34
An Introduction To Two – Port Networks The University of Tennessee Electrical and Computer Engineering Knoxville, TN wlg
An Introduction ToTwo – Port Networks
The University of TennesseeElectrical and Computer Engineering
Knoxville, TNwlg
Two Port Networks
Generalities: The standard configuration of a two port:
The NetworkInputPort
OutputPort
+
_ _
+V1 V2
I1 I2
The network ?
The voltage and current convention ?
* notes
Two Port Networks
Network Equations:
V1 = z11 I1 + z12 I2
V2 = z21 I1 + z22 I2
I1 = y11 V1 + y12 V2
I2 = y21 V1 + y22 V2
V1 = AV2 - BI2
I1 = CV2 - DI2
V2 = b11 V1 - b12 I1
I2 = b21 V1 – b22 I1
V1 = h11 I1 + h12 V2
I2 = h21 I1 + h22 V2
I1 = g11 V1 + g12 I2
V2 = g21 V1 + g22 I2
ImpedanceZ parameters
AdmittanceY parameters
TransmissionA, B, C, D
parameters
HybridH parameters
* notes
Two Port Networks
Z parameters:
11
11 IV
z =02=I
21
12 IV
z =01=I
12
21 IV
z =02=I
22
22 IV
z =01=I
z11 is the impedance seen looking into port 1when port 2 is open.
z12 is a transfer impedance. It is the ratio of thevoltage at port 1 to the current at port 2 whenport 1 is open.
z21 is a transfer impedance. It is the ratio of the voltage at port 2 to the current at port 1 whenport 2 is open.
z22 is the impedance seen looking into port 2when port 1 is open.
* notes
Two Port Networks
Y parameters:
11
11 VI
y =02=V
21
12 VI
y =01=V
12
21 VI
y =02=V
22
22 VI
y =01=V
y11 is the admittance seen looking into port 1when port 2 is shorted.
y12 is a transfer admittance. It is the ratio of thecurrent at port 1 to the voltage at port 2 whenport 1 is shorted.
y21 is a transfer impedance. It is the ratio of the current at port 2 to the voltage at port 1 whenport 2 is shorted.
y22 is the admittance seen looking into port 2when port 1 is shorted.
* notes
Two Port Networks
Z parameters: Example 1
Given the following circuit. Determine the Z parameters.
8
20 20 Ω
ΩΩ
Ω
10
+
_
+
_
V1 V2
I1 I2
Find the Z parameters for the above network.
Two Port Networks
Z parameters: Example 1 (cont 1)
For z11 :
Z11 = 8 + 20||30 = 20 Ω
For z22 :
For z12 :
Z22 = 20||30 = 12 Ω
21
12 IV
z = 01=I
8
20 20 Ω
ΩΩ
Ω
10
+
_
+
_
V1 V2
I1 I2
22
1 83020
2020xI
xxIV =
+=
Therefore:88
2
212 ==
IxIz Ω = 21z
Two Port Networks
Z parameters: Example 1 (cont 2)
The Z parameter equations can be expressed inmatrix form as follows.
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2
1
128820
II
VV
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2221
1211
2
1
II
zzzz
VV
Two Port Networks
Z parameters: Example 2 (problem 18.7 Alexander & Sadiku)
You are given the following circuit. Find the Z parameters.
1Ω
1Ω
4 Ω
2 Ω
2Vx
+
-Vx
+ +
_ _
V1 V2
I1 I2
Two Port Networks
Z parameters: Example 2 (continue p2)
11
11 IV
z =02=I
1Ω
1Ω
4 Ω
2 Ω
2Vx
+
-Vx
+ +
_ _
V1 V2
I1 I2
626
62
11xxxxxx VVVVVV
I++
=+
+=
23
1xV
I = ; but 11 IVVx −=
Substituting gives;
( )2
3 111
I VI −= or Ω==
35
111
1 zIV
Z21 = -0.667 Ω
Z12 = 0.222 Ω
Z22 = 1.111 Ω
Other Answers
Two Port Networks
Transmission parameters (A,B,C,D):
The defining equations are:
⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
2
1
1
IV
DCBA
IV
2
1
VVA=
I2 = 0 2
1
IVB−
= V2 = 0
2
1
VI
C =I2 = 0 2
1
IID−
= V2 = 0
Two Port Networks
Transmission parameters (A,B,C,D):
Example Given the network below with assumed voltage polarities andCurrent directions compatible with the A,B,C,D parameters.
+
_
+
_
R1
R2V1 V2
I1 -I2
We can write the following equations.
V1 = (R1 + R2 )I1 + R2 I2
V2 = R2 I1 + R2 I2
It is not always possible to write 2 equations in terms of the V’s and I’s Of the parameter set.
1
1R
2
1R
2
21
RRR +
Two Port Networks
Transmission parameters (A,B,C,D):
Example (cont.)
V1 = (R1 + R2 )I1 + R2 I2
V2 = R2 I1 + R2 I2
From these equations we can directly evaluate the A,B,C,D parameters.
2
1
VVA=
I2 = 02
1
IVB−
= V2 = 0
2
1
VI
C =I2 = 0 2
1
IID−
= V2 = 0
=
=
=
=
Later we will see how to interconnect two of these networks together for a final answer* notes
Two Port Networks
Hybrid Parameters: The equations for the hybrid parameters are:
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2221
1211
2
1
VI
hhhh
IV
1
111 I
Vh =V2 = 0
2
112 V
Vh =I1 = 0
1
221 I
Ih =V2 = 0 2
222 V
Ih =I1 = 0
* notes
Two Port Networks
Hybrid Parameters: The following is a popular model used to representa particular variety of transistors.
+ +
_ _
V1 V2
+
_
K1
K4K3V1K2V2
I1 I2
We can write the following equations:
DVCII
BVAIV
212
211
+=
+=
* notes
4
1K
K2
K3
K1
Two Port Networks
Hybrid Parameters:
DVCII
BVAIV
212
211
+=
+=
We want to evaluate the H parameters from the above set of equations.
1
111 I
Vh =V2 = 0 2
112 V
Vh =I1 = 0
1
221 I
Ih =V2 = 0
2
222 V
Ih =I1 = 0
=
=
=
=
2
1R
1
- 1
1R
Two Port Networks
Hybrid Parameters: Another example with hybrid parameters.
Given the circuit below.
+
_
+
_
R1
R2V1 V2
I1 -I2The equations for the circuit are:
V1 = (R1 + R2 )I1 + R2 I2
V2 = R2 I1 + R2 I2
The H parameters are as follows.
1
111 I
Vh =2
112 V
Vh =
1
221 I
Ih =2
222 V
Ih =
V2 =0
V2 =0
I1 =0
I1 =0
=
=
=
=
Two Port Networks
Modifying the two port network:
Earlier we found the z parameters of the following network.
8
20 20 Ω
ΩΩ
Ω
10
+
_
+
_
V1 V2
I1 I2
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2
1
128820
II
VV
* notes
Two Port Networks
Modifying the two port network:
We modify the network as shown be adding elements outside the two ports
8
20 20 Ω
ΩΩ
Ω
10
+
_
+
_
V1 V2
I1 I2
+
_10 v
6Ω
4 Ω
We now have:
V1 = 10 - 6I1
V2 = - 4I2
Two Port Networks
Modifying the two port network:We take a look at the original equations and the equations describingthe new port conditions.
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2
1
128820
II
VV V1 = 10 - 6I1
V2 = - 4I2
So we have,
10 – 6I1 = 20I1 + 8I2
-4I2 = 8I1 + 12I2
* notes
-0.2273
0.4545
0
10
168
826
Two Port Networks
Modifying the two port network:
Rearranging the equations gives,
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
II
⎥⎦⎤
⎢⎣⎡
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−1
2
1
II
Two Port Networks
Y Parameters and Beyond:
Given the following network.
+
_
+
_V1 V2
I1I2
s1
1 Ω
1 Ω
s
(a) Find the Y parameters for the network.
(b) From the Y parameters find the z parameters
Two Port Networks
11
11 VI
y =02=V
21
12 VI
y =01=V
12
21 VI
y =02=V 2
222 V
Iy =
01=V
I1 = y11 V1 + y12 V2
I2 = y21 V1 + y22 V2
+
_
+
_V1 V2
I1I2
s1
1 Ω
1 Ω
s
To find y11
⎥⎦⎤
⎢⎣⎡
+=
+=
122)
12
2( 111 s
Is
sIV so = s + 0.5
11
11 VI
y =02=V
We use the above equations toevaluate the parameters from thenetwork.
Y Parameter Example
short
Two Port NetworksY Parameter Example
12
21 VI
y =
02=V +
_
+
_V1 V2
I1
s1
1 Ω
1 Ω
s
I2
21 2IV −=
We see
= 0.5 S
12
21 VI
y =
Two Port NetworksY Parameter Example
+
_
+
_V1 V2
I1
s1
1 Ω
1 Ω
s
I2
21
12 VI
y =01=V
21
12 VI
y =
12 2IV −=
We have
= 0.5 S
short
22
22 VI
y =01=V
We have
)2(2
22 +=
ssIV s
y 15.022 +=
To find y12 and y21 we reversethings and short V1
Two Port NetworksY Parameter Example
Summary:
Y = ⎥⎦
⎤⎢⎣
⎡+−
−+=⎥
⎦
⎤⎢⎣
⎡s
syyyy
15.05.05.05.0
2221
1211
Now suppose you want the Z parameters for the same network.
Two Port Networks
Going From Y to Z Parameters
For the Y parameters we have:
VYI =
For the Z parameters we have:
IZV =
From above; IZIYV == −1
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
ΔΔ
−
Δ
−
Δ− ==
Y
y
Y
yY
y
Y
y
zz
zzYZ
1121
1222
2221
12111 YY det=Δ
Therefore
where
Two Port Parameter Conversions:
Two Port Parameter Conversions:
To go from one set of parameters to another, locate the set of parametersyou are in, move along the vertical until you are in the row that containsthe parameters you want to convert to – then compare element for element
22
11 hz HΔ
=
Interconnection Of Two Port Networks
Three ways that two ports are interconnected:
* Parallel
* Series
* Cascade
[ ] [ ] [ ]ba
yyy +=
[ ] [ ] [ ]ba
zzz +=
[ ] [ ] [ ]ba
TTT =
ya
yb
za
zb
Ta Tb
parametersY
parametersZ
parametersABCD
Interconnection Of Two Port Networks
Consider the following network:
R1
R2
R1
R2T1 T2
Referring to slide 13 we have;
+
_
+
_
V1 V2
I1 I2
1
2VV
Find
⎥⎦
⎤⎢⎣
⎡−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ +
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡ +
=⎥⎦
⎤⎢⎣
⎡
2
2
2
121
2
12
21
1
1
1111
2
IV
R
RR
RR
R
RR
RR
IV
⎥⎦
⎤⎢⎣
⎡−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ +
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡ +
=⎥⎦
⎤⎢⎣
⎡
2
2
2
121
2
12
21
1
1
1111
2
IV
R
RR
RR
R
RR
RR
IV
Interconnection Of Two Port Networks
Multiply out the first row:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
⎟⎟⎠
⎞⎜⎜⎝
⎛ ++
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡+
⎟⎟⎠
⎞⎜⎜⎝
⎛ += )( 211
2
212
2
12
2
211
IRRR
RRV
RR
RRR
V
Set I2 = 0 ( as in the diagram)
22211
22
1
2
32 RRRR
RVV
+= Can be verified directly
by solving the circuit
End of Lesson
Basic Laws of Circuits
Two-Port Networks
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