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Two-Port Networks
One-Port Networks
• Definitions
• Impedance Parameters
+
• Admittance Parameters
v
• Hybrid Parameters
-
• Transmission Parameters
i1
One-Port
Network
i'1
• A pair of terminals at which a signal (voltage or current) may
enter or leave is called a port
• Cascaded Two-Port Networks
• Examples
• A network having only one such pair of terminals is called a
one-port network
• Applications
• No connections may be made to any other nodes internal to the
network
• By KCL, we therefore have i1 = i1
• We discussed in ECE 221 how one-port networks may be modeled
by their Thévenin or Norton equivalents
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
1
J. McNames
Two-Port Networks: Definitions & Requirements
+
i1
i2
Two-Port
Network
v1
-
i'1
I1(s)
+
V1(s)
-
ECE 222
Two-Port Networks
Ver. 1.11
2
+
Two-Port
Network
-
Ver. 1.11
V2(s)
-
• If the network contains dependent sources, one or more of the
equivalent resistors may be negative
• The analysis methods we will discuss require the following
conditions be met
1. Linearity
2. No independent sources inside the network
3. No stored energy inside the network (zero initial conditions)
4. i1 = i1 and i2 = i2
Portland State University
Two-Port Networks
I2(s)
+
• Two-port networks are used to describe the relationship between a
pair of terminals
J. McNames
ECE 222
Two-Port Networks: Defining Equations
v2
i'2
Portland State University
• Generally, the network is analyzed in the s domain
• Each two-port has exactly two governing equations that can be
written in terms of any pair of network variables
• Like Thévenin and Norton equivalents of one-ports, once we know
a set of governing equations we no longer need to know what is
inside the box
3
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
4
Impedance Parameters
+
I1(s)
Impedance Parameter Measurements
+
Two-Port
Network
V1(s)
V2(s)
-
+
I2(s)
I1(s)
V1
z11
=
V2
z21
V2 = z21 I1 + z22 I2
z12
z22
I1
I2
V1
V2
• We can also calculate the impedance parameters after making two
sets of measurements
Two-Port Networks
Ver. 1.11
5
J. McNames
Impedance Parameter Measurements Continued
+
V1(s)
I2(s)
I1(s)
z11
z21
If the left port is an open circuit (I1 = 0), then we can easily solve for
the other two impedance parameters:
V1 V2 z22 =
z12 =
I2 I1 =0
I2 I1 =0
ECE 222
Two-Port Networks
Ver. 1.11
6
V1(s)
Ver. 1.11
8
+
Two-Port
Network
-
= z11 I1 + z12 I2
= z21 I1 + z22 I2
Portland State University
Two-Port Networks
Impedance Parameter Measurements Summarized
-
V1
V2
ECE 222
+
V2(s)
-
J. McNames
Portland State University
+
Two-Port
Network
= z11 I1 + z12 I2
= z21 I1 + z22 I2
If the right port is an open circuit (I2 = 0), then we can easily solve
for two of the impedance parameters:
V1 V2 z
=
z11 =
21
I1 I2 =0
I1 I2 =0
• Relationship can be written in terms of the impedance parameters
ECE 222
-
• Alternatively, if the circuit in the box is known, V1 and V2 can be
calculated based on circuit analysis
Portland State University
V2(s)
-
• Suppose the currents and voltages can be measured
J. McNames
V1(s)
-
V1 = z11 I1 + z12 I2
+
Two-Port
Network
Ver. 1.11
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J. McNames
I2(s)
-
V1 =
I1 I2 =0
V2 =
I1 z12
z22
I2 =0
Portland State University
V2(s)
ECE 222
V1 =
I2 I1 =0
V2 =
I2 I1 =0
Two-Port Networks
Impedance Parameter Equivalent
V1(s)
200 Ω
I2(s)
I1(s)
+
Example 1: Impedance Parameters
z11
z22
z12 I2
z21 I1
-
+
+
40 Ω
I1
I2
V2(s)
V1
-
+
500 Ω
800 Ω
V2
1 kΩ
-
V1
V2
= z11 I1 + z12 I2
= z21 I1 + z22 I2
-
Find the z parameters of the circuit.
• Once we know what the impedance parameters are, we can model
the behavior of the two-port with an equivalent circuit.
• Notice the similarity to Thévenin and Norton equivalents
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
9
J. McNames
Example 1: Workspace
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
10
Example 2: Parameter Conversion
I1(s)
I2(s)
+
+
Two-Port
Network
V1(s)
V2(s)
-
-
V1
= z11 I1 + z12 I2
V2
= z21 I1 + z22 I2
In general, the two defining equations can be written in terms of any
pair of variables. For example, rewrite the defining equations in terms
of the voltages V1 and V2 .
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
11
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
12
Example 2: Workspace
J. McNames
Portland State University
ECE 222
Example 2: Workspace Continued
Two-Port Networks
Ver. 1.11
13
J. McNames
Portland State University
Impedance & Admittance Parameters
I1(s)
V1(s)
-
Impedance Parameters
V2 = z21 I1 + z22 I2
Admittance Parameters
I1 = y11 V1 + y12 V2
I2 = y21 V1 + y22 V2
I1
y11
=
I2
y21
J. McNames
Portland State University
V2(s)
V1(s)
-
-
ECE 222
z12
z22
y12
y22
Ver. 1.11
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I2(s)
+
Hybrid Parameters
V1
z11
=
V2
z21
V1 = z11 I1 + z12 I2
I1(s)
+
Two-Port
Network
Two-Port Networks
Hybrid Parameters
I2(s)
+
ECE 222
I1
I2
V1
V2
Two-Port Networks
V1 =
h11 I1 + h12 V2
I2 =
h21 I1 + h22 V2
Inverse Hybrid Parameters
I1 = g11 V1 + g12 I2
V2 = g21 V1 + g22 I2
Ver. 1.11
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J. McNames
Portland State University
+
Two-Port
Network
V2(s)
-
V1
h11
=
I2
h21
I1
g11
=
V2
g21
ECE 222
h12
h22
g12
g22
I1
V2
V1
I2
Two-Port Networks
Ver. 1.11
16
Transmission Parameters
I1(s)
I2(s)
+
I1(s)
+
V1(s)
Two-Port
Network
-
Transmission Parameters
V1 = a11 V2 − a12 I2
V1
a11
=
I1
a21
I1 = a21 V2 − a22 I2
Inverse Transmission Parameters
V2 = b11 V1 − b12 I1
V2
b11
=
I2
b21
I2 = b21 V1 − b22 I1
J. McNames
Transmission Parameter Conversion
Portland State University
ECE 222
b12
a22
b12
b22
+
V2(s)
V1(s)
-
-
I2(s)
V2
−I2
V1
−I1
V2
−I2
=A
=B
Two-Port Networks
V2
−I2
Ver. 1.11
+
Two-Port
Network
V2(s)
-
• Altogether there are 6 sets of parameters
• Each set completely describes the two-port network
• Any set of parameters can be converted to any other set
• We have seen one example of a conversion
• A complete table of conversions is listed in the text (Pg. 933)
• You should have a copy of this in your notes for the final
17
J. McNames
Example 3: Two-Port Measurements
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
18
Ver. 1.11
20
Example 3: Workspace
The following measurements were taken from a two-port network.
Find the transmission parameters.
Port 2 Open
V1
= 150 cos(4000t) V applied
I1
= 25 cos(4000t − 45◦ ) A measured
V2
= 1000 cos(4000t + 15◦ ) V measured
Port 2 Shorted
J. McNames
V1
=
30 cos(4000t) V applied
I1
=
1.5 cos(4000t + 30◦ ) A measured
I2
=
0.25 cos(4000t + 150◦ ) A measured
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
19
J. McNames
Portland State University
ECE 222
Two-Port Networks
Example 4: Two-Port Analysis
Example 4: Workspace
800 Ω
i1
40 Ω
800 Ω
160 Ω
+
+
v1
v3
-
-
200 Ω
i2
16.2 v3
i1
40 Ω
i2
160 Ω
+
+
+
v2
v1
v3
-
-
-
+
200 Ω
16.2 v3
v2
-
Find the hybrid parameters for the circuit shown above.
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
21
J. McNames
Portland State University
Example 4: Workspace Continued
ECE 222
Two-Port Networks
Ver. 1.11
22
Example 5: Two-Port Measurements
The following measurements were taken from a two-port network.
Find the transmission parameters.
Port 1 Open
Port 1 Shorted
V1
= 1 mV
I1
= −0.5 µA
V2
= 10 V
I2
=
I2
= 200 µA
V2
=
Hint: b = b11 b22 − b12 b21 , a11 =
a22 = b11b .
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
23
J. McNames
Portland State University
b22
b ,
a12 =
ECE 222
80 µA
5V
b12
b ,
a21 =
b21
b ,
Two-Port Networks
and
Ver. 1.11
24
Example 5: Workspace
Example 6: Two-Port Analysis
i1
R1
R3
v+(t)
R4
i2
v-(t)
v1(t)
C1
+
R2
v2(t)
-
C2
Find an expression for the transfer function, h11 , z11 , g12 , g22 , a11 ,
and y21 .
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
25
J. McNames
Portland State University
Example 6: Workspace
i1
v1(t)
R1
R3
v+(t)
R4
v-(t)
C1
i1
i2
R1
R3
v1(t)
v2(t)
v+(t)
R4
v-(t)
C1
26
Ver. 1.11
28
i2
v2(t)
-
C2
Portland State University
Ver. 1.11
+
R2
-
J. McNames
Two-Port Networks
Example 6: Workspace Continued (1)
+
R2
ECE 222
C2
ECE 222
Two-Port Networks
Ver. 1.11
27
J. McNames
Portland State University
ECE 222
Two-Port Networks
Cascaded Two-Port Networks
Example 6: Workspace Continued (2)
i1
R1
R3
v+(t)
R4
v-(t)
v1(t)
C1
I1(s)
i2
+
+
R2
Two Port
Network
A
V1(s)
v2(t)
-
-
I1B (s)
I2A (s)
+
+
V2A (s)
V1B (s)
-
-
I2(s)
+
Two Port
Network
B
V2(s)
-
C2
• Two networks are cascaded when the output of one is the input
of the other
• Note that V2A = V1B and −I2A = I1B
• The transmission parameters take advantage of these properties
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
29
J. McNames
Portland State University
Cascaded Two-Port Networks
I1(s)
I2A (s)
+
Two Port
Network
A
V1(s)
-
V1
=
I1
V2A
−I2A
a11
a21
=
a12
a22
V1B
I1B
A
+
V2A (s)
V1B (s)
-
-
V2A
−I2A
V1
=
I1
I2(s)
Two Port
Network
B
a11
a21
V1B
=
I1B
a11
a21
a12
a22
a12
a22
a11
a21
a12
a22
A
I1(s)
+
+
V2(s)
V1(s)
-
-
B
V2
−I2
B
V2
−I2
I2A (s)
Two Port
Network
A
Portland State University
ECE 222
Ver. 1.11
30
Two-Port Networks
V2
=
I2
b11
b21
Ver. 1.11
I1B (s)
+
+
V2A (s)
V1B (s)
-
-
I2(s)
+
Two Port
Network
B
V2(s)
-
The inverse transmission parameters are also convenient for cascaded
networks.
b12
b22
V1B
V2A
=
−I1B
I2A
J. McNames
Two-Port Networks
Cascaded Two-Port Networks Continued
I1B (s)
+
ECE 222
31
J. McNames
A
V1B
−I1B
V2
=
I2
Portland State University
V2A
=
I2A
b11
b21
b12
b22
ECE 222
A
b11
b21
b12
b22
b11
b21
b12
b22
B
B
Two-Port Networks
V1
−I1
V1
−I1
Ver. 1.11
32
Cascaded Systems: Two-Port Networks versus H(s)
I1(s)
+
V1(s)
-
I2A (s)
Two Port
Network
A
I1B (s)
+
+
V2A (s)
V1B (s)
-
-
I2(s)
Two Port
Network
B
+
V2(s)
-
• Two-ports and transfer functions H(s) are closely related
• H(s) only relates the input signal to the output signal
• Two-ports relate both voltages and currents at each port
• You cannot cascade H(s) unless the circuits are active
• Two-port networks have no such restriction
• Two-ports are used to design passive filters
• However, two-ports are more complicated than H(s)
J. McNames
Portland State University
ECE 222
Two-Port Networks
Ver. 1.11
33