Fundamentals of
Electrical Engineering
COMPILED BY RAKHIM AIBAT
INTRODUCTION
❑So far covered: Ohm’s law and Kirchhoff’s laws, node and mesh analyses
❑Major advantage: we can analyze a circuit without tampering with its original
configuration
❑Major disadvantage: for a large, complex circuit, tedious computation is involved
❑growth in areas of application of electric circuits has led to an evolution from
simple to complex circuits
❑To handle the complexity, engineers have developed some theorems to
simplify circuit analysis (applicable to linear circuits)
❑Thevenin’s and Norton’s theorems
❑superposition
❑source transformation
❑maximum power transfer
Linearity Property
❑is a combination of both the homogeneity property and the additivity property
❑homogeneity: if the input (excitation) is multiplied by a constant, then the output (response) is
multiplied by the same constant
❑Additivity: the response to a sum of inputs is the sum of the responses to each input applied separately
❑resistor is a linear element because the voltage-current relationship satisfies both the homogeneity and
the additivity properties
❑a circuit is linear if it is both additive and homogeneous
❑A linear circuit consists of only linear elements, linear dependent sources, and independent sources
❑Note that since P = I2*R = V2/R the relationship between power and voltage (or current) is nonlinear
❑ theorems covered in this chapter are not applicable to power
Example
For the circuit, find I0 when Vs = 12V and Vs = 24V
Example
Assume I0 = 1A and use linearity to find the actual value of I0 in the circuit
If I0 = 1A then:
voltage on 5Ω resistor is V5Ω = I*R = 5*1 = 5V,
voltage on 3Ω resistor is V3Ω = I*R = 3*1 = 3V,
voltage on 4Ω resistor which is also V1 can be found by applying KVL
V1 = V5Ω + V3Ω = 5+3 = 8V or
Superposition
❑The superposition principle states that the voltage across (or current through) an element in a
linear circuit is the algebraic sum of the voltages across (or currents through) that element due to
each independent source acting alone
❑We consider one independent source at a time while all other independent sources are turned off
❑ Replace every voltage source by 0 V (or a short circuit)
❑ Replace very current source by 0 A (or an open circuit)
❑Dependent sources are left intact because they are controlled by circuit variables
❑Steps to Apply Superposition Principle
1. Turn off all independent sources except one source and find the output (voltage or current) due to that
active source
2. Repeat step 1 for each of the other independent sources
3. Find the total contribution by adding algebraically all the contributions due to the independent sources
❑It may very likely involve more work, however it helps reduce a complex circuit to simpler circuits
through replacement of voltage sources by short circuits and of current sources by open circuits
Example
Use the superposition theorem to find v in the circuit
By current division
To obtain V1 we set the current source to zero
(replace it with open circuit)
Or by voltage division
To obtain V2 we set the voltage source to zero
(replace it with short circuit)
Example
Find I0 in the circuit using superposition
where I0’ and I0’’ are due to the 4-A current
source and 20-V voltage source respectively
Example
Find I0 in the circuit using superposition
where I0’ and I0’’ are due to the 4-A current
source and 20-V voltage source respectively
Problem
For the circuit, use the superposition theorem to find i
Problem
2
For the circuit, use the superposition theorem to find i
3
1
Source Transformation
❑Source transformation is another tool for simplifying circuits
❑Basic to these tools is the concept of equivalence
❑equivalent circuit is one whose v-i characteristics are identical with the original circuit
❑A source transformation is the process of replacing a voltage source vs in series with a
resistor R by a current source is in parallel with a resistor R, or vice versa
If the sources are turned off, the equivalent resistance at terminals a-b in both circuits is R
When terminals a-b are short-circuited, the short-circuit current flowing from a to b is
isc = vs/R and isc = is . Thus, in order for the two circuits to be equivalent
Example
Use source transformation to find V0 in the circuit
Problem
Find Vx using source transformation
Problem
Find Vx using source transformation
Thevenin’s Theorem
❑Often particular element in a circuit is variable (usually
called the load) while other elements are fixed
❑Each time the variable element is changed, the entire
circuit has to be recalculated
❑Thevenin’s theorem provides a technique by which the
fixed part of the circuit is replaced by an equivalent circuit
❑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 sources are turned off
Derivations of Thevenin’s Theorem
Consider the linear circuit, that contains resistors and dependent and independent sources.
We have access to the circuit via terminals a and b, through which current from an external
source is applied. Our objective is to ensure that the voltage-current relation at terminals a and b
is identical to that of the Thevenin equivalent. Suppose the linear circuit contains two
independent voltage sources and two independent current sources.
When external source turned off , I = 0, V = B0. Thus, B0 = Voc = VTh, So
When all the internal sources are turned off, B0 = 0.
The circuit can then be replaced by an equivalent resistance Req = Rth
Finding the Thevenin resistance
❑CASE 1: If the network has no dependent sources - we turn off all independent sources, Rth is
the input resistance of the network looking between terminals a and b
❑CASE 2: If the network has dependent sources
❑we turn off all independent sources
❑dependent sources are not to be turned off
❑ because they are controlled by circuit variables
❑apply a voltage source V0 at terminals a and b and determine the resulting current I0, then
❑alternatively, we may insert a current source I0 at terminals a-b and find the terminal voltage V0
❑we may assume any value of V0 (or I0)
Often Rth can take a negative value
It implies that the circuit is supplying power
This is possible in a circuit with dependent sources
Thevenin equivalent circuit
Example
Find the Thevenin equivalent circuit of the circuit shown, to the left of the terminals a-b.
Then find the current through RL if RL = 6Ω, 16Ω, 36Ω
Now take out RL
and find Voc which
is equal to Vth
Turn off all independent sources
We ignore the 1Ω resistor since no
current flows through it
Find the Thevenin equivalent of the circuit at terminals a-b.
To find Rth we turn off independent source but
leave the dependent source, then insert voltage
source V0=1V at terminals a-b, then calculate
I0, Rth = V0/I0
To get Vth, we find Voc
Norton’s Theorem
❑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 shortcircuit current through the terminals and RN is the input or equivalent resistance at the
terminal when the independent sources are turned off
When external source
is turned off
When all internal
independent sources
are turned off
Find the Norton equivalent circuit of the circuit at terminals a-b.
Set the independent sources equal to
zero to find Rn
To find In we short-circuit
terminals a-b. We ignore the
5Ω resistor because it has been
short-circuited
Alternatively, we may determine In
from Vth/Rth. We obtain Vth as the
open-circuit voltage across terminals ab
Problem
Using Norton’s theorem, find Rn and In of the circuit at terminals a-b
Problem
Using Norton’s theorem, find Rn and In of the circuit at terminals a-b
To find Rn we turn the independent
voltage source off and connect a
voltage source of 1V to the terminals ab. We ignore the 4Ω resistor because it
is short-circuited. Hence, Ix = 0.
Maximum Power Transfer
❑There are applications in areas such as communications where it is desirable to maximize the
power delivered to a load
❑Maximum power is transferred to the load when the load resistance equals the Thevenin
resistance as seen from the load (RL = RTh).
Example
Find the value of RL for maximum power transfer in the circuit. Find the maximum power.
Home Work
From book Fundamentals of Electric Circuits (FIFTH EDITION) by Charles K. Alexander and
Matthew N. O. Sadiku solve:
Chapter 4, Problems section (pages 162-170):
Problems – 4.1, 4.3, 4.7, 4.12, 4.19, 4.20, 4.23, 4.31, 4.33, 4.39, 4.40, 4.47, 4.48, 4.66, 4.67, 4.69