Block A Unit 3 outline
 One port network
> Thevenin transformation
> Norton transformation
> Source transformation
 Two port network
> Hybrid (H) parameters
> Reciprocity theorem
One port networks: G. Rizzoni, “Fundamental of EE”, Chapter 3.6 – 3.7
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One-Port Network
Represented as:
A one port network is simply a two
terminal device (that we can think of
as a block or black box), which can be
described by its I-V (current-voltage)
characteristic.
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One-port
Network
+
v
–
2
Thevenin’s Theorem
Any one-port network composed of ideal voltage and current
sources, and linear resistors, can be represented by an equivalent
circuit consisting of an ideal voltage source VT in series with an
equivalent resistance RT
One-port
network
VT
+
-
RT
This is an extremely power method is you want to find the load
condition for maximum power transfer.
i.e. what is the RL you need to achieve maximum power transfer
for a given one port network?
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Norton’s Theorem
Any one-port network composed of ideal voltage and current
sources, and linear resistors, can be represented by an equivalent
circuit consisting of an ideal current source IN in parallel with
an equivalent resistance RN
One-port
network
IN
RN
This is an extremely power method is you want to find the load
condition for maximum power transfer.
i.e. what is the RL you need to achieve maximum power transfer
for a given one port network?
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Finding RN or RT
Applies to both Thevenin & Norton
Method Summary

Remove the load (assumption: you need to know what the
load is). Never forget this step!

Zero all independent voltage & current sources
> Short-circuit voltage sources (0V across source)
> Open-circuit current sources (0A through source)

Calculate the total resistance seen across the load terminals
to obtain equivalent resistance RN or RT
>
Combine resistors in series and parallel (REF previous unit)
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Zeroing the Sources
That is as if the source never existed
Voltage V
s
Source
Current I
N
Source
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+
-
No voltage
=> Short
No current
=> Open
6
Illustration of finding RN or RT
R1
vs
+
-
R2
Step 1: Remove load
R3
RL
Step 3
Compute total resistance between
the terminals
Thevenin Resistance: RT = R3 +
R1||R2
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vs
+
R1
R3
R2
-
Step 2: Zero all
independent sources
R3
R1
R2
7
Equivalent R: Example 1
R1
VS
+
-
R2
R2
R1
R2
Rth = R1 || R2
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Rth
VS
+
-
R1
Rth
R2
Rth = R2
8
Equivalent R: Example 2
R1
R2
IS
R2
Rth
R2
Rth = R2
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IS
R1
R1
Rth
Rth = R1 + R2
9
Finding Thevenin voltage VT
Definition of VT - The Thevenin equivalent voltage is
equal to the open-circuit voltage present at the load
terminals (with the load removed)
Method Summary
 Remove load (never forget this step)
 Define open circuit voltage Voc across the
open load terminals
 Solve for Voc using any preferred method to
obtain the Thevenin voltage VT = Voc
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Finding Thevenin VT
R1
vs
+
-
Step 1: Remove load!
R3
R2
RL
vs
+
-
R1
R3
R2
+
Voc
-
Step 2: Define open circuit voltage Voc across the open load terminals
Step 3: Solve for Voc using any preferred method
Note that current through R3 = 0 due to open circuit (ie no voltage drop across R3)
Therefore voc = voltage across R2
Voltage divider rule: voc = [R2 / (R1 + R2)] vs
*Thevenin voltage vT = voc
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Thevenin example 1
Problem 3.55
Find the Thevenin equivalent circuit as seen by the resistor RL
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Thevenin example 1 solution
Mesh 1:
1Ω
1kΩ
3Ω
1kΩ
10 = 2000i1 - (1000)(0.01)
Rth = 504Ω
 i1 = 10mA
Mesh 2:
Note that with RL removed, no
current runs through 3Ω resistor
-V = (1001)(0.01) - (1000)(0.01)
 V = -10mV
Vth = V = -10V
504Ω
-10mV
+
This slide is meant to blank
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Thevenin example 2
Problem 3.52
Find the voltage across the 3Ω resistor by replacing the remainder with the its
Thevenin equivalent
V1
+
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Thevenin example 2 solution
Equivalent resistance seen by 3Ω resistor is simply 4Ω resistor
Mesh 1:
V1 = 2(4+2) - i (2)
Vth = (1.5)(2) - 8.5 = -5V
 2i + V1 = 12
4Ω
Mesh 2:
3 = i(2+2) - 2 (2)
 i = 1.75A
-5V
+
-
 V1 = 8.5V
V3Ω = 3/(3+4) * -5 = -5.14V
This slide is meant to blank
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Finding Norton current IN
Definition of IN - The Norton equivalent current is
equal to the short-circuit current that would flow if the
load were replaced by a short circuit
Method Summary
 Replace load with short circuit (SC)
 Define SC current isc to be Norton equivalent
current
 Solve for isc using any preferred method to
obtain IN = isc
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Finding Norton IN (1)
R1
vs
+
-
A
R2
Step 1: Short circuit load
R3
RL
vs
+
-
R3
R1
R2
isc
Step 2: Define short circuit current isc
Step 3: Solve for isc using any preferred method
Apply KCL at node A:
vs  va va va
  0
R1
R2 R3
R2 R3
va 
vs
R1R2  R2 R3  R3 R1
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isc 
va
R3
17
Norton example 1
Problem 3.51 (modified)
Find the Norton equivalent circuit to the left of the 3Ω resistor
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Norton example 1 solution
1Ω
5Ω
4Ω
RN = 3.22Ω
Norton current is given by the current through 1Ω resistor
 1 || 4 
VR1  
36  4.966V


1
||
4

5


I N  I R1  VR1 R1  4.966A
This slide is meant to blank
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Norton example 2
Problem 3.53
Find the Norton equivalent circuit to the left of the 2Ω resistor
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Norton example 2 solution
1Ω
1Ω
3Ω
Rth = 4.75Ω
3Ω
KCL at node 1:
2  V1  1  V1
KCL at node 2:
3  V1  V2  1
7V1  3V2  6
V1  V2  1  2  V2
3
3V1  4V2  6
Solve for V2: V2 = -1.263V
IN = V2 / 3 = -0.42A
This slide is meant to blank
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Source Transformation
We can transform between
the two equivalent circuits,
observing each time that:
RT
VT
+
IN
-
RT
VT = IN RT
For example using one of the previous circuits as shown below:
+
VS
-
R1
R2
R3
R3
R1
R2
I1
VS and R1 form a Thevenin circuit which we can transform to a Norton circuit defined by
current source I1 = VS/R1 and parallel resistance R1. Note that R1 and R2 and be combined.
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Transformation illustration
R3
R1||R2
I1
+
V1
R1||R2
R3
-
The parallel combination of R1 & R2 (R1||R2) and current source I1 form a new Norton
circuit (with a different value of parallel resistance than in the first instance), which can
in turn be transformed back into a Thevenin circuit as follows. This Thevenin circuit
comprises of a series resistor R1||R2, and a voltage source of V1 = I1(R1||R2).
Finally, it will now become obvious that this Thevenin resistor is in series with R3, and
therefore can be easily combined. Therefore, in summary we see that
VT = V1 = [R2/(R1+R2)]VS and RT = R1||R2 + R3 (same result as before)
Note that V1 is NOT the same as VS. Hence the transformed circuit values
are not to be confused with the original values – never make this mistake!
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Source Transformation: Proof
Comes directly from the theorem:
Thevenin to Norton – Any network can be composed of a current source in parallel with
a resistor; this would therefore also include a Thevenin circuit
Norton to Thevenin – Any network can be composed of a voltage source in series with a
resistor; this would therefore also include a Norton circuit
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Two-port network
Looking back
A port refers to a pair of terminals through which a current may enter or leave a network
We have focused only on one-port networks so far, where we consider the voltage across or
current through a single pair of terminal
The rest of this unit deals with two-port networks
A two-port network is an electrical network with two separate
ports for input and output
We will see examples of two-port networks (op amps and
transistor circuits) later on in this course
Like for a one-port network, knowing the parameters of a twoport network enables use to treat it as a “black-box” placed
within a larger network.
In a two-port network, we need to relate V1, V2, I1, I2
The terms relating these currents and voltages are known as parameters
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Impedance parameters
As shown in the figure, a two-port network may be voltage-driven or current-driven.
The terminal voltages and currents represent 4 variables, of which two are independent.
The terminal voltages can be related to the terminal currents
V1 = z11I1 + z12I2
V2 = z21I1 + z22I2
The voltages V1 and V2 are the
dependent variables in this case
It can also be expressed in matrix form:
V1   z11
V    z
 2   21
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Z12   I1 
z22   I 2 
These z terms are known as impedance
parameters (since there are defined by
V over I), or simply z parameters.
26
Handling parameter subscripts
11: Input from port 1 and output back to port 1
21: Input from port 1 and output to port 2
12: Input from port 2 and output to port 1
22: Input from port 2 and output back to port 2
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z parameters
The values of the parameters can be obtained by setting:
I1 = 0 (input port open-circuited) or
I2 = 0 (output port open-circuited)
Therefore:
z 11 
z 21 
V1
I1
V2
I1
,
z 12 
I 2 0
, z 22 
I 2 0
V1
I2
I1 0
V2
I2
I1 0
z11: Open-circuit input impedance
z12: Open-circuit transfer impedance from port 1 to port 2
z21: Open-circuit transfer impedance from port 2 to port 1
z22: Open-circuit output impedance
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Deriving z parameters
We can see from the definition of the z parameters,
we obtain z11 and z21 by connecting V1 to port 1
and leaving port 2 open-circuited (I2 = 0)
[Referring to Figure (a)]:
z11 = V1/I1, z21 = V2/I1
Likewise, we obtain z12 and z22 by connecting V2
to port 2 and leaving port 1 open-circuited (I1 = 0)
[Referring to Figure (b)]:
z12 = V1/I2, z22 = V2/I2
z11 and z22 are known as driving-point impedances
z21 and z12 are known as transfer impedances
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Z parameter example 1
Determine the Z-parameters for the following circuit
We first apply V1 at port 1 and open-circuit port 2 (I2 = 0)
[Figure (a)]
z11 = V1/I1 = (20 + 40)I1/I1 = 60Ω
z21 = V2/I1 = 40I1/I1 = 40Ω
Next we apply V2 at port 2 and open-circuit port 1 (I1 = 0)
[Figure (a)]
z12 = V1/I2 = 40I2/I2 = 40Ω
z22 = V2/I1 = (40 + 30)I2/I2 = 70Ω
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Z parameter example 2
Determine the Z-parameters for the following circuit
z11 = 14, z12 = z21 = z22 = 6Ω
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Hybrid parameters
The z parameters do not exist for all two-port networks, so there is a need to develop an
alternative set of parameters to describe such types of two-port networks. This particular
set of parameters is based on making V1 and I2 the dependent variables.
V1 = h11I1 + h12V2
I2 = h21I1 + h22V2
These h terms are known as hybrid parameters,
since there are a hybrid (mix) of ratios, or simply
h parameters.
This set of parameters is very useful for describing electronic devices such as
transistors (which we will cover in Block C); namely because they are easier to
measure compared to z or y parameters.
The h parameters are defined as: h 11 
h 21 
EE2301: Block A Unit 3
V1
I1
V1
, h 12 
V2
V 0
I2
I1
, h 22 
2
V2  0
I2
V2
I1 0
I1 0
32
h parameters
We can see that:
h11 represents an impedance; h12 represents a voltage ratio
h21 represents a current ratio; h22 represents an admittance
This is why they are known as hybrid (which means a mixture) parameters
h11: Short-circuit input impedance
To find h11 and h21:
h12: Open-circuit reverse voltage gain
Short-circuit port 2 and apply I1 to port 1
h21: Open-circuit forward current gain
h22: Short-circuit output admittance
To find h22 and h12:
Open-circuit port 1 and apply I2 to port 2
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h parameter example 1
Find the h parameters for the following circuit.
To find h11 and h21:
Short-circuit port 2 and apply I1 to port 1 [Fig (a)]
h11 = V1/I1 = (2 + 3||6)I1/I1 = 4Ω
h21 = I2/I1 = -2/3 (current divider)
To find h12 and h22:
Open-circuit port 1 and apply V2 to port 2 [Fig (b)]
h12 = V1/V2 = 6/(6+3) = 2/3 (voltage divider)
h22 = I2/V2 = I2/(3+6)I2 = 1/9S
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h parameter example 2
Find the h parameters for the following circuit.
h11 = 1.2Ω, h12 = 0.4, h21, = -0.4, h22 = 0.4S
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