3. An electric circuit is an interconnection of electrical
elements.
Electric circuit of radio transmitter.
4. Electric Current
The time rate of flow of charge through any cross-section is called current.
•Current is a scalar quantity.
• It's S.I. unit is ampere (A) and C.G.S. unit is emu and is called biot (Bi).
•According to electron theory, it was assumed that current flowed from positive
terminal to the negative terminal of the cell via the circuit.
6. Voltage
Voltage(or potential difference) energy required to move a unit charge through a element,
measured in volts(V).
V= dw/dq
1 volt = 1joule/coulomb = 1 newton-meter/ coulomb.
The voltage Vab between two points a and b in an electric circuit is the energy needed
to move a unit charge from a to b ; mathematically,
Polarity of voltage Vab.
7. Circuit Element
The most important active elements are voltage and current sources .
There are two types of sources: independent and dependent.
Independent Source – Is an active element that
provide a specified voltage or current that
Is completely independent of other circuit element.
Dependent Source – Is an active element
in which the source quantity is controlled by
Another voltage or current.
Symbol for Independent Current source
Symbol for Dependent (a)Voltage Source (b)Current Source
8. Basic Laws
Ohm’s Law
States that the voltage V across a resistor is directly proportional to the current i flowing
Through the resistor.
That is , V ∝ i
V= iR
Resistance
An element denotes its ability to resist the flow of electric current, it is measured in
Ohms(Ω).
R= v/i
SI unit, 1Ω= 1V/A
Conductance
Is the ability of an element to conduct electric current; it is measured in mhos (℧) or
Siemens(S).
G= 1/R SI unit ,
= i/v 1S = 1℧= 1A/V
9. Short Circuit
Is a circuit element with resistance approaching zero.
An element with R= 0 is known as short circuit, as shown
in figure (a)
v= iR=0
Open Circuit
Is a circuit element with resistance approaching
infinity.
An element with R= ∞ is known as open circuit, as
shown in figure (b)
i= lim R→∞ v/R =0
(a)
(b)
10. Kirchhoff’s Laws
Kirchhoff gave two laws to solve complex circuits, namely:
1. KIRCHHOFF’S CURRENT LAW (KCL)
States that the algebraic sum of currents entering a node (or a closed boundary) is zero.
The sun of the current entering a node is equal to the sun of the current leaving the node.
I₁+ I₄ = I₂ + I₃
The sum of currents flowing towards any junction in an electrical circuit is equal
to the sum of currents flowing away from that junction. Kirchhoff’s current law is
also called junction rule.
Or I total= I₁ – I₂ + I₃
11. 2. KIRCHHOFF’S VOLTAGE LAW ( KVL )
States that the algebraic sun of all voltages around a closed path(or loop) is zero.
Sum of voltage drops = Sum of voltage rises.
-Vab+ V₁+ V₂- V₃= 0
Or Vab= V₁+ V₂- V₃
12. Sign Convention
A rise in potential should be considered positive and fall in potential should be
considered negative.
• Thus if we go from the positive terminal of the battery to the negative terminal, there is
fall of potential and the e.m.f. Should be assigned negative sign. fig(i)
•It may be noted that the sign of e.m.f. is independent of the direction of current through
the branch under consideration.
Fig(ii)
•When current flows through a resistor, there is a voltage drop across it. If we go through
the resistor in the same direction as the current, there is a fall in potential because current
flows from higher potential to lower potential. Hence this voltage drop should be assigned
negative sign fig(i).
•It may be noted that sign of voltage drop depends on the direction of current and is
independent of the polarity of the e.m.f. Of source in the circuit under consideration fig(ii)
13. Series Resistor (Voltage Division)
The equivalent resistance of any number of resistors connected in series is the sum of
the individual resistance.
Notice that the source voltage v is divided among the resistor in direct proportion to their
Resistance; the large the resistance, the larger the voltage drop. This is called the principle
of Voltage division and the circuit in fig is called voltage divider.
Eq. ckt
14. Parallel Resistor (Current Division)
The equivalent resistance of two parallel resistors is equal to the product of their
resistance divided by their sum.
Eq. ckt
16. Branch - Represents a single element such as a voltage source or a resistor.
Fig have 5 branches namely 10-V, 2-A and the three resistors.
Node - Is the point of concentration between
two or more branches. Indicated by dot in circuit.
Loop - Is any closed path in a circuit.
A network with b branches, n nodes and l independent loops will satisfy the fundamental
theorem of network topology;
b = l+n-1
Two or more elements are in series if they exclusively share a single node and consequently
carry the same current.
Two or more elements are in parallel if they are connected to the same two nodes and
Consequently have the same voltage across them.
Network Terminology
17. Active Element - Is one which supplies electrical energy to the circuit. In fig
E1 and E2 are active elements.
Passive Elements - Is one which receives electrical energy and then either
converts it into heat(resistance) or stores in an electric field(capacitance) or
magnetic field(inductance). In fig R1,R2 andR3 passive elements.
Active and Passive Networks - An active network is that which
contains active elements as well as passive elements. On the other hand, a passive
network is that which contains passive elements only.
Linear Circuit - A linear circuit is one whose parameters(eg. Resistance) are
constant i.e. they do not change with current or voltage.
19. In this method, Kirchhoff’s voltage law is applied to a network to write mesh equations
in terms of mesh currents instead of branch currents.
Maxwell’s Mesh Current Method
Mesh ABDA.
– I₁R₁ – (I₁ – I₂) R₂ + E₁ = 0
or I₁(R₁ + R₂) – I₂R₂ = E₁ →(i)
Mesh BCDB.
– I₂R₃ – E₂ – (I₂ – I₁) R₂ = 0
or – I₁R₂ + (R₂ + R₃) I₂ = – E₂ →(ii)
Solving eq. (i) and eq. (ii) simultaneously, mesh currents I₁ and I₂ can be found out.
Once the mesh currents are known, the branch currents can be readily obtained. The
advantage of this method is that it usually reduces the number of equations to solve a
network problem.
20. Nodal Analysis
The branch currents in the circuit can be found by Kirchhoff’s laws or Maxwell’s mesh
current method. There is another method, called nodal analysis for determining
branch currents in a circuit.
Nodal analysis essentially aims at choosing a reference node in the network and then
finding the unknown voltages at the independent nodes w.r.t. reference node. For a
circuit containing N nodes, there will be N–1 node voltages, some of which may be
known if voltage sources are present.
In this method, one of the nodes is taken as the reference node. The potential of all
the points in the circuit are measured w.r.t. This reference node. In fig A,B,C and D
are four nodes and the node D is taken as *reference node. The fixed voltage nodes
are called dependent nodes. In fig A and C are fixed. And node B is independent
node.
21. Superposition
The superposition principle states that the voltage across(or current through) an
element in a linear circuit is the algebraic sum of the voltage across ( or current
through) that element due to each independent source acting alone.
Explanation: Find v by superposition in the ckt Fig.
v =v₁ + v₂
Now applying KVL to the loop in Fig
12i₁ - 6= 0 => i₁ =0.5 A
Thus, v₁= 4i₁= 2 V
Now using voltage division,
To get v₂ we set voltage source to 0 , now using current division,
(a) Calculating V₁
(b)Calculating V₂
22. Source Transformation
Process of replacing a voltage source Vs in series with a resistor R by a current source Is
In parallel with resistance R, or vice versa.
Transformation of Independent source.
Transformation of Dependent source.
23. Thevenin’s Theorem
States that a linear two-terminal circuit can be replaced by an equivalent circuit
consisting of a voltage source Vth in series with a resistor Rth, where Vth is the open-
circuit voltage at the terminals and Rth is the input or equivalent resistance at the
terminals when the independent source are turned off.
Any linear, bilateral network having terminals A and B can be replaced by a single source of e.m.f. VTh in
series with a single resistance RTh.
(i) The e.m.f. VTh is the voltage obtained across terminals A and B with load, if any removed i.e. it is open-
circuited voltage between terminals A and B.
(ii) The resistance RTh is the resistance of the network measured between terminals A and B with load
removed and sources of e.m.f. replaced by their internal resistances. Ideal voltage sources are replaced
with short circuits and ideal current sources are replaced with open circuits.
24. Explanation
Consider the circuit shown in Fig. (i). As far as the circuit behind terminals AB is concerned, it can be
replaced by a single source of e.m.f. VTh in series with a single resistance RTh as shown in Fig (ii).
(i)Finding Vth - The e.m.f. Vth is the voltage across terminal AB with load(i.e. ) removed
as shown in Fig(ii). With disconnected there is no current in R₂ and Vth is the voltage
Appearing across R₃.
25. (ii)Finding Rth – To find Rth, remove the load and replace the battery by a shot-circuit
because its internal resistance is assumed zero. Then resistance between terminals A and B Is equal to
Rth as shown in Fig(i). Obviously at the terminal AB in Fig(i), R₁ and R₃ are in parallel and this parallel
combination is in series with R₂.
When load is connected between terminals A and B in Fig(ii), then current in
Is given by ,
26. Norton’s Theorem
States that a linear two-terminal circuit can be replaced by an equivalent circuit consisting
of a current source In in parallel with a resistor Rn, where In is the short-circuit current
through the terminals and Rn is the input or equivalent resistance at the terminal when the
independent source are turned off.
Any linear, bilateral network having two terminals A and B can be replaced by a current source of
current output In in parallel with a resistance Rn.
(i) The output In of the current source is equal to the current that would flow through AB when A and B
are short-circuited.
(ii) The resistance Rn is the resistance of the network measured between A and B with load removed
and the sources of e.m.f. replaced by their internal resistances. Ideal voltage sources are replaced
with short circuits and ideal current sources are replaced with open circuit.
27. Explanation
Fig illustrates the application of Norton’s theorem. As far as the circuit behind terminals
AB is concerned [See Fig.(i)], it can be replaced by a current source In in parallel with a
resistance Rn as shown in Fig.(iv). The output In of the current generator is equal to the
current that would flow through AB when terminals A and B are short-circuited as shown
in Fig.(ii). The load on the source when terminals AB are short-circuited is given by ;
28. To find Rn, remove the load and replace battery by a short because its
internal Resistance is assumed zero. Fig(iii)
Thus the values of In and Rn are known. The Norton equivalent circuit will be as shown
In Fig(iv).
29. Reciprocity theorem
In any linear, bilateral network, if an e.m.f E acting in a branch X causes a current I in
branch Y, then the same e.m.f. E located in branch Y will cause a current I in branch X.
However, currents in other parts of the network will not remain the same .
Consider the circuit shown in Fig.(i). The e.m.f. E (=100 V) acting in the branch FAC produces a current
I amperes in branch CDF and is indicated by the ammeter. According to reciprocity theorem, if the
e.m.f. E and ammeter are interchanged* as shown in Fig.(ii), then the ammeter reading does not
change i.e. the ammeter now connected in branch FAC will read I amperes. In fact, the essence of this
theorem is that E and I are interchangeable. The ratio E/I is constant and is called transfer
resistance (or impedance in case of a.c. system).
Explanation.
30. Compensation theorem
The compensation theorem states that any resistance R in a branch of a network in
which current I is flowing can be replaced, for the purpose of calculations, by a
voltage equal to – IR.
Explanation
Let us illustrate the compensation theorem with a numerical example. Consider the
circuit shown in Fig. (i). The various branch currents in this circuit are :
31. Now suppose that the resistance of the right branch is increased to
20 Ω i.e. ∆R = 20 – 10 = 10 Ω
and a voltage V = – I3 ∆R = – 1 × 10 = – 10 V
is introduced in this branch and voltage source replaced by a short ( internal
resistance is assumed zero). The circuit becomes as shown in Fig.(ii).
The compensating currents produced by this voltage are also indicated. When these
compensating currents are algebraically added to the original currents in their
respective branches, the new branch currents will be as shown in Fig. The
compensation theorem is useful in bridge and potentiometer circuits, where a slight
change in one resistance results in a shift from a null condition.
32. Delta/star transformation
Consider three resistors , and connected in delta to three terminals A, B and C as shown in
Fig.(i). Let the equivalent star-connected network have resistances Ra, Rb and Rc. Since the two
arrangements are electrically equivalent, the resistance between any two terminals of one network is
equal to the resistance between the corresponding terminals of the other network.
Let us consider the terminals A and B of the two networks.
Resistance between A and B for star = Resistance between A and B for delta
..........(i)
33. Note: To find the star resistance that connects to a terminal A, divide the product of
the two delta resistors connected to A by the sum of the delta resistor.
Subtracting eq(ii) from eq(i) and adding the result to eq(iii), we have,
.........(ii)
.........(iii)
..........(iv)
..........(v)
..........(vi)
34. Star/Delta Transformation
Now let us consider how to replace the star-connected network of Fig.(ii) by the
equivalent delta-connected network of Fig.(i).
Dividing eq(iv) by eq(v) we have,
Dividing eq(iv) by (vi), we have,
