Lecture 36: Thevenin's and Norton's Theorems for Transformed Circuits
Thévenin's and Norton's Theorems and Equivalent One-Port Transformed Networks
In this section, we review the basic Thévenin and Norton equivalent circuits for one-port networks as
applied to transformed circuits.
10.3.1 Thévenin Equivalent Circuit
Consider the following linear single-port network containing basic circuit elements and independent and
controlled sources.
i (t )
+
v (t )
circuit
-
Thévenin's theorem for resistive circuits states that a resistive one-port network is equivalent to a voltage
source in series with a resistance. The open-circuit source voltage voc ( t ) is measured with the terminals
open, and the Thévenin resistance is measured indirectly by measuring the shot-circuit current isc ( t )
and by computing Rth  voc (t ) isc (t ) . Another way to obtain the Thévenin resistance is to "kill" all
independent sources inside the network (i.e., replace voltage sources by short-circuits and current
sources by open circuits) and to measure the resistance using an external voltage source at the input
port.
Since a transformed circuit is linear and similar to a resistive network, Thévenin's theorem applies as well
for such a circuit. The concept of open-circuit voltage remains the same, but instead of a Thévenin
resistance, we have a Thévenin impedance Zth ( s) in the Laplace domain.
Example: Consider the following circuit initially at rest.
I( s)
Ls
Is ( s)
+
1
Cs
R
V( s)
-
The open-circuit voltage is the voltage at the node connecting all three passive elements. It is equal to the
source current multiplied by the equivalent impedance formed by the capacitance and the resistance
connected in parallel:
1
Voc ( s) 
R
Cs
R  Cs1
Is ( s) 
R
Is ( s)
RCs  1
(10.31)
The Thévenin impedance can be obtained by killing the current source. From the input port, we get an
equivalent impedance formed by the capacitance and the resistance connected in parallel:
Zth ( s) 
R
Cs
R
1
Cs

R
RCs  1
(10.32)
The Thévenin equivalent circuit is shown below.
I( s)
+
Voc ( s) 
-
R
RCs  1
R
Is ( s)
RCs  1
+
V( s)
-
10.3.2 Norton Equivalent Circuit
Consider again our linear single-port network containing basic circuit elements and independent and
controlled sources.
i (t )
+
v (t )
circuit
-
Norton's theorem for resistive circuits states that a resistive one-port network is equivalent to a current
source in parallel with a (Thévenin) resistance. The short-circuit source current isc ( t ) is determined by
shorting the terminals, and the Thévenin resistance is measured as indicated before.
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Norton's theorem also applies to transformed circuits. The Thévenin impedance Zth ( s) is the same as
discussed above for the Thévenin equivalent circuit. For our example above, we get a short-circuit current
which is just the source current.
I( s)
Ls
Is ( s)
+
1
Cs
R
V( s)
-
I( s)
+
Is ( s)
R
RCs  1
V( s)
-
Note that in this example the inductance L appears in neither the Thévenin nor the Norton equivalent
circuit. Its only function is to generate an internal node voltage between itself and the current source, and
this node voltage does not affect the one-port dynamics.
Initial conditions can be treated as current or voltage sources internal to the network, so one can find a
Thévenin or Norton equivalent to a one-port circuit with initial conditions.
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