Lectures .pdf

Abdelmomen Mahgoub
Assistant Professor
Cairo University

D. Abdelmomen Mahgoub – Circuits
▪ “Fundamentals of electric circuits” Charles Alexander
2

Materials can be classified according to their ability
to carry electric current to
▪ Conductors
▪ Insulators
▪ Semiconductors
▪ Superconductors
4

▪ Materials that have very few free electrons in their valence
band, because they are tightly connected
▪ With very high voltage applied, the electromotive force is great
such that the electrons will leave the atoms, causing insulators
break down
▪ Examples: air, glass, plastics
5

▪ Electrons in their valence band are loosely bonded to the atom
▪ Example: Cu, AL, Gold
▪ Cu is the most used conductor
▪ Al is cheaper with less conductivity
6

▪ Materials with typically four valence band electrons
▪ Neither good insulators nor a good conductor
▪ Used in the fabrication of electronic devices such as transistors,
opamps, diodes
7

▪ A perfect material that can carry a very high current with zero
(almost) resistance
▪ Achieved at zero kelvin for all materials
▪ However some materials can be superconductors at higher
temperatures
8

The energy converted per unit electrical charge is known as the
electromotive force (e.m.f.). The electrical energy which is
converted into heat when a unit charge moves from one point to
another in a circuit is known as the potential difference (p.d.)
between the two points.
9
𝑒𝑚𝑓 𝑜𝑟 𝑝𝑑 =
𝑒𝑛𝑒𝑟𝑔𝑦
𝑐ℎ𝑎𝑟𝑔𝑒
=
𝑑𝑊
𝑑𝑞

The e.m.f. of a supply source, and the p.d. across a circuit
element are shown
10

▪ The time rate of flow of electric charge through a conductor or
circuit element
▪ The unit is in (A) which is equivalent to (C/sec)
11

▪ If the current does not change with time, but remains constant,
we call it a direct current (dc). There is a general case called
Unidirectional current
12

▪ A time-varying current is represented by the symbol i . A
common form of time-varying current is the sinusoidal current
or alternating current (ac)
13

Generally speaking, the instantaneous value of an electrical unit,
e.g. current, is represented by a lower-case letter e.g., i, and the
average value (and the r.m.s. or effective value of an alternating
unit) is represented by a capital letter e.g., I. Peak or maximum
values carry subscripts, e.g., IP, Im
14

Conventionally, current flows from a positive charge to a
negative (or less positive) charge, and the direction of flow is
shown by an arrow on the circuit. Electron flow is in the
opposite direction.
15

▪ Power is defined as: the time rate of expending or absorbing
energy, measured in watts (W).
16

▪ The power absorbed or supplied by an element is the product
of the voltage across the element and the current through it.
▪ If the power has a + sign, power is being delivered to or
absorbed by the element. If the power has a - sign, power is
being supplied by the element.
17

▪ How do we know?
▪ By the direction of current and voltage as shown
18

▪ the law of conservation of energy must be obeyed in any
electric circuit. For this reason, the algebraic sum of power in a
circuit, at any instant of time, must be zero
19

▪ Energy is the capacity to do work, measured in joules (J).
20

▪ There are two types of elements found in electric circuits:
passive elements and active elements.
▪ An active element is capable of generating energy while a
passive element is not.
▪ Examples of passive elements are resistors, capacitors, and
inductors.
▪ Typical active elements include generators, batteries, and
operational amplifiers.
21

The most important active elements are voltage or current
sources that generally deliver power to the circuit connected to
them. There are two kinds of sources: independent and
dependent sources.
22

An ideal independent source is an active element that provides a
specified voltage or current that is completely independent of
other circuit elements.
23

▪ An ideal independent voltage source delivers to the circuit
whatever current is necessary to maintain its terminal voltage.
▪ An ideal independent current source delivers to the circuit
whatever voltage is necessary to maintain the designated
current.
24

An ideal dependent (or controlled) source is an active element in
which the source quantity is controlled by another voltage or
current.
25

Types of dependent sources, namely:
1. A voltage-controlled voltage source (VCVS).
2. A current-controlled voltage source (CCVS).
3. A voltage-controlled current source (VCCS).
4. A current-controlled current source (CCCS).
26

27

Basic Laws and Methods of analysis 48

66

78

Example

118

119

120

Transient analysis in DC circuit (first order)

❖ Capacitor Charging Equations
For the simple R-C circuit the voltage differential
equation at the transient period is given by:
Transient of DC Circuits 143
)
(
)
(
)
(
)
(
)
(
)
(
t
V
dt
t
dV
RC
E
t
V
t
Ri
E
t
V
t
V
E
c
c
c
c
c
R
+
=
+
=
+
=

tr
ss
c V
V
t
V +
=
)
(
Where:
Vss = the steady state voltage which is constant
with time.
Vtr = the transient voltage which is a decaying
voltage with time.
To get Vss we put 0
)
(
=
dt
t
dVc
ss
V
E =
This Equation is a first order differential equation and
has a solution given by:

The Vtr could be assumed in the following form:

/
t
tr e
A
V −
=
Where :
A is a constant
RC
=

 :
by
given
and
constant
time
the
is

/
)
( t
ss
c Ae
V
t
V −
+
=

To get the constant ‘A’ we use the initial conditions
Let at t = 0 the voltage has an initial value of Vo
ss
o
ss
o
V
V
A
A
V
V
−
=
+
=

/
)
(
)
( t
ss
o
ss
c e
V
V
V
t
V −
−
+
=
Hence:

 /
/
)
1
(
)
( t
o
t
ss
c e
V
e
V
t
V −
−
+
−
=

))
(
(
1
)
(
)
(
)
(
t
V
E
RC
dt
t
dV
dt
t
dV
RC
t
V
E
c
c
c
c
−
=
+
=
dt
RC
E
t
V
t
dV
dt
RC
t
V
E
t
dV
c
c
c
c
1
)
(
)
(
1
)
(
)
(
−
=
−
=
−

 −
=
−
t
t
V
V c
c
dt
RC
E
t
V
t
dV
c
0
)
(
1
)
(
)
(
0
t
RC
c E
t
V
t
V
V
c 1
)}
)
(
{ln(
)
(
0
−
=
−
t
RC
E
V
E
t
V
o
c 1
}
)
(
ln{ −
=
−
−

t
RC
o
c
e
E
V
E
t
V
1
)
( −
=
−
− t
RC
o
c e
E
V
E
t
V
1
)
(
)
(
−
−
=
−

t
RC
o
c e
E
V
E
t
V
1
)
(
)
(
−
−
+
=


 /
/
1
1
)
1
(
)
(
)
1
(
)
(
t
o
t
ss
c
t
RC
o
t
RC
c
e
V
e
V
t
V
e
V
e
E
t
V
−
−
−
−
+
−
=

+
−
=

RC
E
Vss =
=
 

t
Vc(t)
Vo
Vss
The voltage as a function of time

5
The voltage reaches its
steady state value at
t ≈ 
5

t
Vc(t)
Vss
If Vo = 0
)
1
(
)
( /
t
ss
c e
V
t
V −
−
=

5
The voltage reaches its
steady state value at
t ≈ 
5

dt
t
dV
C
t
i c
c
)
(
)
( =

/
)
(
)
( t
o
ss
c e
R
V
V
t
i −
−
=
R
t
i
t
V
E
t
V c
C
R )
(
)
(
)
( =
−
=

/
)
(
)
( t
o
R e
V
E
t
V −
−
=

If Vo = 0

/
)
( t
c e
R
E
t
i −
=

/
)
( t
R Ee
t
V −
=
)
1
(
)
( /
t
c e
E
t
V −
−
=

Joule
t
V
C
t
W
t
dV
t
V
C
t
W
dt
t
p
t
dV
t
V
C
dt
t
p
dt
t
dV
C
t
V
t
V
t
i
t
p
C
C
C
C
C
C
C
C
)
(
2
1
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
=
=
=
=
=
=



❖ Capacitor Discharging Equations
)
(
)
(
0
)
(
)
(
0
)
(
)
(
0
t
V
dt
t
dV
RC
t
V
t
Ri
t
V
t
V
c
c
c
c
c
R
+
=
+
=
+
=
For the simple R-C circuit the voltage differential
equation at the transient period is given by:
This Equation is a first order differential equation and
has a solution given by:

/
)
( t
ss
c Ae
V
t
V −
+
=

To get Vss we put 0
)
(
=
dt
t
dVc
ss
V
=
0

/
)
( t
c Ae
t
V −
=

To get the constant ‘A’ we use the initial conditions
Let at t = 0 the voltage has an initial value of Vo
A
Vo =

/
)
( t
o
c e
V
t
V −
=
Hence:

dt
t
dV
C
t
i c
c
)
(
)
( =

/
)
( t
o
c e
R
V
t
i −
−
=
R
t
i
t
V
t
V c
C
R )
(
)
(
)
( =
−
=

/
)
( t
o
R e
V
t
V −
−
=

Example
The capacitor is initially uncharged. Close the switch
at t = 0 s.
a. Determine the expression for vC.
b. Determine the expression for iC.
c. Determine capacitor current and voltage at t= 5ms.

RTh = 200 Ω ETh = 40 V
V0 = 0 Vss = ETh = 40 V
Sec
RC 01
.
0
10
*
50
*
200 6
=
=
= −

)
1
(
40
)
( 100t
c e
t
V −
−
=
t
c e
t
i 100
2
.
0
)
( −
=
V
Vc 7
.
15
)
005
.
0
( =

A
ic 121
.
0
)
005
.
0
( =


Example
The switch has been in position A for a
long time. At t = 0, the switch moves
to B. Determine v(t) for t > 0 and
calculate its value at t = 1 s and 4 s.
V
VTh 30
=

AC Circuits 161

AC Circuits 162

AC Circuits 163

AC Circuits 164

AC Circuits 165

AC Circuits 166

AC Circuits 167

AC Circuits 168

AC Circuits 169

AC Circuits 170

AC Circuits 171

AC Circuits 172

AC Circuits 173

AC Circuits 174

AC Circuits 175

AC Circuits 176

AC Circuits 177
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