Chapter 14 Two-port networks
14.1 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-ports
terminated two-ports, two-ports in cascade, two-ports in series,
two-ports in parallel
14-1
電路學講義第14章
14.1 Two-ports and impedance parameters
Basics
1. Two-port network
• a four-terminal network with input port and output port
• the network characteristics is completely described by v1 , i1 , v2 , i2
• a useful method to analyze filter, amplifier,….
• can be extended to multi-port networks
No independent sources are
in the two-port network and
load.
i1 = i3 , i2 = i4
v21 and v43 are not concerned
14-2
電路學講義第14章
2. O.C. impedance parameters
sources : i1 , i2 , responses : v1 , v2
⎧V 1 = z11 I 1 + z12 I 2
⎡V 1 ⎤
⎡I1 ⎤
in s − domain : ⎨
→ ⎢ ⎥ = [z ]⎢ ⎥
=
+
V
z
I
z
I
⎣I 2 ⎦
21 1
22 2
⎩ 2
⎣V 2 ⎦
⎡ z11 z12 ⎤
[z ] ≡ ⎢
⎥ : o.c. impedance parameter matrix
z
z
⎣ 21 22 ⎦
z11 =
z 22 =
z12 =
z 21 =
V1
I1
V2
I2
V1
I2
V2
I1
: input impedance at port 1 with port 2 o.c.
I 2 =0
: input impedance at port 2 with port 1 o.c.
I 1 =0
: reverse transfer impedance with port 1 o.c.
I 1 =0
: forward transfer impedance with port 2 o.c.
I 2 =0
14-3
電路學講義第14章
Discussion
1. Most two-port networks are three-terminal networks.
2. Equivalent circuit expressed in z-parameters
⎧V 1 = z11 I 1 + z12 I 2
⎨
⎩V 2 = z 21 I 1 + z 22 I 2
⎡V 1 ⎤ ⎡ z11
⎢V ⎥ = ⎢ z
⎣ 2 ⎦ ⎣ 21
z12 ⎤ ⎡ I 1 ⎤
z 22 ⎥⎦ ⎢⎣ I 2 ⎥⎦
14-4
電路學講義第14章
3. Use definition to determine the z-parameter of a two-port network.
4. Ex. 14.1 find z-parameters of a “symmetrical” network
z11 = z 22
z12 = z21
sources : I 1 , I 2
⎡V 1 ⎤ ⎡ z11
⎢V ⎥ = ⎢ z
⎣ 2 ⎦ ⎣ 21
z11 =
z 21 =
V1
I1
V2
I1
z12 ⎤ ⎡ I 1 ⎤
z22 ⎥⎦ ⎢⎣ I 2 ⎥⎦
, z12 =
I 2 =0
V1
I2
, z 22 =
I 2 =0
V2
I2
I 1 =0
I 1 =0
1
1
1
R + sR 2C
z11 = R //( + R) =
=
=
= z 22
sC
1
1
1
sRC
sC
1
2
+
+
+
1
R R+
R 1 + sRC
sC
V
R
sRC
z12 = 1
, V 1 I =0 =
V 2 I =0 =
I 2 z22
1
1
1
I 2 I =0
1 + sRC
R+
1
sC
sRC
sRC R + sR 2C
sR 2C
→ z12 =
z 22 =
=
= z21
1 + sRC
1 + sRC 1 + 2sRC 1 + 2sRC
14-5
電路學講義第14章
5. Ex. 14.2 find z-parameters of an “active” network
sources : i1 , i2
⎡ v1 ⎤ ⎡ z11
⎢v ⎥ = ⎢ z
⎣ 2 ⎦ ⎣ 21
z11 =
v1
i1
v
z 21 = 2
i1
z12 ⎤ ⎡ i1 ⎤
z 22 ⎥⎦ ⎢⎣i2 ⎥⎦
, z12 =
i2 = 0
i2
v1
i2
v
, z 22 = 2
i2
=0
i1 = 0
(1)port − 2 o.c., i2 = 0
v1 = i1 (10 + 50)
→ z11 =
v1
= 60
i1
vx = 50i1 , v2 = −3v x + vx
= −2vx = −100i1
i1 = 0
→ z21 =
v2
= −100
i1
14-6
(2)port − 1 o.c., i1 = 0
v2 = −2vx , vx = 50i2
→ z 22 =
v2
= −100 ≠ z11
i2
v1 = vx = 50i2
→ z12 =
v1
= 50 ≠ z12
i2
電路學講義第14章
6. Reciprocal circuit
reciprocal network :
reciprocal network :
V 1oc V 2oc
=
→ z12 = z21
I2
I1
I 1sc I 2 sc
=
→ y12 = y21
V2
V1
電路學講義第14章
14-7
7. Any linear network containing no controlled sources is a reciprocal
network.
∵node equation and mesh equation have symmetrical forms
8. Ex. 14.3 T-network
⎡V ⎤ ⎡ z
sources : I 1 , I 2 ⎢ 1 ⎥ = ⎢ 11
⎣V 2 ⎦ ⎣ z 21
z11 =
V1
I1
, z12 =
I 2 =0
V1
I2
z12 ⎤ ⎡ I 1 ⎤
z 22 ⎥⎦ ⎢⎣ I 2 ⎥⎦
, z 21 =
I 1 =0
V2
I1
, z22 =
I 2 =0
V2
I2
I 1 =0
(1) I 2 = 0, z11 =
V1
V
= z11 − z12 + z12 , z21 = 2 = z12
I1
I1
(2) I 1 = 0, z22 =
V2
V
= z11 − z12 + z12 , z12 = 1
I2
I2
Z a = z11 − z12 , Z c = z12
Z b = z 22 − z12
14-8
電路學講義第14章
14.2 Admittance, hybrid, and transmission parameters
Basics
1. Not all two-ports posses meaningful or measurable z-parameters.
→other parameters
2. Admittance parameter
[ y ] ≡ ⎡⎢
y11
⎣ y21
y11 =
I1
V1 V
y12 ⎤
: s.c. admittance parameter matrix
⎥
y22 ⎦
: input admittance at port 1 with port 2 s.c.
2 =0
I2
y
=
: input admittance at port 2 with port 1 s.c.
22
sources : V 1 , V 2 , responses : I 1 , I 2
V 2 V =0
1
I
y
V
y
V
=
+
⎧ 1
11 1
12 2
I1
⎨
y
=
: reverse transfer admittance with port 1 s.c.
I
y
V
y
V
=
+
12
21 1
22 2
⎩ 2
V 2 V =0
1
⎡V 1 ⎤
⎡I1 ⎤
→ ⎢ ⎥ = [ y ]⎢ ⎥
I2
V
I
y
=
: forward transfer admittance with port 2 s.c.
⎣ 2⎦
⎣ 2⎦
21
V 1 V =0
2
電路學講義第14章
14-9
3. Hybrid parameter
[h] ≡ ⎡⎢
h12 ⎤
: hybrid parameter matrix
⎥
h22 ⎦
h11
⎣h21
h11 =
h22 =
sources : I 1 , V 2 , responses : V 1 , I 2
⎧V 1 = h11 I 1 + h12V 2
⎨
⎩ I 2 = h21 I 1 + h22V 2
h12 =
⎡ I1 ⎤
⎡V 1 ⎤
→ ⎢ ⎥ = [h]⎢ ⎥
⎣I 2 ⎦
⎣V 2 ⎦
h21 =
V1
I1 V
I2
V2
V1
V2
I2
I1
=
1
: input admittance at port 1 with port 2 s.c.
y11
=
1
: input impedance at port 2 with port 1 o.c.
z 22
2 =0
I 1 =0
: reverse voltage ratio with port 1 o.c.
I 1 =0
: forward current ratio with port 2 s.c.
V 2 =0
14-10
電路學講義第14章
4. Transmission parameter
⎡A B⎤
[T ] ≡ ⎢
⎥ : transmission parameter matrix
C
D
⎦
⎣
A=
V1
V2
: reverse voltage ratio with port 2 o.c.
I 2 =0
V1
B
=
: reverse transfer impedance with port 2 s.c.
sources : V 2 , I 2 , responses : V 1 , I 1
− I 2 V =0
2
⎧V 1 = AV 2 − B I 2
I1
⎨
C
: reverse transfer admittance with port 2 o.c.
=
I
C
V
D
I
=
−
2
2
⎩ 1
V 2 I =0
2
⎡V 1 ⎤ ⎡ A B ⎤ ⎡ V 2 ⎤
→⎢ ⎥=⎢
I1
⎥ ⎢− I ⎥
I
C
D
D
: forward current ratio with port 2 s.c.
=
⎣ 1⎦ ⎣
⎦⎣ 2 ⎦
− I 2 V =0
2
⎡V2 ⎤
= [T ]⎢
⎥
⎣− I 2 ⎦
14-11
電路學講義第14章
Discussion
1. Equivalent circuit expressed in y-parameters
sources : V 1 ,V 2
⎧ I 1 = y11V 1 + y12V 2
⎨
⎩ I 2 = y21V 1 + y22V 2
⎡ I 1 ⎤ ⎡ y11 y12 ⎤ ⎡V 1 ⎤
⎥ ⎢V ⎥
⎢I ⎥ = ⎢ y
y
22 ⎦ ⎣ 2 ⎦
⎣ 2 ⎦ ⎣ 21
I
y11 = 1 = y11 + y12 − y12
V1
(1)V 2 = 0,
1
I
y21 = 2 = −
= y12
V1
− 1 y12
I2
= y22 + y12 − y12
V2
(2)V 1 = 0,
1
I
y12 = 1 = −
= y12
V2
− 1 y12
y22 =
14-12
電路學講義第14章
2. Ex. 14.5 find y-parameters of an “active” network
sources : V 1 ,V 2
⎡ I 1 ⎤ ⎡ y11
⎢I ⎥ = ⎢ y
⎣ 2 ⎦ ⎣ 21
(1)port − 2 s.c.,V 2 = 0
(2)port − 1 s.c.,V 1 = 0
s
I1
s
I
=
V
→
y
=
=
1
11
y12 ⎤ ⎡V 1 ⎤ 1
V 1 40
40
y22 ⎥⎦ ⎢⎣V 2 ⎥⎦
I 2 = 3I 1 − I 1 = 2 I 1 = 2
→ y21 =
s
V1
40
I2
s
=
V 1 20
14-14
I 1 = −V 2
s
I
s
→ y12 = 1 = −
≠ y21
40
40
V2
V2
V
= 2I 1 + 2
10
10
2−s
sV
V
V2
=− 2 + 2 =
20 10
20
2−s
I
→ y22 = 2 =
20
V2
I 2 = 3I 1 − I 1 +
電路學講義第14章
3. H-parameters are applied to transistor because they are measured
physical quantities.
ib
ic
sources : i1 , v2 , responses : v1 , i2
vce
vbe
⎡v1 ⎤ ⎡ h11 h12 ⎤ ⎡ i1 ⎤
⎧v1 = h11i1 + h12 v2
→
=⎢
⎨
⎢
⎥
⎥⎢ ⎥
⎩i2 = h21i1 + h22v2
⎣i2 ⎦ ⎣h21 h22 ⎦ ⎣v2 ⎦
⎡vbe ⎤ ⎡ hie hre ⎤ ⎡ ib ⎤
CE → ⎢ ⎥ = ⎢
⎥⎢ ⎥
h
h
i
⎣ c ⎦ ⎣ fe oe ⎦ ⎣vce ⎦
4. Equivalent circuit expressed in h-parameters
14-14
電路學講義第14章
5. Ex. 14.6 find h-parameters of an “active” network
sources : I 1 ,V 2
(1)port − 2 s.c.,V 2 = 0
responses : V 1 , I 2
s
40
40
V
→ h11 = 1 =
I1
s
⎧V 1 = h11 I 1 + h12V 2
⎨
⎩ I 2 = h21 I 1 + h22V 2
⎡V 1 ⎤ ⎡ h11 h12 ⎤ ⎡ I 1 ⎤
→⎢ ⎥=⎢
⎥⎢ ⎥
⎣ I 2 ⎦ ⎣h21 h22 ⎦ ⎣V 2 ⎦
I1 = V 1
I 2 = 3I 1 − I 1 = 2 I 1
→ h21 =
(2)port − 1 o.c., I 1 = 0
V
V 1 = V 2 → h12 = 1 = 1
V2
V 2 = 10 I 2
→ h22 =
I2
= 0.1
V2
I2
=2
I1
14-15
電路學講義第14章
6. Ex. 14.7 find ABCD-parameters of an “active” network
sources : V 2 , I 2
responses : V 1 , I 1
⎧V 1 = AV 2 − B I 2
⎨
⎩ I 1 = CV 2 − D I 2
⎡V ⎤ ⎡ A B ⎤ ⎡ V 2 ⎤
→ ⎢ 1⎥ = ⎢
⎥⎢
⎥
⎣ I 1 ⎦ ⎣C D ⎦ ⎣− I 2 ⎦
(1)port − 2 o.c., I 2 = 0
V
V
I 1 = 3I 1 + 2 , I 1 = − 2
10
20
1
I
→C = 1 = −
20
V2
40
2
I 1 = (1 − )V 2
s
s
2
V
→ A = 1 = 1−
V2
s
V1 =V 2 +
14-16
(2)port − 2 s.c.,V 2 = 0
I 1 = 3I 1 − I 2 ,2 I 1 = I 2
I
1
→D= 1 =−
−I2
2
40
20
V1 =
I1 =
I2
s
s
V
20
→B= 1 =−
s
−I2
電路學講義第14章
7. All the 2-port parameters are related as given in Table 14.2.
8. Conversion between z-parameters and y-parameters
z ⎤⎡ I1 ⎤
⎡V ⎤ ⎡ z
⎡V ⎤
⎡I1 ⎤ ⎡I1 ⎤
−1 ⎡V ⎤
z − parameter ⎢ 1 ⎥ = ⎢ 11 12 ⎥ ⎢ ⎥ = [z ]⎢ ⎥ → ⎢ ⎥ = [z ] ⎢ 1 ⎥ = [ y ]⎢ 1 ⎥
⎣I 2 ⎦ ⎣I 2 ⎦
⎣V 2 ⎦ ⎣ z 21 z 22 ⎦ ⎣ I 2 ⎦
⎣V 2 ⎦
⎣V 2 ⎦
z12 ⎤
⎡ z22
−
⎢ ∆
∆z ⎥
−1
z
⎥, ∆ z = [z ] = z11 z22 − z12 z21
⇒ [ y ] = [z ] = ⎢
z
z
11 ⎥
⎢− 21
⎢⎣ ∆ z
∆ z ⎥⎦
⎡ y22
⎢ ∆
−1
[z ] = [ y ] = ⎢ yy
⎢− 21
⎢ ∆y
⎣
y12 ⎤
∆y ⎥
⎥, ∆ = [ y ] = y11 y22 − y12 y21
y11 ⎥ y
∆ y ⎥⎦
−
14-17
電路學講義第14章
9. Derivation of h-parameters from z-parameters
⎡V 1 ⎤ ⎡ z11
z − parameter ⎢ ⎥ = ⎢
⎣V 2 ⎦ ⎣ z 21
⎧ V 1 = z11 I 1 + z12 I 2 ...(1)
⎨
⎩V 2 = z 21 I 1 + z22 I 2 ...(2)
( 3)
(1) →V 1 = z11 I 1 + z12 (−
⎡ ∆z
⎢ z
[h] = ⎢ z22
⎢− 21
⎢⎣ z22
z12 ⎤ ⎡ I 1 ⎤
⎡V 1 ⎤ ⎡ h11 h12 ⎤ ⎡ I 1 ⎤
, h − parameter ⎢ ⎥ = ⎢
⎥⎢ ⎥
z 22 ⎥⎦ ⎢⎣ I 2 ⎥⎦
⎣ I 2 ⎦ ⎣h21 h22 ⎦ ⎣V 2 ⎦
(2) → I 2 = −
z21
1
I1 +
V 2 ...(3)
z22
z22
∆
z21
z
1
I1 +
V 2 ) = z I 1 + 12 V 2 , ∆ z = z11 z22 − z12 z21
z22
z22
z 22
z 22
z12 ⎤
z22 ⎥
⎥
1 ⎥
z22 ⎥⎦
14-18
電路學講義第14章
10. Derivation of z-parameters from T-parameters
⎡V 1 ⎤ ⎡ z11 z12 ⎤ ⎡ I 1 ⎤
⎡V ⎤ ⎡ A B ⎤ ⎡ V 2 ⎤
,
parameter
z
T − parameter ⎢ 1 ⎥ = ⎢
=⎢
−
⎥⎢ ⎥
⎥
⎢
⎥
⎢
⎥
⎣ I 1 ⎦ ⎣C D ⎦ ⎣− I 2 ⎦
⎣V 2 ⎦ ⎣ z21 z22 ⎦ ⎣ I 2 ⎦
⎧V 1 = AV 2 − B I 2 ...(1)
1
D
(2) → V 2 = I 1 + I 2 ...(3)
⎨
C
C
⎩ I 1 = CV 2 − D I 2 ...(2)
( 3)
∆
1
A
D
(1) →V 1 = A( I 1 + I 2 ) − B I 2 = I 1 + T I 2 , ∆T = AD − BC
C
C
C
C
⎡ A ∆T ⎤
⎢
⎥
[z ] = ⎢ C1 CD ⎥
⎢
⎥
⎣C C ⎦
11. For a reciprocal 2-port network,
z12 = z21 , y12 = y21 , h12 = −h21 , ∆T = AD − BC = 1
14-19
電路學講義第14章
14.3 Circuit analysis with two-ports
Basics
1. Terminated two-ports using z-parameters
⎧ V 1 = z11 I 1 + z12 I 2 ...(1) ⎧V s = Z S I 1 + V 1...(3)
,⎨
⎨
⎩V 2 = z 21 I 1 + z 22 I 2 ...(2) ⎩V 2 = − Z L I 2 ........(4)
( 4)
(2) →− Z L I 2 = z21 I 1 + z22 I 2 , z 21 I 1 = −( Z L + z 22 ) I 2
⇒ current transfer function H i ≡
(4)
(1) →V 1 = z11 I 1 −
(5)
(2) →V 2 = z21 (
I2
− z21
=
I 1 Z L + z22
z
z12
1
V 2 , I 1 = V 1 + 12 V 2 ...(5)
z11Z L
ZL
z11
z
z
1
V 1 + 12 V 2 ) − 22 V 2
ZL
z11
z11Z L
z 21
V
z 21Z L
z21Z L
z11
⇒ voltage transfer function H v ≡ 2 =
=
=
V 1 1 − z12 z21 + z22 z11Z L − z12 z 21 + z11 z22 ∆ z + z11Z L
z11Z L Z L
電路學講義第14章
14-20
⎧ V 1 = z11 I 1 + z12 I 2 ...(1) ⎧V s = Z S I 1 + V 1...(3)
,⎨
⎨
...(
2
)
V
z
I
z
I
=
+
21 1
22 2
⎩ 2
⎩V 2 = − Z L I 2 ........(4)
Hi ≡
I2
− z 21
...(6)
=
I 1 Z L + z22
(6)
(1) →V 1 = z11 I 1 − z12
− z 21
I1
Z L + z22
⇒ equivalent input impedance Z i ≡
V 1 z11Z L + z11 z22 − z12 z 21 ∆ z + z11Z L
=
=
I1
Z L + z 22
Z L + z 22
(7)
V s = 0, (3) → V 1 = − Z S I 1...(7), (1) →− Z S I 1 = z11 I 1 + z12 I 2 → I 1 = −
(8)
(2) →V 2 = z21 (−
z12
I 2 ...(8)
Z S + z11
z12
I 2 ) + z 22 I 2
Z S + z11
⇒ equivalent output impedance Z o ≡
V2
I2
=
V s =0
z 22 Z S + z11 z22 − z12 z 21 ∆ z + z 22 Z S
=
Z S + z11
Z S + z11
14-21
電路學講義第14章
2. Cascade connection using T-parameters
⎡ V 2a ⎤
⎡V 1b ⎤
⎡V 1 ⎤
⎢ I ⎥ = [T ]a ⎢ − I ⎥ = [T ]a ⎢ I ⎥
⎣ 1⎦
⎣ 2a ⎦
⎣ 1b ⎦
⎡V2 ⎤
= [T ]a [T ]b ⎢
⎥
−
I
⎣ 2⎦
⇒ [T ]cas = [T ]a [T ]b
3. Series connection using z-parameters ⎡V 1 ⎤ ⎡V 1a ⎤ ⎡V 1b ⎤
⎢V ⎥ = ⎢V ⎥ + ⎢V ⎥
⎣ 2 ⎦ ⎣ 2 a ⎦ ⎣ 2b ⎦
⎡I1 ⎤
⎡I1 ⎤
= [z ]a ⎢ ⎥ + [z ]b ⎢ ⎥
⎣I 2 ⎦
⎣I 2 ⎦
⎡I1 ⎤
= ([z ]a + [z ]b ) ⎢ ⎥
⎣I 2 ⎦
⇒ [z ]ser = [z ]a + [z ]b
14-22
電路學講義第14章
4. Parallel connection using y-parameters
⎡ I 1 ⎤ ⎡ I 1a ⎤ ⎡ I 1b ⎤
⎢I ⎥ = ⎢I ⎥ + ⎢I ⎥
⎣ 2 ⎦ ⎣ 2 a ⎦ ⎣ 2b ⎦
⎡V 1 ⎤
⎡V 1 ⎤
= [ y ]a ⎢ ⎥ + [ y ]b ⎢ ⎥
⎣V 2 ⎦
⎣V 2 ⎦
⎡V ⎤
= ([ y ]a + [ y ]b ) ⎢ 1 ⎥
⎣V 2 ⎦
⇒ [ y ]par = [ y ]a + [ y ]b
14-23
電路學講義第14章
Discussion
1. Relations of terminated two-ports in terms of z- y- h- and Tparameters 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
20 ⎤
⎡ 2
−
−
1
⎡A B⎤ ⎢ s
s ⎥
=
From ex.14.7 ⎢
⎥ ⎢
1⎥
⎣C D ⎦ ⎢− 1
− ⎥
2⎦
⎣ 20
H i Table14.2 − 1 CZ L + D
I2
I2
=
=
=
H ( s) =
V 1 Zi I 1 Zi
CZ L + D AZ L + B
−1
−1
− 0.4s
=
= 2
AZ L + B (1 − 2 )2.5s − 20 s − 2s − 8
s
s
− 0.4s
=
( s − 4)(s + 2)
=
14-24
電路學講義第14章
3. Ex. 14.10 find RL to give Ai=Iout/Is=-25 from transistor h-parameters
Ai =
⎡1000
10−3 ⎤
[h] = ⎢
−3 ⎥
×
50
0
.
1
10
⎣
⎦
∆ h = 0.05
Z i ( s) =
V 1 ∆ h + h11YL
=
I1
h22 + YL
H i ( s) =
I2
h Y
= 21 L
I 1 h22 + YL
Rs
I out − I 2 I 1
=
= −Hi
Is
I1 I s
Rs + Z i
Z i ( s) =
Ai =
0.05RL + 1
50
, H i ( s) =
0.1RL + 1
0.1RL + 1
− 100
= −25 → RL = 4kΩ
3 + 0.25RL
14-25
電路學講義第14章
4. Ex. 14.11 find Ai of two amplifiers of ex.14.10 in cascade
⎡1000
10 −3 ⎤
[h] = ⎢
, [T ]a = [T ]b
−3 ⎥
×
50
0
.
1
10
⎣
⎦
− h11 ⎤
−3
h21 ⎥ ⎡ − 10
⎥=
− 1 ⎥ ⎢⎣− 2 ×10−6
h21 ⎥⎦
− 20 ⎤
⎥
− 0.02⎦
⎤
⎥
440 ×10 −6 ⎦
−1
ARL + B
= 961, H i ( s) =
= −1645
Z i ( s) =
CRL + D
CRL + D
→ [T ]cas = [T ]a [T ]b
Ai =
⎡ 41×10−6
=⎢
−6
0
.
042
×
10
⎣
⎡ − ∆h
⎢ h
= ⎢ 21
⎢ − h22
⎢⎣ h21
0.42
Rs
I out − I 2 I 1
=
= −Hi
= 1100
14-26
Is
I1 I s
Rs + Z i
電路學講義第14章
5. Bridged-T connection
⎡ y11
[y] = ⎢
⎣ y21
y12 ⎤
I1
I2
y
y
=
=
,
,
11
22
y22 ⎥⎦
V 1 V =0
V2
2
, y12 =
V 1 =0
I1
V2
, y21 =
V 1 =0
I2
V1 V
2 =0
y11 = y22 = YF , y12 = y21 = −YF
⎡ Y + y11b
⇒ [y] = ⎢ F
⎣− YF + y21b
− YF + y12b ⎤
YF + y22b ⎥⎦
14-27
電路學講義第14章
6. Ex.14.12 find Hv=V2/V1 of a high frequency transistor
1
⎡1
⎤
⎡
+
0
sC
⎢R
⎥
⎢
⎡ sC − sC ⎤
1
Ri
i
⎢
⎥
⎢
[
]
[
]
=
⇒
=
ZF =
, [ y ]a = ⎢
,
y
y
⎥ b
1⎥
sC
⎢g
⎢ − sCg
⎣− sC sC ⎦
m
m
⎢⎣
⎢⎣
Ro ⎥⎦
sC − g m
s − gm / C
− y21
Hv =
=
=
y22 + YL sC + 1 / Ro + 1 / RL s + G / C
⎤
− sC ⎥
⎥
1⎥
sC +
Ro ⎥⎦
G = 1 / Ro + 1 / RL = 1 / Ro // RL
14-28
電路學講義第14章