Linearity
A linear circuit is one whose output is linearly
related (or directly proportional) to its input.
Circuit Theorems
Homogeneity (scaling) property
Additive property
EEE110 Electric Circuits
v=iR
kv=kiR
v1 = i1 R and v2 = i2 R
→ v = (i1 + i2) R = v1 + v2
Anawach Sangswang
Dept. of Electrical Engineering
KMUTT
Note: power relationship
is nonlinear
2
Superposition
Example
It states that the voltage across (or current
through) an element in a linear circuit is the
algebraic sum of the voltage across (or
currents through) that element due to EACH
independent source acting alone
The principle of superposition helps us to
analyze a linear circuit with more than one
independent source by calculating the
contribution of each independent source
separately
By assuming Io = 1 A, use linearity to find the
actual value of Io in the circuit shown below.
Answer: Io = 3A
3
4
Superposition
Superposition
We consider the effects of 8A and 20V one by
one, then add the two effects together for
final vo
1.
2.
3.
Steps to apply superposition principle
Turn off all independent sources except one
source. Find the output (voltage or current)
due to that active source using nodal or
mesh analysis.
Repeat step 1 for each of the other
independent sources.
Find the total contribution by adding
algebraically all the contributions due to the
independent sources.
5
Superposition
6
Example 4.3
Two things have to be kept in mind:
When we say turn off all other independent
sources:
Independent voltage sources are replaced
by 0 V (short circuit) and
Independent current sources are replaced
by 0 A (open circuit).
Dependent sources are left intact because
they are controlled by circuit variables.
Use the superposition theorem to find v in
the circuit shown below.
3A is discarded
by open-circuit
6V is
discarded by
short-circuit
7
*Refer to text book, answer v = 10V
8
Source Transformation
Example
Use superposition to find vx in the circuit
2A is discarded by
open-circuit
20 Ω
10 V
10V is discarded
by open-circuit
20 Ω
v1
+
−
4Ω
0.1v1
v2
2A
Dependent
source is
unchanged
4Ω
An equivalent circuit is one whose v-i
characteristics are identical with the original
circuit.
It 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.
0.1v2
(b)
10
9
*Refer to text book, answer Vx = 12.5V
Source Transformation
Example
+
+
-
-
(a) Independent source transformation
+
+
-
-
(b) Dependent source transformation
• The arrow of the
current source is
directed toward
the positive
terminal of the
voltage source.
• The source
transformation is
not possible when
R = 0 for voltage
source and R = ∞
for current source.
11
Find io in the circuit shown below using source
transformation.
*Refer to textbook, answer io = 1.78A
12
Thevenin’s Theorem
Thevenin’s Theorem
It states that a linear two-terminal circuit (Fig. a) can be
replaced by an equivalent circuit (Fig. b) consisting of a
voltage source VTH in series with a resistor RTH,
Equivalent circuit
Open-circuit voltage
Resistance at terminals
where
• VTH is the open-circuit voltage
at the terminals.
• RTH is the input or equivalent
resistance at the terminals
when the independent
sources are turned off.
Case1: no dependent source
Turn off all independent sources
Rth = input resistance between terminals a & b
13
Example
14
Thevenin’s Theorem
Find the Thevenin’s equivalent circuit and the
current through RL = 6 Ω, 16 Ω, 36 Ω
Case2: with dependent sources
Turn off all independent sources
Apply vo to find Rth = vo/io or
Apply io to find Rth = vo/io
RTh = 4Ω
−32 + 4i1 + 12(i1 − i2 ) = 0, i2 = −2
i1 = 0.5
VTh = 30V
15
16
Example
Example
Find the Thevenin equivalent
circuit
Loop1 −2v + 2(i − i ) = 0
Loop2 4i + 2(i − i ) + 6(i − i ) = 0
x
2
vx = −4i2
Loop3
1
i3 = − A
6
RTh =
1
2
VTh
4(i2 − i1 ) + 2(i2 − i3 ) + 6i2 = 0
−2vx + 2(i3 − i2 ) = 0
10
i2 =
3
2
1
2
3
i1 = −3i2
VTh = voc = 6i2 = 20V
6(i3 − i2 ) + 2i3 + 1 = 0
i0 = −i3 =
i1 = 5 A
The Thevenin equivalent is
1
A
6
1V
= 6Ω
io
17
Norton’s Theorem
18
Example
Find the Norton equivalent circuit of the
circuit shown below.
It states that a linear two-terminal circuit
can be replaced by an equivalent circuit of
a current source IN in parallel with a
resistor RN,
Where
• IN is the short circuit current through
the terminals.
io =
• RN is the input or equivalent resistance
at the terminals when the independent
sources are turned off.
ix =
The Thevenin’s and Norton equivalent circuits are related by a source transformation.
19
isc =
1v
= 0.2 A
5Ω
RN =
vo
1v
=
= 5Ω
io 0.2 A
10
= 2.5 A
4
10
+ 2ix = 2 + 5 = 7 A = I N
5
20
Maximum Power Transfer
Example
Find the maximum
power transferred
to the load
RTh RTh = 2 + 3 + 6 //12 = 9
VTh : mesh
If the entire circuit is replaced by
its Thevenin equivalent except for
the load, the power delivered to
the load is:
2
 VTh 
 RL
P = i 2 RL = 
 RTh + RL 
−12 + 18i1 − 12i2 = 0, i2 = −2 A
For maximum power dissipated
in RL, Pmax, for a given RTH,
and VTH,
the voltage
−12 + 6i1 + 3i2 + VTh = 0
2
RL = RTH
⇒
Pmax =
VTh
4 RL
The power transfer profile with different RL
21
Pmax
i1 = −
2
A
3
VTh = 22V
VTh2
222
=
= 13.44W
4 RL 4 × 9
22