THÉVENIN & NORTON EQUIVALENT CIRCUITS
Thévenin Equivalent Circuit
• Any linear 2-terminal (1-port) and active network can be
replaced by an equivalent circuit consisting of an independent
voltage source in series with a resistor.
network
of
sources
and
resistors
≡
–+
VTh
RTh
RL
iL+
vL
–
a
b
RL
iL+
vL
–
a
b
Thévenin equivalent circuitActual circuit
Norton Equivalent Circuit
• Any linear 2-terminal (1-port) network of independent voltage
sources, independent current sources, and linear resistors can be
replaced by an equivalent circuit consisting of an independent current
source in parallel with a resistor without affecting the operation of the
rest of the circuit.
Norton equivalent circuit
network
of
sources
and
resistors
≡RL
iL+
vL
–
a
b
a
RL
iL+
vL
–
iN
b
RN
Determination of Thévenin/Norton Equivalent
Calculate the open-circuit voltage, voc
Calculate the short-circuit current, isc
Note that isc is in the direction of the
open-circuit voltage drop across the
terminals a,b !
sc
oc
Th
ocTh
i
vR
vV
network
of
sources
and
resistors
a
b
+
voc
–
network
of
sources
and
resistors
a
b
isc
Alternative Method of Calculating RTH
For a network containing only independent sources
and linear resistors:
1. Set all independent sources to zero
voltage source short circuit
current source open circuit
2. Find equivalent resistance Req between the terminals by inspection
Or, set all independent sources to zero
1. Apply a test voltage source VTEST
2. Calculate ITEST
TEST
TEST
ThI
VR
network of
independent
sources and
resistors, with
each source
set to zero
Req
network of
independent
sources and
resistors, with
each source
set to zero
ITEST
–+
VTEST
Finding IN
IN ≡ isc = VTh/RTh
Analogous to calculation of Thevenin Equivalent Circuit:
• Find open-circuit voltage and short-circuit current
• Or, find short-circuit current and Norton (Thevenin) resistance
Finding IN and RN
• We can derive the Norton equivalent circuit from a Thévenin equivalent
circuit simply by making a source transformation:
RLRN
iL
iN
+
vL
–
a
b
–+
RL
iL+
vL
–
vTh
RTh
sc
Th
Th
N
sc
oc
ThN ; i
R
vi
i
vRR
a
b
Answer: VTh= 10V, RTH = 1W
Find Thevenin’s and Norton equivalent circuits between the terminals of Ro.
Exercise
Find Thevenin’s equivalent circuit between the terminals of Ro.
Exercise
Open Circuit to find VTH
Short Circuit to find IN
Maximum Power Transfer
Objectives
• To find the resistance that absorb maximum power from an
active circuit.
• To calculate the maximum power that can be transferred from
a circuit to a resistance.
13
• Suppose that a general network N
has a resistive load as shown.
LL
L
dPSolve 0 to find where P is max imum
dR
Now we might consider two questions:
• For what value of RL is maximum power delivered to RL?
• What is the maximum power that can be delivered to RL?
To answer these questions
1) Replace N by a Thevenin Equivalent Circuit
2) Determine a general expression for power to RL
3)
independent
sources,
dependent sources,
and resistors
Network N
RL
Maximum Power Transfer Theorem
A resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit.
L
2
LTh
ThL
2L R
RR
VRip
LTh
LThL2
LTh
4LTh
LThL2
LTh2Th
02
02
RR
RRRRR
RR
RRRRRV
dR
dp
L
–+
VTh
RTh
RL
iL+
vL
–
Thévenin equivalent circuit
To find the value of RL for which p is
maximum, set to 0:
Power absorbed by load resistor:
LdR
dp
Maximum Power Transfer Theorem
• The relationship between PL and RL can be illustrated by the graph
shown below.
Maximum Power Transfer Theorem
Example
• Find Ro for maximum power and Pmax.
Example
Solution
Solution
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