Equivalent circuit of a simple voltage source (battery)
+
-
A
+
9V
-
9V
battery
Ri
RL
B
Equivalent circuit of an electronic voltage source
A
B
Stabilized voltage source is a fairly complex electronic circuit
A
It would be very
convenient to having a
simple equivalent
circuit of it:
+
VTH
-
RTH
RL
B
Open-circuit voltage of the voltage source
+
-
A
+
9V
-
Ri
RL
B
9V
battery
First, let us disconnect RL. This corresponds to an “open-circuit” condition
(RL >>Ri). The current I = 0. The voltage at the battery output can be
measured:
VOC= VB
Zeroing the source
+
-
A
+
9V
-
Ri
RL
B
9V
battery
Next we can short-circuit the load. The current ISC = VB/Ri – can be measured.
The Ri can be found as VOC/ISC = Ri
We can get the same results by zeroing the source.
The voltage source becomes a short-circuit.
The resistance seen from the output terminals = Ri
Zeroing a voltage source
A
A
B
B
VS = 0
If the voltage of the voltage source VS = 0, then the voltage VAB = 0.
The potentials of the point A and point B are equal.
Compare to the wire connecting points A and B:
No matter what current is flowing through the wire, VAB = 0.
The voltage source with ZERO voltage is equivalent to a wire (shortcircuit) connecting its terminals.
Zeroing a current source source
A
A
B
B
IS = 0
If the current of the current source IS = 0, then
the current between the point A and point B is zero.
No matter what the voltage VAB is, the current between A and B is zero.
The current source with ZERO current is equivalent to a break
(an open circuit) between its terminals.
Thevenin equivalent of a voltage source
Any linear
circuit
containing
voltage and
current sources
and resistors
A
A
+
VTH
RTH
-
B
Output terminals
(to be connected to the load)
According to the Thevenin theorem, the equivalent voltage source
and internal resistance of ANY linear circuit are:
VTH = VAB (open-circuit voltage)
RTH = VTH/IAB where IAB is the short-circuit current
or RTH = RAB with all the voltage and current sources ZEROED.
RL
B
Example 1:
convert current source into voltage source
+
IS
RP
VTH
RTH
-
1) find the open-circuit voltage between the terminals:
VOC = IS * RP
2) find the input resistance with the current source zeroed:
RIN = RP
Equivalent (Thevenin) voltage source parameters:
VTH = VOC = IS*RP
RTH = RP
Example 2
1
VS1
IS2
First, find the open-circuit voltage between terminals a and b:
VTH = VAB (open-circuit)
Applying nodal analysis to the node “1”:
G11=1/5+1/20 = 0.25; (there is no current through the 4-Ohm
resistor when the circuit is open);
Is=25/5+3 = 8 A
V1oc=G^(-1)*Is = (1/0.25)*8 = 32;
VTH = Vab(open circuit) = 32 V
Example 2 (cont.)
1
VS1
IS2
Next, let us find the resistance between a and b terminals when all
the sources are set to zero.
Rab = 5//20 +4 = 4 + 4 = 8 Ohm
RTH = 8 Ohm
Example 2 (cont.)
1
+
VTH
a
RTH
-
VS1
IS2
The circuit on the left can be replaced with the Thevenin equivalent voltage
source having the equivalent open-circuit voltage VTH = 32 V and
equivalent resistance RTH = 8 Ohm
If the load resistance RL connected to the terminals a and b is to be
optimized for the maximum power dissipated in it, then RLopt = RTH.
In the above circuit, the optimal load resistance,
RLopt = 8 Ohm
b
Norton equivalent current source
Any linear
circuit
containing
voltage and
current sources
and resistors
A
a
+
INt
-
RNt
b
B
According to the Norton theorem, any circuit can be represented by the
equivalent current source, with the current source value
INt = ISC
and internal shunting resistance,
RNt = Routput @ [zeroed sources]
Example 3:
convert voltage source into current source
+
VS
RS
IN
RN
-
1) find the output resistance with the voltage source zeroed:
Rout = RS
2) find the short-circuit current between the terminals:
ISC = VS/RS;
Equivalent (Norton) current source parameters:
IN = ISC = VS/RS
RN = ROUT = RS