two-port networks - khon kaen universityeestaff.kku.ac.th/~jamebond/182304/two port.pdf · two-port...
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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 tτ τ τ= + ≥∫For LTI network
0( ) ( ) ( ) ( ) 0
t
OCv t e t h t i d tτ τ τ= + − ≥∫In frequency domain
( ) ( ) ( ) ( )OCV s E s Z s I s= +
0
No independent sources
iD(t)
VD
vd(t)
+
-
vD(t)
+
-
t
va
VA
0t
vA=V
A+v
a
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 pointQ
VD
ID
di
dv
vD (V)
iD (mA)
t
t
ID
0.699 0.6995 0.7 0.7005 0.7011.38
1.40
1.42
1.44
1.46
1.48
1.50
1.52
VD
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
32xx
xIi 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+ v
d
iD = I
D+ i
d
rd
id+
-
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
d
d
gr
=
iD
vD
V
10 kΩmA93.0
k10
V7.0V10=
Ω−
≈DI
V7.0≈DV
Ω=×
== 8.53mA93.0
mV252
D
Td
I
nVr
)100sin(µV5.53)100sin(mV01k10
ttr
rv
d
dd =
Ω+=
)100sin(V5.53V7.0 tvD µ+≈
Example If 10V 10mV sin(100 )V t= + find Dv
Two-port networks
LTI one ports
1V
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
1Iβ
1I
inZ
Fig. 2
211
Z
VIII in
in +−== β 2(1 )in inV Z Iβ= + 2(1 )inZ Zβ= +
Example 2
Determine the output impedance of the circuit in Fig. 3
1Iβ
1I
outZ
outI
outV
Fig. 3
1 11
(1 ) outout
VI I I
Zβ β= − − = + 1
1
outout
out
V ZZ
I β= =
+
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.
1V 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
( )g=i v 1 1 1 2
2 2 1 2
( , )
( , )
i g v v
i g v v
=
=or
( )r=v 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 2I
10.5VFig 5.
1 1 1 2 1 2 1 2 1 2 2( ) ( )I Y V 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 parameter
Example 4
Compute the y-parameter of the circuit in Fig.6
1V 2V
1I 2I1I
1VFig.6
1 1 1 1 1 1 1 2
1ˆ 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 Va
I V
a a
−
= −
11 12
21 22 2
12,
1 2,
y ya
y ya a
= = −
= − =
Y parameter analysis of terminated two-port
1V 2V
1I 2I
LY
Fig. 9 Terminated two-port
=
2
1
2221
1211
2
1
V
V
yy
yy
I
I
Y-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( )in
L
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 2
10 10 104 ( ) (4 )V I I I I I
s s s= + + = + +
2 2 1 2 1 2
10 10 103 ( ) (3 )V I I I I I
s s s= + + = + +
11 12
21 22
10 4 10
10 3 10
s
z z s sZ
z z s
s s
+
= = +
1V
1I+ -24I
0.1F
3
+ +
- -2V
2I
In frequency domain
The Y parameter
Example 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= − + + + +
313 1 2
1 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
RR
RRR
RR
RRR
RRR
zz
zz
Z parameter analysis of terminated two-port
1V 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= = =
+ + + +i i i
1I
inZ
sv
sR
11z
12 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
sV k1V2V
1I2I
outI
outV
inZ 2inZFig 16
Z parameter analysis of terminated two-port
Solution12 21
2 1122
66 6790.8 1.0258 10
1.0262 106793.5 16
3,159
in
L
z zZ z
z Z= −
+
× ⋅= ⋅ −
+= Ω
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 10
3506667 1224.7
687.9
in
L
z zZ z
z Z= −
+
×= +
+= Ω
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=
+ +
− = + + = −
Ω=== 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= = =
+ + + +i i i
0.902 225.6
Z parameter analysis of terminated two-port
The 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
outI
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
0.5 350
14.276 k
out
s
z zZ z
R z= −
+
×= +
+= Ω
2 1 // 2 1.7542 ks outR Z k= = Ω6
6
6790.8 1.0258 106793.5
1754.24 1.0262 10
16.93
outZ⋅ ×
= −+ ×
= Ω
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 V
LZ
22h
21 1h I
1V 2V
2I
Fig 17 Hybrid parameter model
The h-parameter
Example 10
Determine the h-parameter of the two-port circuit shown in Fig. 18
1V 2V
2I1I
2V
Fig. 18
21ˆ1
Va
V = 1 2I aI= −
2 2 2 1 2ˆ R
V V RI I Va
= − = +
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-parameter
Example 10
Find the h-parameter of the circuit in Fig. 19 assuming L1=L2=M=1H
1V 2V
2I1I
2V
2L1L
1I
Fig. 19
In frequency domain
2111ˆ sMIIsLV +=
111ˆ 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
1
1 1 11
2 2 2
1 0
(1 ) 1
V sL sM IsL
I sM sL VsM
−+ −
= − + −
The h-parameter
1
1 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
=
1−=Z Y and 1−=Y 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 z
y y Z Z
y y z z
Z Z
− ∆ ∆= − ∆ ∆
22 12
11 12
21 22 21 11
y y
z z Y Y
z 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
− − = −
1
1 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 parameters
It can be shown that for the terminated two-port with h-parameter the following equations can be derived
122
212 I
Yh
hV
L+−=
L
inYh
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
( )VS
s 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 at
the output port
1 2
1 2
V VA B
I IC D
= −
or
Transmission parameter
Example 11
1V 2V
2I1I
2V 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.
1V2V 3V 4V
1I 2I− 3I 4I−
−=
2
21
1
1T
I
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
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
top related