Two-port networks Review of one ports Various two-port descriptions Terminated nonlinear two-ports...
72
Two-port networks Review of one ports Various two-port descriptions Terminated nonlinear two-ports Impedance and admittance matrices of two- ports Other two-port parameter matrices The hybrid matrices The transmission matrices
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Transcript of Two-port networks Review of one ports Various two-port descriptions Terminated nonlinear two-ports...
Two-port networks
Review of one ports Various two-port descriptions Terminated nonlinear two-ports Impedance and admittance matrices of two-ports Other two-port parameter matrices The hybrid matrices The transmission matrices
1-port 2-port 2-port 2-port 1-port
Thevenin’s Equivalent Circuit
Norton’s Equivalent Circuit
Nv
i
NOv
i
OCe
0( ) ( ) ( , ) ( ) 0
t
OCv t e t h t i d t For LTI network
0( ) ( ) ( ) ( ) 0
t
OCv t e t h t i d t In frequency domain
( ) ( ) ( ) ( )OCV s E s Z s I s
0
No independent sources
i D (t)
V D
vd(t)
+
-
vD (t)
+
-
tva
V A
0 t
vA = V A + va
0
|va |p
Nonlinear one port
0.750.5 0.55 0.6 0.65 0.7
0
1
1.5
2
2.5
3
3.5
0.5
vD (V)
iD (mA)
t
tBias point Q
VD
ID
di
dv
vD (V )
i D (m A )
t
tI D
0 .6 9 9 0 .6 9 9 5 0 .7 0 .7 0 0 5 0 .7 0 11 .3 8
1 .4 0
1 .4 2
1 .4 4
1 .4 6
1 .4 8
1 .5 0
1 .5 2
V D
For small dv
dv
di
TD nVVSD eII /
Td
TdTD
TdDTD
nVvD
nVvnVVS
nVvVS
nVvSD
eI
eeI
eIeIi
/
//
/)(/
For DC bias
For DC bias + small signal
...!3!2
132
xx
xex
From Taylor’s series expansion
...
!3!21
32 xxxIi DD
Where /d Tx v nV
dT
DDDD v
nV
IIxIi 1
For 1 or d Tx v nV
d
dd
T
Dd r
vv
nV
Ii
di
T
D
nV
I
vD = V D + vd
i D = I D + i d
r d
i d+
-
vd
+
-
Td
D
nVr
I
0.750.5 0.55 0.6 0.65 0.7
0
1
1.5
2
2.5
3
3.5
0.5
vD (V)
iD (mA)
t
tBias point
Q
VD
ID
Slope at Q point = 1
dd
gr
i D
vD
V
10 kmA93.0
k10
V7.0V10
DI
V7.0DV
8.53mA93.0
mV252
D
Td I
nVr
)100sin(μV5.53)100sin(mV01k10
ttr
rv
d
dd
)100sin(V5.53V7.0 tvD μ
Example If 10V 10mVsin(100 )V t find Dv
Two-port networks
LTI one ports
One port network1V
+
-
1I
inZ inY
1
1
I
VZ in
1
1in
IY
V
Input impedance Input admittance
Fig. 1
Two-port networks
Example 1
Determine the input impedance of the circuit in Fig. 2
Z2
Z3
1I
1I
inZ
Fig. 2
211 Z
VIII in
in 2(1 )in inV Z I 2(1 )inZ Z
Example 2
Determine the output impedance of the circuit in Fig. 3
Z1
Z3
1I
1I
outZ
outI
outV
+
-
Fig. 3
1 11
(1 ) outout
VI I I
Z 1
1out
outout
V ZZ
I
Two-port networks Circuits can be considered by theirs terminal variables Voltages and currents are terminal’s variables Complex circuit can be analyzed more easily. There are many kinds of two port parameters.
Two port network1V 2V
+
-
+
-
1I 2I
Fig. 4 A two port network
Common-Emitter (CE) Fixed-Bias Configuration
Removing DC effects of VCC and Capacitors
Small signal equivalent circuit
Hybrid equivalent model re equivalent model
Various two-port descriptions
( )gi v 1 1 1 2
2 2 1 2
( , )
( , )
i g v v
i g v v
or
( )rv i 1 1 1 2
2 2 1 2
( , )
( , )
v r i i
v r i i
or
Port currentPort voltage
1 1 1 2
2 2 1 2
( , )
( , )
v h i v
i h i v
Or hybrid
Two-port networks
The Y parameter
2
1
2221
1211
2
1
V
V
yy
yy
I
I
The admittance or Y parameter of a two port network is defined by
1 11 1 12 2
2 21 1 22 2
I y V y V
I y V y V
or in scalar form
The Y parameter
2 1
2 1
1 111 12
1 20 0
2 221 22
1 20 0
V V
V V
I Iy y
V V
I Iy y
V V
The Y parameters can found from
These parameters are call short-circuited admittance parameters
The Y parameter
Example 3
Determine the admittance parameters from the circuit in Fig 5.
1V 2V+
-
+
-
1I 2IY2
Y1 Y310.5V
Fig 5.
1 1 1 2 1 2 1 2 1 2 2( ) ( )I YV Y V V Y Y V Y V
2 1 3 2 2 2 1 2 1 2 3 20.5 ( ) (0.5 ) ( )I V Y V Y V V Y V Y Y V
1 2 21 1
2 2 32 20.5
Y Y YI V
Y Y YI V
11 1 2 12 2
21 2 22 2 3
,
0.5 ,
y Y Y y Y
y Y y Y Y
The Y parameterExample 4
Compute the y-parameter of the circuit in Fig.6
1V 2V+
-
+
-
1I 2I1
11
1:a1̂I
1̂V+
- Fig.6
1 1 1 1 1 1 1 21ˆ ˆ( ) 2 2I V V V V V V Va
2 1 1 1 1 1 22
1 1 1 2ˆ ˆ ˆ( )I I V V V V Va a a a
1 1
2 22
12
1 2
I VaI V
a a
11 12
21 22 2
12,
1 2,
y y a
y ya a
Y parameter analysis of terminated two-port
Two port network1V 2V
+
-
+
-
1I 2I
LY
Fig. 9 Terminated two-port
2
1
2221
1211
2
1
V
V
yy
yy
I
IY-parameter equations
22 VYI L
11 12 11
21 22 20 L
y y VI
y y Y V
Y parameter analysis of terminated two-port
From Crammer’s rules
21122211
122
2221
1211
22
121
1 )(
)(0
yyYyy
IYy
Yyy
yy
Yy
yI
VL
L
L
L
The input admittance Yin
12 2111
11 22( )inL
y yY y
y y Y
and
21 1 22 2
212 1
22
( )L
L
y V y Y V
yV V
y Y
Y parameter analysis of terminated two-port
12 211 11 1 12 2 11 1
22 L
y yI y V y V y V
y Y
Gain: 2 21
1 22 L
V y
V y Y
1I
inY
+
-
sv
sR
11y 12 2y VLY22y21 1y V
+ +
--
1V 2V
2I
Fig 10 Terminated two-port Y-parameter model
Two-port networks
The Z parameter
1 11 12 1
2 21 22 2
V z z I
V z z I
The impedance or Z parameter of a two port network is defined by
1 11 1 12 2
2 21 1 22 2
V z I z I
V z I z I
or in scalar form
The Z parameter
2 1
2 1
1 111 12
1 20 0
2 221 22
1 20 0
I I
I I
V Vz z
I I
V Vz z
I I
The Z parameters can be found from
These parameters are call open circuit impedance parameters
The Z parameterExample 6Determine the impedance parameters from the circuit in Fig 11
Fig 11.
1 2 1 2 1 210 10 10
4 ( ) (4 )V I I I I Is s s
2 2 1 2 1 210 10 10
3 ( ) (3 )V I I I I Is s s
11 12
21 22
10 4 10
10 3 10
sz z s sZz z s
s s
1V
1I+ -24I
0.1F
3
+ +
- -2V
2I
In frequency domain
The Y parameterExample 7
Compute the z-parameter of the circuit in Fig.12
Fig.12
1 1 1 1 3V R I R I
2 3 2 3 3V R I R I
1V
1I R2
+ +
- -2V
2I
R3R13I
1 1 3 2 1 2 3 30 ( )R I R I R R R I 31
3 1 21 2 3 1 2 3
RRI I I
R R R R R R
The Z parameter2
1 311 1 1 2
1 2 3 1 2 3
1 2 3 1 31 2
1 2 3 1 2 3
( )
( )
R RRV R I I
R R R R R R
R R R R RI I
R R R R R R
2
1 3 32 1 3 2
1 2 3 1 2 3
1 3 3 1 21 2
1 2 3 1 2 3
( )
( )
R R RV I R I
R R R R R R
R R R R RI I
R R R R R R
321
213
321
31
321
31
321
321
2121
1211
)()
))(
RRR
RRR
RRR
RRRRR
RR
RRR
RRR
zz
zz
Z parameter analysis of terminated two-port
Two port network1V 2V
+
-
+
-
1I 2I
LZ
Fig. 14 Terminated two-port
1 11 12 1
2 21 22 2
V z z I
V z z I
Z-parameter equations
2 2LV Z I
11 12 11
21 22 20 L
z z IV
z z Z I
Z parameter analysis of terminated two-port
From Crammer’s rules
1 12
22 22 11
11 12 11 22 12 21
21 22
0 ( )
( )L L
L
L
V z
z Z z Z VI
z z z z Z z z
z z Z
The input impedance Zin
12 2111
22in
L
z zZ z
z Z
and
21 1 22 2
212 1
22
( )L
L
z I z Z I
zI I
z Z
Z parameter analysis of terminated two-port
12 211 11 1 12 2 11 1
22 L
z zV z I z I z I
z Z
Gain: 2 1 2 21 21
1 22 22
in L L
s s in s L in L in s
ZV V V Z z Z z
V V V Z Z z Z Z z Z Z Z
1I
inZ
+
-
sv
sR
11z12 2z I
LZ
22z
21 1z I+ +
--
1V 2V
2I
+
-+
-
Fig 15 Terminated two-port Z-parameter model
Z parameter analysis of terminated two-portExample 9
The circuit in Fig 16 is a two-stage transistor amplifier. The Z-parameters for each stage are
6
2 6
1.0262 10 6,790.8Z
1.0258 10 6,793.5
1 6
350 2.667Z
10 6,667
Determine a) The input impedance and 2inZ inZb) The overall voltage gain
c) Check the matching of the load and output impedance
Z1
Stage 1
Z2
Stage 2
0.5
sV+- 162k
+ +
- -
1V2V
1I 2IoutI
+
-outV
inZ 2inZ Fig 16
Z parameter analysis of terminated two-portSolution
12 212 11
22
66 6790.8 1.0258 10
1.0262 106793.5 16
3,159
inL
z zZ z
z Z
21
22 22
616(1.0258 10 )
(16 6793.5)3,159
0.7629
out L
L in
V Z z
z Z ZV
Z parameter analysis of terminated two-port
12 2111
22 1
62.667 10350
6667 1224.7687.9
inL
z zZ z
z Z
2 1 21
1 22
61224.7 10
1224.7 6667 75 687.9
203.4
L
L s ins
V Z z
Z z Z ZV
7.12243159//2000//2 21 inL ZkZ
2 1 2 21 21
1 22 22
in L L
s s in s L in L in s
ZV V V Z z Z z
V V V Z Z z Z Z z Z Z Z
0.902 225.6
Z parameter analysis of terminated two-portThe overall voltage gain
VV
V
V
V
V
V
VA
s
out
s
outVS
/2.155
)4.203(7629.0
2
2
Out put impedance
02
2
sV
out I
VZ
The detail is left to the student to show that
12 2122
11out
s
z zZ z
R z
Z parameter analysis of terminated two-port
Therefore the load is closely matched to the output impedance
12 211 22
11
62.667 106667
0.5 35014.276k
outs
z zZ z
R z
2 1 // 2 1.7542ks outR Z k 6
6
6790.8 1.0258 106793.5
1754.24 1.0262 1016.93
outZ
The h-parameter (Hybrid parameter)H-parameter is the combination of Z and Y parameter defined by
1 11 12 1
2 21 22 2
V h h I
I h h V
1 11 1 12 2
2 21 1 22 2
V h I h V
I h I h V
or in scalar form
H-parameter is commonly used in transistor modeling.
The h-parameter
2
2
1
1
1 12 2111 11
1 11 220
2 21 2121
1 11 220
2 12 2122 22
2 11 220
1 12 1212
2 11 220
1
1
V
V
I
I
V z zh z
I y z
I y zh
I y z
I y yh y
V y z
V y zh
V y z
The h parameters can found from
The h-parameter
1I
inZ
+
-
sv
sR11h
12 2h VLZ
22h
21 1h I+ +
--
1V 2V
2I
+
-
Fig 17 Hybrid parameter model
The h-parameterExample 10
Determine the h-parameter of the two-port circuit shown in Fig. 18
+ +
--1V 2V
2I1I 1:a+
2̂V-
R
Fig. 18
21ˆ1V
aV 1 2I aI
2 2 2 1 2ˆ RV V RI I V
a
1 1 22
1RV I V
aa
2 2 2 22
1 2
ˆ ˆ
10
V V V VI
R R R
I Va
2
12
2
1
01
1
V
I
a
aa
R
I
V
The h-parameterExample 10
Find the h-parameter of the circuit in Fig. 19 assuming L1=L2=M=1H
+ +
--1V 2V
2I1I
+
2̂V-
1
2L1L
1̂I
1
M
Fig. 19
In frequency domain
2111ˆ sMIIsLV
111̂ VII
1 1 2 1 1(1 )sL V sMI sL I
The h-parameter
2 2 2 1 2 2 1 1ˆ ˆ ( )V sL I sMI sL I sM I V
2 2 2ˆV V I
2 2 2 1 1(1 ) ( )V sL I sM I V
1 2 2 1 2(1 )sMV sL I sMI V
In matrix form
2
11
2
1
2
1
1
0
)1(
1
V
I
sM
sL
I
V
sLsM
sMsL
11 1 11
2 2 2
1 0
(1 ) 1
V sL sM IsL
I sM sL VsM
The h-parameter
11 1
2 2
1 0
(1 ) 1
V Is s s
I Vs s s
With L1=L2=M=1 H
2
1
112
1
V
I
ss
ss
s
The inverse hybrid parameter (g- parameter)g-parameter is defined by
1 11 12 1
2 21 22 2
I g g V
V g g I
1 11 1 12 2
2 21 1 22 2
I g V g I
V g V g I
or in scalar form
g-parameter is an alternative form of hybrid representation.
2
2
1
1
1 12 21 2211 11
1 11 220
2 21 21 2121
1 11 220
2 12 21 1122 22
2 11 220
1 12 12 1212
2 11 220
11 22 12 21
1
1
I
I
V
V
I y y hg y
V z y h
V z y hg
V z y h
V z z hg z
I z y h
I z y hg
I z y h
h h h h h
where
The g parameters can found from
Inverse hybrid parameter model
Conversion of Two-port parameters
I YV
V ZYV
Two port parameters can be converted to any form as follows
From
2
1
2221
1211
2
1
V
V
yy
yy
I
I
And 1 11 12 1
2 21 22 2
V z z I
V z z I
1Z Y and 1Y Z
V ZI
11 22 12 21
11 22 12 21
Z z z z z
Y y y y y
22 12
11 12
21 22 21 11
z zy y Z Zy y z z
Z Z
22 12
11 12
21 22 21 11
y yz z Y Yz z y y
Y Y
where
Conversion of Two-port parametersFrom y to h
2
1
2221
1211
2
1
V
V
yy
yy
I
I
11 1 1 12 2
21 1 2 22 2
y V I y V
y V I y V
11 1 12 1
21 2 22 2
0 1
1 0
y V y I
y I y V
11 11 12 1
2 21 22 2
0 1
1 0
V y y I
I y y V
Conversion of Two-port parameters
1 12 1
2 21 11 22 12 21 211
11V y I
I y y y y y Vy
Hence
12
11 1111 12
21 22 21 12 2122
11 11
1 y
y yh h
h h y y yy
y y
Conversion of Two-port parametersIt can be shown that for the terminated two-port with h-parameter the following equations can be derived
122
212 I
Yh
hV
L
Lin Yh
hhh
I
VZ
22
211211
1
1
2 12 2122
2 11out
s
V h hZ h
I h Z
2 21
1 22( )L in
V h
V h Y Z
and2 1 2 21
1 22
1
( )VSs s L in s
V V V hA
V V V h Y Z Z
Transmission parameter The t-parameter or transmission parameters are used in power systemand it is called ABCD parameter. The transmission parameter is defined by
1 11 12 2
1 21 22 2
V t t V
I t t I
This means that the power flows into the input port and flow out to theload from the output port.
t-parameter can be calculated from
2 2
2 2
1 111 12
2 20 0
1 121 22
2 20 0
I V
I V
V Vt t
V I
I It t
V I
Open or short circuit atthe output port
1 2
1 2
V VA B
I IC D
or
Transmission parameterExample 11
+ +
--1V 2V
2I1I 1:a+
2̂V-
R
Fig 20
Determine the t-parameter of the circuit shown in Fig 20.
)(1ˆ1
2221 RIVa
Va
V
21 aII
2
21
1
1
0 I
V
aI
VaR
a
Transmission parameter
One of the most importance characteristics of the two-port circuit witht-parameter is to determine the overall cascade parameter.
T1 T2
+ + + +
----1V 2V 3V 4V
1I 2I 3I 4I
2
21
1
1 TI
V
I
V3 4
23 4
TV V
I I
3232 , IIVV
1 41 2
1 4
T TV V
I I
Therefore
2 1
2 1
V VA B
I IC D
Inverse Transmission parameter
1 1
1 1
2 2
1 10 0
2 2
1 10 0
I V
I V
V VA B
V I
I IC D
V I
Interconnection of two-port network Two port networks can be connected in series parallel or
cascaded Series and parallel of two-port have 4 configurations
Series input-series output (Z-parameter) Series input-parallel output (h-parameter) Parallel in put-series output (g or h-1-parameter) Parallel input-parallel output (Y-parameter)
With proper choice of parameters the combined parameters can be added together.
Interconnection of two-port network
Z1
Z2
Z=Z1+Z2
V1
+ + +
+ +
-
- -
- -
V11
V12
V21
V22
H1
H2
H=H1+H2
V1V2
+ ++
+
- -
-
-
V11
V12
G1
G2
G=G1+G2
V2
+ ++
+-
-
-
V21
V21
Y1
Y2
Y=Y1+Y2
V1V2
+ +
- -
Example Bridge-T network
N1 // N2
For network N2
For network N1
41
0 1
ZT
Y-parameters of the bridge-t network are
