which has the
symbol of the letter, Υ (wye) and a Delta connected network which has the symbol of a triangle, Δ (delta). If a
3-phase, 3-wire supply or even a 3-phase load is connected in one type of configuration, it can be easily
transformed or changed it into an equivalent configuration of the other type by using either the Star Delta
Transformation or Delta Star Transformation process.

A resistive network consisting of three impedances can be connected together to form a T or "Tee"
configuration but the network can also be redrawn to form a Star or Υ type network as shown below.

T-connected and Equivalent Star Network

As we have already seen, we can redraw the T resistor network to produce an equivalent Star or Υ type
network. But we can also convert a Pi or π type resistor network into an equivalent Delta or Δ type network as
shown below.

Pi-connected and Equivalent Delta Network.

Having now defined exactly what is a Star and Delta connected network it is possible to transform the Υ into
an equivalent Δ circuit and also to convert a Δ into an equivalent Υ circuit using a the transformation process.
This process allows us to produce a mathematical relationship between the various resistors giving us a Star
Delta Transformation as well as a Delta Star Transformation.

These transformations allow us to change the three connected resistances by their equivalents measured
between the terminals 1-2, 1-3 or 2-3 for either a star or delta connected circuit. However, the resulting
networks are only equivalent for voltages and currents external to the star or delta networks, as internally the
voltages and currents are different but each network will consume the same amount of power and have the same
power factor to each other.

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Delta Star Transformation

To convert a delta network to an equivalent star network we need to derive a transformation formula for
equating the various resistors to each other between the various terminals. Consider the circuit below.

Delta to Star Network.

Compare the resistances between terminals 1 and 2.

Resistance between the terminals 2 and 3.

Resistance between the terminals 1 and 3.

This now gives us three equations and taking equation 3 from equation 2 gives:

Then, re-writing Equation 1 will give us:

Adding together equation 1 and the result above of equation 3 minus equation 2 gives:

From which gives us the final equation for resistor P as:

Then to summarize a little the above maths, we can now say that resistor P in a Star network can be found as
Equation 1 plus (Equation 3 minus Equation 2) or Eq1 + (Eq3 - Eq2).

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Similarly, to find resistor Q in a star network, is equation 2 plus the result of equation 1 minus equation 3
or Eq2 + (Eq1 - Eq3) and this gives us the transformation of Q as:

and again, to find resistor R in a Star network, is equation 3 plus the result of equation 2 minus equation 1
or Eq3 + (Eq2 - Eq1) and this gives us the transformation of R as:

When converting a delta network into a star network the denominators of all of the transformation formulas are
the same: A + B + C, and which is the sum of ALL the delta resistances. Then to convert any delta connected
network to an equivalent star network we can summarized the above transformation equations as:

Delta to Star Transformations Equations

If the three resistors in the delta network are all equal in value then the resultant resistors in the equivalent star
network will be equal to one third the value of the delta resistors, giving each branch in the star network as:
RSTAR = 1/3RDELTA

Example No1

Convert the following Delta Resistive Network into an equivalent Star Network.

Star Delta Transformation

We have seen above that when converting from a delta network to an equivalent star network that the resistor
connected to one terminal is the product of the two delta resistances connected to the same terminal, for
example resistor P is the product of resistors A and B connected to terminal 1. By rewriting the previous
formulas a little we can also find the transformation formulas for converting a resistive star network to an
equivalent delta network giving us a way of producing a star delta transformation as shown below.

Star to Delta Network.

The value of the resistor on any one side of the delta, Δ network is the sum of all the two-product combinations
of resistors in the star network divide by the star resistor located "directly opposite" the delta resistor being
found. For example, resistor A is given as:

K. Adisesha Page 18


with respect to terminal 3 and resistor B is given as:

with respect to terminal 2 with resistor C given as:

with respect to terminal 1.

By dividing out each equation by the value of the denominator we end up with three separate transformation
formulas that can be used to convert any Delta resistive network into an equivalent star network as given below.

Star Delta Transformation Equations

Star Delta Transformations allow us to convert one circuit type of circuit connection to another in order for us
to easily analyise a circuit and one final point about converting a star resistive network to an equivalent delta
network. If all the resistors in the star network are all equal in value then the resultant resistors in the equivalent
delta network will be three times the value of the star resistors and equal, giving: RDELTA = 3RSTAR

SEMICONDUCTOR DEVICES

Semiconductor Basics

If Resistors are the most basic passive component in electrical or electronic circuits, then we have to consider
the Signal Diode as being the most basic "Active" component. However, unlike a resistor, a diode does not
behave linearly with respect to the applied voltage as it has an exponential I-V relationship and therefore can
not be described simply by using Ohm's law as we do for resistors. Diodes are basic unidirectional
semiconductor devices that will only allow current to flow through them in one direction only, acting more like
a one way electrical valve, (Forward Biased Condition). But, before we have a look at how signal or power
diodes work we first need to understand the semiconductors basic construction and concept.

Diodes are made from a single piece of Semiconductor material which has a positive "P-region" at one end and
a negative "N-region" at the other, and which has a resistivity value somewhere between that of a conductor and
an insulator. But what is a "Semiconductor" material?, firstly let's look at what makes something either a
Conductor or an Insulator.

Resistivity

The electrical Resistance of an electrical or electronic component or device is generally defined as being the
ratio of the voltage difference across it to the current flowing through it, basic Ohm´s Law principals. The
problem with using resistance as a measurement is that it depends very much on the physical size of the
material being measured as well as the material out of which it is made. For example, If we were to increase the
length of the material (making it longer) its resistance would also increase. Likewise, if we increased its
diameter (making it fatter) its resistance would then decrease. So we want to be able to define the material in
such a way as to indicate its ability to either conduct or oppose the flow of electrical current through it no matter
what its size or shape happens to be. The quantity that is used to indicate this specific resistance is called
Resistivity and is given the Greek symbol of ρ, (Rho). Resistivity is measured in Ohm-metres, ( Ω-m ) and is
the inverse to conductivity.

If the resistivity of various materials is compared, they can be classified into three main groups, Conductors,
Insulators and Semi-conductors as shown below.

Resistivity Chart

K. Adisesha Page 19


Notice also that there is a very small
margin between the resistivity of the
conductors such as silver and gold,
compared to a much larger margin
for the resistivity of the insulators
between glass and quartz. The
resistivity of all the materials at any
one time also depends upon their
temperature.

Conductors

From above we now know that Conductors are materials that have a low value of resistivity allowing them to
easily pass an electrical current due to there being plenty of free electrons floating about within their basic atom
structure. When a positive voltage potential is applied to the material these "free electrons" leave their parent
atom and travel together through the material forming an electron drift. Examples of good conductors are
generally metals such as Copper, Aluminium, Silver or non metals such as Carbon because these materials have
very few electrons in their outer "Valence Shell" or ring, resulting in them being easily knocked out of the
atom's orbit. This allows them to flow freely through the material until they join up with other atoms, producing
a "Domino Effect" through the material thereby creating an electrical current.
Generally speaking, most metals are good conductors of electricity, as they have very small resistance values,
usually in the region of micro-ohms per metre with the resistivity of conductors increasing with temperature
because metals are also generally good conductors of heat.

Insulators

Insulators on the other hand are the exact opposite of conductors. They are made of materials, generally non-
metals, that have very few or no "free electrons" floating about within their basic atom structure because the
electrons in the outer valence shell are strongly attracted by the positively charged inner nucleus. So if a
potential voltage is applied to the material no current will flow as there are no electrons to move and which
gives these materials their insulating properties. Insulators also have very high resistances, millions of ohms per
metre, and are generally not affected by normal temperature changes (although at very high temperatures wood
becomes charcoal and changes from an insulator to a conductor). Examples of good insulators are marble, fused
quartz, p.v.c. plastics, rubber etc.
Insulators play a very important role within electrical and electronic circuits, because without them electrical
circuits would short together and not work. For example, insulators made of glass or porcelain are used for
insulating and supporting overhead transmission cables while epoxy-glass resin materials are used to make
printed circuit boards, PCB's etc.

Semiconductor Basics
Semiconductors materials such as silicon (Si), germanium (Ge) and gallium arsenide (GaAs), have electrical
properties somewhere in the middle, between those of a "conductor" and an "insulator". They are not good
conductors nor good insulators (hence their name "semi"-conductors). They have very few "fee electrons"
because their atoms are closely grouped together in a crystalline pattern called a "crystal lattice". However, their
ability to conduct electricity can be greatly improved by adding certain "impurities" to this crystalline structure
thereby, producing more free electrons than holes or vice versa. By controlling the amount of impurities added
to the semiconductor material it is possible to control its conductivity. These impurities are called donors or
acceptors depending on whether they produce electrons or holes respectively. This process of adding impurity
atoms to semiconductor atoms (the order of 1 impurity atom per 10 million (or more) atoms of the
semiconductor) is called Doping.
The most commonly used semiconductor basics material by far is silicon. Silicon has four valence electrons in
its outermost shell which it shares with its neighbouring silicon atoms to form full orbital's of eight electrons.
The structure of the bond between the two silicon atoms is such that each atom shares one electron with its
neighbour making the bond very stable. As there are very few free electrons available to move around the

K. Adisesha Page 20


silicon crystal, crystals of pure silicon (or germanium) are therefore good insulators, or at the very least very
high value resistors.
Silicon atoms are arranged in a definite symmetrical pattern making them a crystalline solid structure. A crystal
of pure silica (silicon dioxide or glass) is generally said to be an intrinsic crystal (it has no impurities) and
therefore has no free electrons. But simply connecting a silicon crystal to a battery supply is not enough to
extract an electric current from it. To do that we need to create a "positive" and a "negative" pole within the
silicon allowing electrons and therefore electric current to flow out of the silicon. These poles are created by
doping the silicon with certain impurities.

The diagram above shows the structure and lattice of a 'normal' pure
crystal of Silicon.

N-type Semiconductor Basics

In order for our silicon crystal to conduct electricity, we need to introduce an impurity atom such as Arsenic,
Antimony or Phosphorus into the crystalline structure making it extrinsic (impurities are added). These atoms
have five outer electrons in their outermost orbital to share with neighbouring atoms and are commonly called
"Pentavalent" impurities. This allows four out of the five orbital electrons to bond with its neighbouring silicon
atoms leaving one "free electron" to become mobile when an electrical voltage is applied (electron flow). As
each impurity atom "donates" one electron, pentavalent atoms are generally known as "donors".

Antimony (symbol Sb) or Phosphorus (symbol P), are frequently used as a pentavalent additive as they have
51 electrons arranged in five shells around their nucleus with the outermost orbital having five electrons. The
resulting semiconductor basics material has an excess of current-carrying electrons, each with a negative
charge, and is therefore referred to as an "N-type" material with the electrons called "Majority Carriers" while
the resulting holes are called "Minority Carriers".

When stimulated by an external power source, the electrons freed from the silicon atoms by this stimulation are
quickly replaced by the free electrons available from the doped Antimony atoms. But this action still leaves an
extra electron (the freed electron) floating around the doped crystal making it negatively charged. Then a
semiconductor material is classed as N-type when its donor density is greater than its acceptor density, in other
words, it has more electrons than holes thereby creating a negative pole.

The diagram above shows the structure and lattice of the donor impurity atom Antimony.

P-Type Semiconductor Basics

If we go the other way, and introduce a "Trivalent" (3-electron) impurity into the crystalline structure, such as
Aluminium, Boron or Indium, which have only three valence electrons available in their outermost orbital, the
fourth closed bond cannot be formed. Therefore, a complete connection is not possible, giving the
semiconductor material an abundance of positively charged carriers known as "holes" in the structure of the
crystal where electrons are effectively missing.

K. Adisesha Page 21


As there is now a hole in the silicon crystal, a neighbouring electron is attracted to it and will try to move into
the hole to fill it. However, the electron filling the hole leaves another hole behind it as it moves. This in turn
attracts another electron which in turn creates another hole behind it, and so forth giving the appearance that the
holes are moving as a positive charge through the crystal structure (conventional current flow). This movement
of holes results in a shortage of electrons in the silicon turning the entire doped crystal into a positive pole. As
each impurity atom generates a hole, trivalent impurities are generally known as "Acceptors" as they are
continually "accepting" extra or free electrons.

Boron (symbol B) is commonly used as a trivalent additive as it has only five electrons arranged in three shells
around its nucleus with the outermost orbital having only three electrons. The doping of Boron atoms causes
conduction to consist mainly of positive charge carriers resulting in a "P-type" material with the positive holes
being called "Majority Carriers" while the free electrons are called "Minority Carriers". Then a semiconductor
basics material is classed as P-type when its acceptor density is greater than its donor density. Therefore, a P-
type semiconductor has more holes than electrons.

The diagram above shows the structure and lattice of the acceptor impurity
atom Boron.
Semiconductor Basics Summary

N-type (e.g. add Antimony)

These are materials which have Pentavalent impurity atoms (Donors) added and conduct by "electron"
movement and are called, N-type Semiconductors.

In these types of materials are:

1. The Donors are positively charged.
2. There are a large number of free electrons.
3. A small number of holes in relation to the number of free electrons.
4. Doping gives:
o positively charged donors.
o negatively charged free electrons.
5. Supply of energy gives:
o negatively charged free electrons.
o positively charged holes.
P-type (e.g. add Boron)

These are materials which have Trivalent impurity atoms (Acceptors) added and conduct by "hole" movement
and are called, P-type Semiconductors.

In these types of materials are:

1. The Acceptors are negatively charged.
2. There are a large number of holes.
3. A small number of free electrons in relation to the number of holes.
4. Doping gives:
negatively charged acceptors.
positively charged holes.
5. Supply of energy gives:
positively charged holes.
negatively charged free electrons.
and both P and N-types as a whole, are electrically neutral on their own.
K. Adisesha Page 22


Antimony (Sb) and Boron (B) are two of the most commony used doping agents as they are more feely
available compared to others and are also classed as metalloids. However, the periodic table groups together a
number of other different chemical elements all with either three, or five electrons in their outermost orbital
shell. These other chemical elements can also be used as doping agents to a base material of either Silicon (S) or
Germanium (Ge) to produce different types of basic semiconductor materials for use in electronic components
and these are given below.

Periodic Table of Semiconductors

Elements Group 13 Elements Group 14 Elements Group 15
3-Electrons in Outer Shell 4-Electrons in Outer Shell 5-Electrons in Outer Shell
(Positively Charged) (Neutrally Charged) (Negatively Charged)
(5) Boron ( B ) (6) Carbon ( C )
(13) Aluminium ( Al ) (14) Silicon ( Si ) (15) Phosphorus ( P )
(31) Gallium ( Ga ) (32) Germanium ( Ge ) (33) Arsenic ( As )
(51) Antimony ( Sb )
The PN junction

In the previous tutorial we saw how to make an N-type semiconductor material by doping it with Antimony and
also how to make a P-type semiconductor material by doping that with Boron. This is all well and good, but
these semiconductor N and P-type materials do very little on their own as they are electrically neutral, but when
we join (or fuse) them together these two materials behave in a very different way producing what is generally
known as a PN Junction.

When the N and P-type semiconductor materials are first joined together a very large density gradient exists
between both sides of the junction so some of the free electrons from the donor impurity atoms begin to migrate
across this newly formed junction to fill up the holes in the P-type material producing negative ions. However,
because the electrons have moved across the junction from the N-type silicon to the P-type silicon, they leave
behind positively charged donor ions (ND) on the negative side and now the holes from the acceptor impurity
migrate across the junction in the opposite direction into the region were there are large numbers of free
electrons. As a result, the charge density of the P-type along the junction is filled with negatively charged
acceptor ions (NA), and the charge density of the N-type along the junction becomes positive. This charge
transfer of electrons and holes across the junction is known as diffusion.

This process continues back and forth until the number of electrons which have crossed the junction have a
large enough electrical charge to repel or prevent any more carriers from crossing the junction. The regions on
both sides of the junction become depleted of any free carriers in comparison to the N and P type materials
away from the junction. Eventually a state of equilibrium (electrically neutral situation) will occur producing a
"potential barrier" zone around the area of the junction as the donor atoms repel the holes and the acceptor
atoms repel the electrons. Since no free charge carriers can rest in a position where there is a potential barrier
the regions on both sides of the junction become depleted of any more free carriers in comparison to the N and
P type materials away from the junction. This area around the junction is now called the Depletion Layer.

The PN junction

The total charge on each side of the junction must be equal and opposite to maintain a neutral charge condition
K. Adisesha Page 23


around the junction. If the depletion layer region has a distance D, it therefore must therefore penetrate into the
silicon by a distance of Dp for the positive side, and a distance of Dn for the negative side giving a relationship
between the two of Dp.NA = Dn.ND in order to maintain charge neutrality also called equilibrium.

PN junction Distance

As the N-type material has lost electrons and the P-type has lost holes, the N-type material has become positive
with respect to the P-type. Then the presence of impurity ions on both sides of the junction cause an electric
field to be established across this region with the N-side at a positive voltage relative to the P-side. The problem
now is that a free charge requires some extra energy to overcome the barrier that now exists for it to be able to
cross the depletion region junction.

This electric field created by the diffusion process has created a "built-in potential difference" across the
junction with an open-circuit (zero bias) potential of:

Where: Eo is the zero bias junction voltage, VT the thermal voltage of 26mV at room temperature, ND and NA
are the impurity concentrations and ni is the intrinsic concentration.

A suitable positive voltage (forward bias) applied between the two ends of the PN junction can supply the free
electrons and holes with the extra energy. The external voltage required to overcome this potential barrier that
now exists is very much dependent upon the type of semiconductor material used and its actual temperature.
Typically at room temperature the voltage across the depletion layer for silicon is about 0.6 - 0.7 volts and for
germanium is about 0.3 - 0.35 volts. This potential barrier will always exist even if the device is not connected
to any external power source.

The significance of this built-in potential across the junction, is that it opposes both the flow of holes and
electrons across the junction and is why it is called the potential barrier. In practice, a PN junction is formed
within a single crystal of material rather than just simply joining or fusing together two separate pieces.
Electrical contacts are also fused onto either side of the crystal to enable an electrical connection to be made to
an external circuit. Then the resulting device that has been made is called a PN junction Diode or Signal Diode.

The Junction Diode

The effect described in the previous tutorial is achieved without any external voltage being applied to the actual
PN junction resulting in the junction being in a state of equilibrium. However, if we were to make electrical
connections at the ends of both the N-type and the P-type materials and then connect them to a battery source,
an additional energy source now exists to overcome the barrier resulting in free charges being able to cross the
depletion region from one side to the other. The behaviour of the PN junction with regards to the potential
barrier width produces an asymmetrical conducting two terminal device, better known as the Junction Diode.

A diode is one of the simplest semiconductor devices, which has the characteristic of passing current in one
direction only. However, unlike a resistor, a diode does not behave linearly with respect to the applied voltage
as the diode has an exponential I-V relationship and therefore we can not described its operation by simply
using an equation such as Ohm's law.

If a suitable positive voltage (forward bias) is applied between the two ends of the PN junction, it can supply
free electrons and holes with the extra energy they require to cross the junction as the width of the depletion

K. Adisesha Page 24


layer around the PN junction is decreased. By applying a negative voltage (reverse bias) results in the free
charges being pulled away from the junction resulting in the depletion layer width being increased. This has the
effect of increasing or decreasing the effective resistance of the junction itself allowing or blocking current flow
through the diode.

Then the depletion layer widens with an increase in the application of a reverse voltage and narrows with an
increase in the application of a forward voltage. This is due to the differences in the electrical properties on the
two sides of the PN junction resulting in physical changes taking place. One of the results produces rectification
as seen in the PN junction diodes static I-V (current-voltage) characteristics. Rectification is shown by an
asymmetrical current flow when the polarity of bias voltage is altered as shown below.

Junction Diode Symbol and Static I-V Characteristics.

But before we can use the PN junction as a practical device or as a rectifying device we need to firstly bias the
junction, ie connect a voltage potential across it. On the voltage axis above, "Reverse Bias" refers to an external
voltage potential which increases the potential barrier. An external voltage which decreases the potential barrier
is said to act in the "Forward Bias" direction.
There are two operating regions and three possible "biasing" conditions for the standard Junction Diode and
these are:
1. Zero Bias - No external voltage potential is applied to the PN-junction.
2. Reverse Bias - The voltage potential is connected negative, (-ve) to the P-type material and
positive, (+ve) to the N-type material across the diode which has the effect of Increasing the
PN-junction width.
3. Forward Bias - The voltage potential is connected positive, (+ve) to the P-type material and
negative, (-ve) to the N-type material across the diode which has the effect of Decreasing the
PN-junction width.
Zero Biased Junction Diode

When a diode is connected in a Zero Bias condition, no external potential energy is applied to the PN junction.
However if the diodes terminals are shorted together, a few holes (majority carriers) in the P-type material with
enough energy to overcome the potential barrier will move across the junction against this barrier potential.
This is known as the "Forward Current" and is referenced as IF
Likewise, holes generated in the N-type material (minority carriers), find this situation favourable and move
across the junction in the opposite direction. This is known as the "Reverse Current" and is referenced as IR.
This transfer of electrons and holes back and forth across the PN junction is known as diffusion, as shown
below.

Zero Biased Junction Diode

K. Adisesha Page 25


The potential barrier that now exists discourages the diffusion of any more majority carriers across the junction.
However, the potential barrier helps minority carriers (few free electrons in the P-region and few holes in the N-
region) to drift across the junction. Then an "Equilibrium" or balance will be established when the majority
carriers are equal and both moving in opposite directions, so that the net result is zero current flowing in the
circuit. When this occurs the junction is said to be in a state of "Dynamic Equilibrium".

The minority carriers are constantly generated due to thermal energy so this state of equilibrium can be broken
by raising the temperature of the PN junction causing an increase in the generation of minority carriers, thereby
resulting in an increase in leakage current but an electric current cannot flow since no circuit has been
connected to the PN junction.

Reverse Biased Junction Diode

When a diode is connected in a Reverse Bias condition, a positive voltage is applied to the N-type material and
a negative voltage is applied to the P-type material. The positive voltage applied to the N-type material attracts
electrons towards the positive electrode and away from the junction, while the holes in the P-type end are also
attracted away from the junction towards the negative electrode. The net result is that the depletion layer grows
wider due to a lack of electrons and holes and presents a high impedance path, almost an insulator. The result is
that a high potential barrier is created thus preventing current from flowing through the semiconductor material.

Reverse Biased Junction Diode showing an Increase in the Depletion Layer

This condition represents a high resistance value to the PN junction and practically zero current flows through
the junction diode with an increase in bias voltage. However, a very small leakage current does flow through
the junction which can be measured in microamperes, (μA). One final point, if the reverse bias voltage Vr
applied to the diode is increased to a sufficiently high enough value, it will cause the PN junction to overheat
and fail due to the avalanche effect around the junction. This may cause the diode to become shorted and will
result in the flow of maximum circuit current, and this shown as a step downward slope in the reverse static
characteristics curve below.

Reverse Characteristics Curve for a Junction Diode

Sometimes this avalanche effect has practical applications in voltage stabilising circuits where a series limiting
resistor is used with the diode to limit this reverse breakdown current to a preset maximum value thereby
producing a fixed voltage output across the diode. These types of diodes are commonly known as Zener Diodes
and are discussed in a later tutorial.

Forward Biased Junction Diode

When a diode is connected in a Forward Bias condition, a negative voltage is applied to the N-type material
and a positive voltage is applied to the P-type material. If this external voltage becomes greater than the value of
K. Adisesha Page 26


the potential barrier, approx. 0.7 volts for silicon and 0.3 volts for germanium, the potential barriers opposition
will be overcome and current will start to flow. This is because the negative voltage pushes or repels electrons
towards the junction giving them the energy to cross over and combine with the holes being pushed in the
opposite direction towards the junction by the positive voltage. This results in a characteristics curve of zero
current flowing up to this voltage point, called the "knee" on the static curves and then a high current flow
through the diode with little increase in the external voltage as shown below.

Forward Characteristics Curve for a Junction Diode

The application of a forward biasing voltage on the junction diode results in the depletion layer becoming very
thin and narrow which represents a low impedance path through the junction thereby allowing high currents to
flow. The point at which this sudden increase in current takes place is represented on the static I-V
characteristics curve above as the "knee" point.

Forward Biased Junction Diode showing a Reduction in the Depletion Layer

This condition represents the low resistance path through the PN junction allowing very large currents to flow
through the diode with only a small increase in bias voltage. The actual potential difference across the junction
or diode is kept constant by the action of the depletion layer at approximately 0.3v for germanium and
approximately 0.7v for silicon junction diodes. Since the diode can conduct "infinite" current above this knee
point as it effectively becomes a short circuit, therefore resistors are used in series with the diode to limit its
current flow. Exceeding its maximum forward current specification causes the device to dissipate more power
in the form of heat than it was designed for resulting in a very quick failure of the device.

Junction Diode Summary

The PN junction region of a Junction Diode has the following important characteristics:

1). Semiconductors contain two types of mobile charge carriers, Holes and Electrons.
2). The holes are positively charged while the electrons negatively charged.
3). A semiconductor may be doped with donor impurities such as Antimony (N-type doping), so that it
contains mobile charges which are primarily electrons.
4). A semiconductor may be doped with acceptor impurities such as Boron (P-type doping), so that it
contains mobile charges which are mainly holes.
5). The junction region itself has no charge carriers and is known as the depletion region.
6). The junction (depletion) region has a physical thickness that varies with the applied voltage.
7).When a diode is Zero Biased no external energy source is applied and a natural Potential Barrier is
developed across a depletion layer which is approximately 0.5 to 0.7v for silicon diodes and
approximately 0.3 of a volt for germanium diodes.
8). When a junction diode is Forward Biased the thickness of the depletion region reduces and the
diode acts like a short circuit allowing full current to flow.
9). When a junction diode is Reverse Biased the thickness of the depletion region increases and the
diode acts like an open circuit blocking any current flow, (only a very small leakage current).

K. Adisesha Page 27


The Signal Diode

The semiconductor Signal Diode is a small non-linear semiconductor devices generally used in electronic
circuits, where small currents or high frequencies are involved such as in radio, television and digital logic
circuits. The signal diode which is also sometimes known by its older name of the Point Contact Diode or the
Glass Passivated Diode, are physically very small in size compared to their larger Power Diode cousins.

Generally, the PN junction of a small signal diode is encapsulated in glass to protect the PN junction, and
usually have a red or black band at one end of their body to help identify which end is the cathode terminal. The
most widely used of all the glass encapsulated signal diodes is the very common 1N4148 and its equivalent
1N914 signal diode. Small signal and switching diodes have much lower power and current ratings, around
150mA, 500mW maximum compared to rectifier diodes, but they can function better in high frequency
applications or in clipping and switching applications that deal with short-duration pulse waveforms.

The characteristics of a signal point contact diode are different for both germanium and silicon types and are
given as:

Germanium Signal Diodes - These have a low reverse resistance value giving a lower forward volt drop
across the junction, typically only about 0.2-0.3v, but have a higher forward resistance value because of
their small junction area.
Silicon Signal Diodes - These have a very high value of reverse resistance and give a forward volt drop
of about 0.6-0.7v across the junction. They have fairly low values of forward resistance giving them
high peak values of forward current and reverse voltage.
The electronic symbol given for any type of diode is that of an arrow with a bar or line at its end and this is
illustrated below along with the Steady State V-I Characteristics Curve.

Silicon Diode V-I Characteristic Curve

The arrow points in the direction of conventional current flow through the diode meaning that the diode will
only conduct if a positive supply is connected to the Anode (a) terminal and a negative supply is connected to
the Cathode (k) terminal thus only allowing current to flow through it in one direction only, acting more like a
one way electrical valve, (Forward Biased Condition). However, we know from the previous tutorial that if we
connect the external energy source in the other direction the diode will block any current flowing through it and
instead will act like an open switch, (Reversed Biased Condition) as shown below.

Forward and Reversed Biased Diode

Then we can say that an ideal small signal diode conducts current in one direction (forward-conducting) and
blocks current in the other direction (reverse-blocking). Signal Diodes are used in a wide variety of applications
such as a switch in rectifiers, limiters, snubbers or in wave-shaping circuits.

K. Adisesha Page 28


Signal Diode Parameters

Signal Diodes are manufactured in a range of voltage and current ratings and care must be taken when choosing
a diode for a certain application. There are a bewildering array of static characteristics associated with the
humble signal diode but the more important ones are.

1. Maximum Forward Current

The Maximum Forward Current (IF(max)) is as its name implies the maximum forward current allowed to flow
through the device. When the diode is conducting in the forward bias condition, it has a very small "ON"
resistance across the PN junction and therefore, power is dissipated across this junction (Ohm´s Law) in the
form of heat. Then, exceeding its (IF(max)) value will cause more heat to be generated across the junction and the
diode will fail due to thermal overload, usually with destructive consequences. When operating diodes around
their maximum current ratings it is always best to provide additional cooling to dissipate the heat produced by
the diode.

For example, our small 1N4148 signal diode has a maximum current rating of about 150mA with a power
dissipation of 500mW at 25oC. Then a resistor must be used in series with the diode to limit the forward current,
(IF(max)) through it to below this value.

2. Peak Inverse Voltage

The Peak Inverse Voltage (PIV) or Maximum Reverse Voltage (VR(max)), is the maximum allowable Reverse
operating voltage that can be applied across the diode without reverse breakdown and damage occurring to the
device. This rating therefore, is usually less than the "avalanche breakdown" level on the reverse bias
characteristic curve. Typical values of VR(max) range from a few volts to thousands of volts and must be
considered when replacing a diode.

The peak inverse voltage is an important parameter and is mainly used for rectifying diodes in AC rectifier
circuits with reference to the amplitude of the voltage were the sinusoidal waveform changes from a positive to
a negative value on each and every cycle.

3. Total Power Dissipation

Signal diodes have a Total Power Dissipation, (PD(max)) rating. This rating is the maximum possible power
dissipation of the diode when it is forward biased (conducting). When current flows through the signal diode the
biasing of the PN junction is not perfect and offers some resistance to the flow of current resulting in power
being dissipated (lost) in the diode in the form of heat. As small signal diodes are nonlinear devices the
resistance of the PN junction is not constant, it is a dynamic property then we cannot use Ohms Law to define
the power in terms of current and resistance or voltage and resistance as we can for resistors. Then to find the
power that will be dissipated by the diode we must multiply the voltage drop across it times the current flowing
through it: PD = VxI

4. Maximum Operating Temperature

The Maximum Operating Temperature actually relates to the Junction Temperature (TJ) of the diode and is
related to maximum power dissipation. It is the maximum temperature allowable before the structure of the
diode deteriorates and is expressed in units of degrees centigrade per Watt, ( oC/W ). This value is linked
closely to the maximum forward current of the device so that at this value the temperature of the junction is not
exceeded. However, the maximum forward current will also depend upon the ambient temperature in which the
device is operating so the maximum forward current is usually quoted for two or more ambient temperature
values such as 25oC or 70oC.

Then there are three main parameters that must be considered when either selecting or replacing a signal diode
and these are:
The Reverse Voltage Rating
The Forward Current Rating
The Forward Power Dissipation Rating

Signal Diode Arrays

When space is limited, or matching pairs of switching signal diodes are required, diode arrays can be very
useful. They generally consist of low capacitance high speed silicon diodes such as the 1N4148 connected
K. Adisesha Page 29


together in multiple diode packages called an array for use in switching and clamping in digital circuits. They
are encased in single inline packages (SIP) containing 4 or more diodes connected internally to give either an
individual isolated array, common cathode, (CC), or a common anode, (CA) configuration as shown.

Signal Diode Arrays

Signal diode arrays can also be used in digital and computer circuits to protect high speed data lines or other
input/output parallel ports against electrostatic discharge, (ESD) and voltage transients. By connecting two
diodes in series across the supply rails with the data line connected to their junction as shown, any unwanted
transients are quickly dissipated and as the signal diodes are available in 8-fold arrays they can protect eight
data lines in a single package.

CPU Data Line Protection

Signal diode arrays can also be used to connect together diodes in either series or parallel combinations to form
voltage regulator or voltage reducing type circuits or to produce a known fixed voltage. We know that the
forward volt drop across a silicon diode is about 0.7v and by connecting together a number of diodes in series
the total voltage drop will be the sum of the individual voltage drops of each diode. However, when signal
diodes are connected together in series, the current will be the same for each diode so the maximum forward
current must not be exceeded.

Connecting Signal Diodes in Series
Another application for the small signal diode is to create a regulated voltage supply. Diodes are connected
together in series to provide a constant DC voltage across the diode combination. The output voltage across the
diodes remains constant in spite of changes in the load current drawn from the series combination or changes in
the DC power supply voltage that feeds them. Consider the circuit below.

Signal Diodes in Series

K. Adisesha Page 30


As the forward voltage drop across a silicon diode is almost constant at about 0.7v, while the current through it
varies by relatively large amounts, a forward-biased signal diode can make a simple voltage regulating circuit.
The individual voltage drops across each diode are subtracted from the supply voltage to leave a certain voltage
potential across the load resistor, and in our simple example above this is given as 10v - (3 x 0.7v) = 7.9v. This
is because each diode has a junction resistance relating to the small signal current flowing through it and the
three signal diodes in series will have three times the value of this resistance, along with the load resistance R,
forms a voltage divider across the supply.
By adding more diodes in series a greater voltage reduction will occur. Also series connected diodes can be
placed in parallel with the load resistor to act as a voltage regulating circuit. Here the voltage applied to the load
resistor will be 3 x 0.7v = 2.1v. We can of course produce the same constant voltage source using a single
Zener Diode. Resistor, RD is used to prevent excessive current flowing through the diodes if the load is
removed.

Freewheel Diodes
Signal diodes can also be used in a variety of clamping, protection and wave shaping circuits with the most
common form of clamping diode circuit being one which uses a diode connected in parallel with a coil or
inductive load to prevent damage to the delicate switching circuit by suppressing the voltage spikes and/or
transients that are generated when the load is suddenly turned "OFF". This type of diode is generally known as a
"Free-wheeling Diode" or Freewheel diode as it is more commonly called.
The Freewheel diode is used to protect solid state switches such as power transistors and MOSFET's from
damage by reverse battery protection as well as protection from highly inductive loads such as relay coils or
motors, and an example of its connection is shown below.

Use of the Freewheel Diode

Modern fast switching, power semiconductor devices require fast switching diodes such as free wheeling diodes
to protect them form inductive loads such as motor coils or relay windings. Every time the switching device
above is turned "ON", the freewheel diode changes from a conducting state to a blocking state as it becomes
reversed biased. However, when the device rapidly turns "OFF", the diode becomes forward biased and the
collapse of the energy stored in the coil causes a current to flow through the freewheel diode. Without the
protection of the freewheel diode high di/dt currents would occur causing a high voltage spike or transient to
flow around the circuit possibly damaging the switching device.

Previously, the operating speed of the semiconductor switching device, either transistor, MOSFET, IGBT or
digital has been impaired by the addition of a freewheel diode across the inductive load with Schottky and
Zener diodes being used instead in some applications. But during the past few years however, freewheel diodes
had regained importance due mainly to their improved reverse-recovery characteristics and the use of super fast
semiconductor materials capable at operating at high switching frequencies.

Other types of specialized diodes not included here are Photo-Diodes, PIN Diodes, Tunnel Diodes and Schottky
Barrier Diodes. By adding more PN junctions to the basic two layer diode structure other types of
semiconductor devices can be made. For example a three layer semiconductor device becomes a Transistor, a
four layer semiconductor device becomes a Thyristor or Silicon Controlled Rectifier and five layer devices
known as Triacs are also available.

Half Wave Rectification

A rectifier is a circuit which converts the Alternating Current (AC) input power into a Direct Current (DC)
output power. The input power supply may be either a single-phase or a multi-phase supply with the simplest of
all the rectifier circuits being that of the Half Wave Rectifier. The power diode in a half wave rectifier circuit
passes just one half of each complete sine wave of the AC supply in order to convert it into a DC supply. Then

K. Adisesha Page 31

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