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電路學講義第14章14-1
Chapter 14 Two-port networks14.1 Two-ports and impedance parameters
two-port concept, impedance parameters, reciprocal networks 14.2 Admittance, hybrid, and transmission parameters
admittance parameters, hybrid parameters, transmission parameters, parameter conversion
14.3 Circuit analysis with two-portsterminated two-ports, two-ports in cascade, two-ports in series, two-ports in parallel
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電路學講義第14章14-2
14.1 Two-ports and impedance parameters Basics1. Two-port network
• a four-terminal network with input port and output port• the network characteristics is completely described by• a useful method to analyze filter, amplifier,….• can be extended to multi-port networks
4231 , iiii ==
2211 ,,, iviv
No independent sources are in the two-port network and load.concernednot are and 4321 vv
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電路學講義第14章14-3
2. O.C. impedance parameters
[ ]
[ ]
o.c. 2port with impedance transfer forward:
o.c. 1port with impedance transfer reverse:
o.c. 1port with 2port at impedanceinput :
o.c. 2port with 1port at impedanceinput :
matrixparameter impedance o.c.:
:domain sin
,:responses,,:sources
01
221
02
112
02
222
01
111
2221
1211
2
1
2
1
2221212
2121111
2121
2
1
1
2
=
=
=
=
=
=
=
=
⎥⎦
⎤⎢⎣
⎡≡
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡→
⎩⎨⎧
+=+=
−
I
I
I
I
IVz
IVz
IVz
IVz
zzzz
z
II
zVV
IzIzVIzIzV
vvii
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電路學講義第14章14-4
Discussion1. Most two-port networks are three-terminal networks.
2. Equivalent circuit expressed in z-parameters
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎩⎨⎧
+=+=
2
1
2221
1211
2
1
2221212
2121111
II
zzzz
VV
IzIzVIzIzV
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電路學講義第14章14-5
3. Use definition to determine the z-parameter of a two-port network.4. Ex. 14.1 find z-parameters of a “symmetrical” network
21
22
2212
222020102
112
22
2
11
212111
11,
211
11
111
1)1//(
11
1
zsRCCsR
sRCCsRR
sRCsRCz
sRCsRCz
zIsRC
sRCV
sCR
RVIVz
zsRC
CsRR
sRCsC
RsC
RR
RsC
Rz
III
=+
=++
+=
+=→
+=
+==
=++
=
++
=
++
=+=
===
2112
2211
zzzz
==
02
222
01
221
02
112
01
111
2
1
2221
1211
2
1
21
12
12
,
,
,:sources
==
==
==
==
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
II
II
IVz
IVz
IVz
IVz
II
zzzz
VV
II
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電路學講義第14章14-6
5. Ex. 14.2 find z-parameters of an “active” network
100
10023,50
60
)5010(0o.c., 2port)1(
1
221
1
21
1
111
11
2
−==→
−=−=+−==
==→
+==−
ivz
ivvvviv
ivz
ivi
x
xxx
122
112
21
112
222
22
1
50
50
100
50,20o.c., 1port)2(
zivz
ivv
zivz
ivvvi
x
xx
≠==→
==
≠−==→
=−==−
02
222
01
221
02
112
01
111
2
1
2221
1211
2
1
21
12
12
,
,
,:sources
==
==
==
==
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
ii
ii
ivz
ivz
ivz
ivz
ii
zzzz
vv
ii
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電路學講義第14章14-7
6. Reciprocal circuit
21121
2
2
1:network reciprocal zzI
VI
V ococ =→=
21121
2
2
1:network reciprocal yyVI
VI scsc =→=
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電路學講義第14章14-8
7. Any linear network containing no controlled sources is a reciprocal network.∵node equation and mesh equation have symmetrical forms8. Ex. 14.3 T-network
2
112121211
2
2221
121
221121211
1
1112
,,0)2(
,,0)1(
IVzzzz
IVzI
zIVzzzz
IVzI
=+−===
==+−===
1222
121211 ,zzZ
zZzzZ
b
ca
−==−=
02
222
01
221
02
112
01
111
2
1
2221
1211
2
121
1212
,,,
,:sources
====
====
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
IIII IVz
IVz
IVz
IVz
II
zzzz
VV
II
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電路學講義第14章14-9
14.2 Admittance, hybrid, and transmission parametersBasics1. Not all two-ports posses meaningful or measurable z-parameters. →other parameters2. Admittance parameter
[ ]
s.c. 2port with admittance transfer forward:
s.c. 1port with admittance transfer reverse:
s.c. 1port with 2port at admittanceinput :
s.c. 2port with 1port at admittanceinput :
matrixparameter admittance s.c.:
01
221
02
112
02
222
01
111
2221
1211
2
1
1
2
=
=
=
=
=
=
=
=
⎥⎦
⎤⎢⎣
⎡≡
V
V
V
V
VIy
VIy
VIy
VIy
yyyy
y
[ ] ⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡→
⎩⎨⎧
+=+=
2
1
2
1
2221212
2121111
2121 ,:responses,,:sources
VV
yII
VyVyIVyVyI
IIVV
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電路學講義第14章14-10
3. Hybrid parameter
[ ]
s.c. 2port with ratiocurrent forward:
o.c. 1port with ratio voltagereverse:
o.c. 1port with 2port at impedanceinput :1
s.c. 2port with 1port at admittanceinput :1
matrixparameter hybrid:
01
221
02
112
2202
222
1101
111
2221
1211
2
1
1
2
=
=
=
=
=
=
==
==
⎥⎦
⎤⎢⎣
⎡≡
V
I
I
V
IIh
VVh
zVIh
yIVh
hhhh
h
[ ] ⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡→
⎩⎨⎧
+=+=
2
1
2
1
2221212
2121111
2121 ,:responses,,:sources
VI
hIV
VhIhIVhIhV
IVVI
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電路學講義第14章14-11
4. Transmission parameter
[ ]
s.c. 2port with ratiocurrent forward:
o.c. 2port with admittance transfer reverse:
s.c. 2port with impedance transfer reverse:
o.c. 2port with ratio voltagereverse:
matrixparameter on transmissi:
02
1
02
1
02
1
02
1
2
2
2
2
=
=
=
=
−=
=
−=
=
⎥⎦
⎤⎢⎣
⎡≡
V
I
V
I
IID
VIC
IVB
VVA
DCBA
T
[ ] ⎥⎦
⎤⎢⎣
⎡−
=
⎥⎦
⎤⎢⎣
⎡−⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡→
⎩⎨⎧
−=−=
2
2
2
2
1
1
221
221
1122
,:responses,,:sources
IV
T
IV
DCBA
IV
IDVCIIBVAV
IVIV
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電路學講義第14章14-12
Discussion1. Equivalent circuit expressed in y-parameters
12122
112
1212222
222
1
12121
221
1212111
111
2
11,0)2(
11,0)1(
yyV
Iy
yyyVIy
V
yyV
Iy
yyyVIy
V
=−
−==
−+===
=−
−==
−+===
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎩⎨⎧
+=+=
2
1
2221
1211
2
1
2221212
2121111
21,:sources
VV
yyyy
II
VyVyIVyVyI
VV
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電路學講義第14章14-14
2. Ex. 14.5 find y-parameters of an “active” network
20
40223
4040
0s.c., 2port)1(
1
221
11112
1
11111
2
sVIy
VsIIII
sVIysVI
V
==→
==−=
==→=
=−
202
202
1020
102
103
4040
0s.c., 1port)2(
2
222
222
21
2112
212
11221
1
sVIy
VsVVs
VIVIII
ysVIysVI
V
−==→
−=+−=
+=+−=
≠−==→−=
=−
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2221
1211
2
1
21,:sources
VV
yyyy
II
VV
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電路學講義第14章14-14
4. Equivalent circuit expressed in h-parameters
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡→
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡→
⎩⎨⎧
+=+=
ce
b
oefe
reie
c
be
vi
hhhh
iv
CE
vi
hhhh
iv
vhihivhihv
ivvi
2
1
2221
1211
2
1
2221212
2121111
2121 ,:responses,,:sources
3. H-parameters are applied to transistor because they are measured physical quantities.
bevcev
bici
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電路學講義第14章14-15
5. Ex. 14.6 find h-parameters of an “active” network
2
23
4040
0s.c., 2port)1(
1
221
1112
1
111
11
2
==→
=−=
==→
=
=−
IIh
IIIIsI
Vh
sVI
V
1.0
10
1
0o.c., 1port)2(
2
222
22
2
11221
1
==→
=
==→=
=−
VIh
IVVVhVV
I
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡→
⎩⎨⎧
+=+=
2
1
2221
1211
2
1
2221212
2121111
21
21
,:responses,:sources
VI
hhhh
IV
VhIhIVhIhV
IVVI
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電路學講義第14章14-16
6. Ex. 14.7 find ABCD-parameters of an “active” network
sVVA
Vs
Is
VV
VIC
VIVII
I
21
)21(40201
20,
103
0o.c., 2port)1(
2
1
2121
2
1
21
211
2
−==→
−=+=
−==→
−=+=
=−
sIVB
Is
Is
V
IID
IIIIIV
20
204021
2,30s.c., 2port)2(
2
1
211
2
1
21211
2
−=−
=→
==
−=−
=→
=−==−
⎥⎦
⎤⎢⎣
⎡−⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡→
⎩⎨⎧
−=−=
2
2
1
1
221
221
11
22
,:responses,:sources
IV
DCBA
IV
IDVCIIBVAVIV
IV
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電路學講義第14章14-17
7. All the 2-port parameters are related as given in Table 14.2.8. Conversion between z-parameters and y-parameters
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ] 211222111121
1222
1
211222111121
1222
1
2
1
2
11
2
1
2
1
2
1
2221
1211
2
1
,
,
parameter
yyyyyyy
yy
yz
zzzzzzz
zz
zy
VV
yVV
zII
II
zII
zzzz
VV
z
y
yy
yy
z
zz
zz
−==∆
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∆∆−
∆−
∆==
−==∆
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∆∆−
∆−
∆==⇒
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡→⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−
−
−
−
-
電路學講義第14章14-18
9. Derivation of h-parameters from z-parameters
[ ]⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
∆
=
−=∆+∆
=+−+=→
+−=→⎩⎨⎧
+=+=
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−
2222
21
22
12
22
21122211222
121
222
221
22
21121111
)3(
222
122
212
2221212
2121111
2
1
2221
1211
2
1
2
1
2221
1211
2
1
1
,)1()1(
)3...(1)2( )2...()1...(
parameter ,parameter
zzz
zz
zh
zzzzVzzI
zV
zI
zzzIzV
Vz
IzzI
IzIzVIzIzV
VI
hhhh
IV
hII
zzzz
VV
z
z
zz
-
電路學講義第14章14-19
10. Derivation of z-parameters from T-parameters
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ ∆
=
−=∆∆
+=−+=→
+=→⎩⎨⎧
−=−=
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡−⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−
CD
C
CCA
z
BCADIC
ICAIBI
CDI
CAV
ICDI
CV
IDVCIIBVAV
II
zzzz
VV
zI
VDCBA
IV
T
T
TT
1
,)1()1(
)3...(1)2( )2...()1...(
parameter ,parameter
212211
)3(
212221
221
2
1
2221
1211
2
1
2
2
1
1
11. For a reciprocal 2-port network, 1,,, 211221122112 =−=∆−=== BCADhhyyzz T
-
電路學講義第14章14-20
14.3 Circuit analysis with two-portsBasics1. Terminated two-ports using z-parameters
Lz
L
L
L
LL
v
LL
LL
Li
LL
L
Ss
ZzZz
zzzzZzZz
Zz
Zzzzzz
VVH
VZzV
ZzzV
zzV
VZz
zVz
IVZzIzV
zZz
IIH
IzZIzIzIzIZ
IZVVIZV
IzIzVIzIzV
11
21
2211211211
21
22
11
2112
11
21
1
2
222
211
121
11212
(5)
211
121
1112
121111
(4)22
21
1
2
2221212221212
)4(
22
11
2221212
2121111
1function transfer voltage
)1()2(
)5...(1,(1)
function ansfer current tr
)(,)2(
)4........()3...(
,)2...()1...(
+∆=
+−=
+−=≡⇒
−+=→
+=−=→
+−
=≡⇒
+−=+=−→
⎩⎨⎧
−=+=
⎩⎨⎧
+=+=
-
電路學講義第14章14-21
11
22
11
2112221122
02
2
222211
12212
)8(
211
1212121111
)7(
11
22
11
22
2112221111
1
1
122
21121111
(6)22
21
1
2
22
11
2221212
2121111
impedanceoutput equivalent
)()2(
)8...()1(),7...()3(,0
impedanceinput equivalent
(1)
)6...(
)4........()3...(
,)2...()1...(
zZZz
zZzzzzZz
IVZ
IzIzZ
zzV
IzZ
zIIzIzIZIZVV
zZZz
zZzzzzZz
IVZ
IzZ
zzIzV
zZz
IIH
IZVVIZV
IzIzVIzIzV
S
Sz
S
S
Vo
S
SSSs
L
Lz
L
Li
L
Li
L
Ss
s+
+∆=
+−+
=≡⇒
++
−=→
+−=→+=−→−=→=
++∆
=+
−+=≡⇒
+−
−=→
+−
=≡
⎩⎨⎧
−=+=
⎩⎨⎧
+=+=
=
-
電路學講義第14章14-22
2. Cascade connection using T-parameters
[ ] [ ]
[ ] [ ]
[ ] [ ] [ ]bacas
ba
b
ba
a
aa
TTTI
VTT
IV
TI
VT
IV
=⇒
⎥⎦
⎤⎢⎣
⎡−
=
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡
2
2
1
1
2
2
1
1
3. Series connection using z-parameters
[ ] [ ]
[ ] [ ]
[ ] [ ] [ ]baser
ba
ba
b
b
a
a
zzzII
zz
II
zII
z
VV
VV
VV
+=⇒
⎥⎦
⎤⎢⎣
⎡+=
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2
1
2
1
2
1
2
1
2
1
)(
-
電路學講義第14章14-23
4. Parallel connection using y-parameters
[ ] [ ]
[ ] [ ]
[ ] [ ] [ ]bapar
ba
ba
b
b
a
a
yyyVV
yy
VV
yVV
y
II
II
II
+=⇒
⎥⎦
⎤⎢⎣
⎡+=
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2
1
2
1
2
1
2
1
2
1
)(
-
電路學講義第14章14-24
Discussion1. Relations of terminated two-ports in terms of z- y- h- and T-
parameters are given in Table 14.3. They are useful in network analysis.
2. Ex. 14.9 given load be a 2.5H inductor, find I2/V1 from T-parameters
)2)(4(4.0
824.0
205.2)21(
11
1)(
21
201
2021 ex.14.7 From
2
2.14
1
2
1
2
+−−
=
−−−
=−−
−=
+−
=
++
+−
====
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−−=⎥
⎦
⎤⎢⎣
⎡
sss
sss
ss
sBAZ
BAZDCZ
DCZZH
IZI
VIsH
ssDCBA
L
L
L
L
Table
i
i
i
-
電路學講義第14章14-25
3. Ex. 14.10 find RL to give Ai=Iout/Is=-25 from transistor h-parameters
[ ]
L
Li
L
Lhi
h
YhYh
IIsH
YhYh
IVsZ
h
+==
++∆
==
=∆
⎥⎦
⎤⎢⎣
⎡
×=
−
−
22
21
1
2
22
11
1
1
3
3
)(
)(
05.0101.050
101000
Ω=→−=+−
=
+=
++
=
+−=
−==
kRR
A
RsH
RRsZ
ZRRH
II
II
IIA
LL
i
Li
L
Li
is
si
ss
outi
42525.03
10011.0
50)(,11.0105.0)(
11
2
-
電路學講義第14章14-26
4. Ex. 14.11 find Ai of two amplifiers of ex.14.10 in cascade
[ ] [ ] [ ]
[ ] [ ] [ ]
1100
16451)(,961)(
1044010042.042.01041
02.01022010
1,101.050101000
1
1
2
66
6
6
3
2121
22
21
11
213
3
=+
−=−
==
−=+
−==
++
=
⎥⎦
⎤⎢⎣
⎡
×××
==→
⎥⎦
⎤⎢⎣
⎡
−×−−−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−∆−
==⎥⎦
⎤⎢⎣
⎡
×=
−−
−
−
−
−
−
is
si
ss
outi
Li
L
Li
bacas
h
ba
ZRRH
II
II
IIA
DCRsH
DCRBARsZ
TTT
hhh
hh
hTTh
-
電路學講義第14章14-27
5. Bridged-T connection
[ ]
[ ] ⎥⎦
⎤⎢⎣
⎡++−+−+
=⇒
−====
====⎥⎦
⎤⎢⎣
⎡=
====
bFbF
bFbF
FF
VVVV
yYyYyYyY
y
YyyYyy
VIy
VIy
VIy
VIy
yyyy
y
2221
1211
21122211
01
221
02
112
02
222
01
111
2221
1211
,
,, ,,2112
-
電路學講義第14章14-28
6. Ex.14.12 find Hv=V2/V1 of a high frequency transistor
[ ] [ ] [ ]
LoLo
m
Lo
m
Lv
om
i
om
ibaF
RRRRGCGsCgs
RRsCgsC
YyyH
RsCsCg
sCR
sCy
Rg
RysCsCsCsC
ysC
Z
///1/1/1//
/1/1
1
1
1
01
,,1
22
21
=+=+−
=++
−=
+−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−
−+=⇒
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=⎥⎦
⎤⎢⎣
⎡−
−==