The z and y parameters of a two-port network do not always exist. ... In fact an
ideal transformer for example does not have z-parameters as it is impossible.
EE 205 Dr. A. Zidouri
Electric Circuits II
Two-Port Circuits
Hybrid and Transmission Parameters Lecture #43
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EE 205 Dr. A. Zidouri The material to be covered in this lecture is as follows: o o o o
The Two-Port Hybrid parameters The Two-Port Transmission parameters Relationships Among the Two-Port Parameters Reciprocal Two-Port Circuits
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EE 205 Dr. A. Zidouri After finishing this lecture you should be able to: ¾ ¾ ¾ ¾
Determine the Two-Port Hybrid Parameters Determine the Two-Port Transmission Parameters Derive all the Other Sets from a Known Set of Parameters Recognize Reciprocal and Symmetric Two-Port Circuits
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EE 205 Dr. A. Zidouri
Hybrid parameters The z and y parameters of a two-port network do not always exist. There is a need for developing another set of parameters.
V1 = h11 I1 + h12V2
Hybrid Parameters (h-parameters):
(43-1)
I 2 = h21 I1 + h22V2
or in matrix form:
⎡V1 ⎤ ⎡ h11 ⎢ I ⎥ = ⎢h ⎣ 2 ⎦ ⎣ 21
h12 ⎤ ⎡ I1 ⎤ ⎡ I1 ⎤ = h [ ] ⎢V ⎥ h22 ⎥⎦ ⎢⎣V2 ⎥⎦ ⎣ 2⎦
(43-2)
The h-parameters are very useful for describing electronic devices such as transistors. It is much easier to measure experimentally the h-parameters of such devices than to measure their z or y parameters. In fact an ideal transformer for example does not have z-parameters as it is impossible to express the voltages in terms of the currents or vice versa. See Fig. 43-1 a
I1
I2
c
1:n v1
N1
N2
v2
1 V1 = V2 n I1 = − nI 2
d b Fig. 43-1 An Ideal Transformer has no z-parameters
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EE 205 Dr. A. Zidouri The values of the parameters can be evaluated by setting I1=0 (input port open-circuited) or V2=0 (output port short-circuited). Thus,
h11 = h21 =
V1 V , h12 = 1 I1 V =0 V2 I =0 2 1
(43-3)
I2 I , h22 = 2 I1 V =0 V2 I =0 2 1
The parameters h11, h12, h21, h22 represent an impedance, a voltage gain, a current gain, and an admittance respectively. This is why the h-parameters are called the hybrid parameters: ¾ h11= short-circuit input impedance ¾ h12= Open-circuit reverse voltage gain ¾ h21= short-circuit forward current gain ¾ h22= Open-circuit output admittance The following example illustrates the determination of the h-parameters for a resistive circuit. Example 43-1 Find the hybrid parameters for the two-port network of Fig. 43-2 Solution: To find h11 and h21, we short circuit the output port and connect a current source I1 to the input port as shown in Fig. 43-3a Fig. 43-2 Circuit for Example 43-1
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EE 205 Dr. A. Zidouri I2
I1
V1
V2=0
V1 = I1 ( 2 + 3 6 ) = 4 I1 hence h11 =
V1 = 4Ω and by I1 I 2 6 2 current division, − I 2 = I1 = I1 hence h21 = 2 = − I1 3 6+3 3
Fig. 43-3a Circuit for Example 43-1
To find h12 and h22, we open circuit the input port and connect a voltage source I2 to the input port as shown in Fig. 43-3b. By voltage division
V 2 6 2 V2 = V2 hence h12 = 1 = and, V2 3 6+3 3 I 1 V2 = I 2 ( 6 + 3) = 9 I 2 hence h22 = 2 = S V2 9
V1 =
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EE 205 Dr. A. Zidouri
Hybrid Parameters (g-parameters):
I1 = g11V1 + g12 I 2 V2 = g 21V1 + g 22 I 2
(43-4)
or in matrix form:
⎡ I1 ⎤ ⎡ g11 ⎢V ⎥ = ⎢ g ⎣ 2 ⎦ ⎣ 21
g12 ⎤ ⎡V1 ⎤ ⎡V1 ⎤ g = [ ] ⎢I ⎥ g 22 ⎥⎦ ⎢⎣ I 2 ⎥⎦ ⎣ 2⎦
(43-5)
The values of the g-parameters can be evaluated by setting V1=0 (input port short-circuited) or I2=0 (output port open-circuited). Thus,
g11 = g 21 =
I1 I , g12 = 1 V1 I =0 I 2 V =0 2 1
(43-6)
V2 V , g 22 = 2 V1 I =0 I 2 V =0 2 1
The g-parameters or inverse hybrid parameters are: ¾ g11= Open-circuit input admittance ¾ g12= Short-circuit reverse current gain ¾ g21= Open-circuit forward voltage gain ¾ g22= Short-circuit output impedance The following example illustrates the determination of the g-parameters. -7-
EE 205 Dr. A. Zidouri Example 43-2 Find the g parameters as a function of s for the circuit of Fig. 43-4 Solution: In the s domain, we have 1H 1H
1F
⇒ sL = s , 1F ⇒
1 1 = sC s
To get g11 and g21, we open circuit the output port and connect a voltage source V1 to the input port as shown in Fig. 43-4a
I 1 V1 or g11 = 1 = V1 s + 1 s +1 V 1 1 By voltage division, V2 = V1 or g 21 = 2 = V1 s + 1 s +1 I1 =
Fig. 43-4 Circuit for Example 43-2
I1
I2=0 1/s
s V1=0
1
V2
I2
Fig. 43-4b Circuit for Example 43-2
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EE 205 Dr. A. Zidouri To get g12 and g22, we short circuit the input port and connect a current source I2 to the output port as shown in Fig. 43-4b.
I 1 1 I 2 or g12 = 1 = − I2 s +1 s +1 V2 1 s s2 + s + 1 ⎛1 ⎞ = + = Also, V2 = I 2 ⎜ + s 1⎟ or g 22 = I 2 s s + 1 s ( s + 1) ⎝s ⎠ 1 ⎤ ⎡ 1 − ⎢ s +1 s +1 ⎥ ⎥ Thus, [ g ] = ⎢ 2 1 s s 1 + + ⎢ ⎥ ⎢ s + 1 s ( s + 1) ⎥ ⎣ ⎦ By current division,
I1 = −
Transmission parameters As we have seen in Equs. (42-5) and (42-6) that we reproduce here for convenience: Transmission Parameters (a-parameters):
Inverse Transmission Parameters (b-parameters):
V1 = a11V2 − a12 I 2
I1 = a21V2 − a22 I 2 V2 = b11V1 − b12 I1 I 2 = b21V1 − b22 I1
(43-7)
(43-8)
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EE 205 Dr. A. Zidouri where we express the input variables in terms of the output variables or the output variables in terms of the input variables. In matrix form:
Where
⎡V1 ⎤ ⎡ a11 a12 ⎤ ⎡ V2 ⎤ ⎡ V2 ⎤ = = T (43-9) ⎢ I ⎥ ⎢a a ⎥ ⎢− I ⎥ [ ] ⎢− I ⎥ 22 ⎦ ⎣ 2⎦ ⎣ 1 ⎦ ⎣ 21 ⎣ 2⎦ ⎡V2 ⎤ ⎡ b11 b12 ⎤ ⎡ V1 ⎤ ⎡ V1 ⎤ = = t (43-10) [ ] ⎢− I ⎥ ⎢ I ⎥ ⎢b b ⎥ ⎢ − I ⎥ ⎣ 2 ⎦ ⎣ 21 22 ⎦ ⎣ 1 ⎦ ⎣ 1⎦ V V V V a11 = 1 b11 = 2 , a12 = − 1 , b12 = − 2 V2 I =0 I 2 V =0 V1 I =0 I1 V =0 2 2 1 1 (43-11)
a21 =
I1 I , a22 = − 1 V2 I =0 I 2 V =0 2 2
b21 =
(43-12)
I2 I , b22 = − 2 V1 I =0 I1 V =0 1 1
The a-parameters represent: ¾ a11= Open-circuit reverse voltage ratio ¾ a12= Negative short-circuit transfer impedance ¾ a21= Open circuit transfer admittance ¾ a22= Negative short-circuit reverse current ratio
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EE 205 Dr. A. Zidouri The b-parameters or inverse hybrid parameters are: ¾ b11= Open-circuit voltage gain ¾ b12= Negative short-circuit transfer impedance ¾ b21= Open circuit transfer admittance ¾ b22= Negative short-circuit current gain These parameters are useful in the analysis of transmission lines (such as cable and fiber) because they express sending-end variables in terms of receiving-end variables. They are also called ABCD parameters (for a) and abcd parameters (for b). They are used in the design of telephone systems, microwave networks, and radars.
Relationships among the Two-Port Parameters If we know one set of parameters we can derive all the other sets from the known set Table 43-1 shows the Parameter Conversion Table Let’s derive the relationship between z and y and between z and a as an example. To derive the relationship between z and y we first solve Eqs. (42-2)
I1 = y11V1 + y12V2
I 2 = y21V1 + y22V2
for V1 and V2.
Then compare the coefficients of I1 and I2 in the resulting expressions to the coefficients of I1 and I2 in Eqs. (42-1)
V1 = z11 I1 + z12 I 2
V2 = z21 I1 + z22 I 2
. From Eqs. (42-2) we have:
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EE 205 Dr. A. Zidouri
I1 y12 y22 y12 I 2 y22 = I − I2 V1 = 1 ∆y y11 y12 ∆y y21 y22
y11 I1 y21 y11 y21 I 2 = I − I2 V2 = 1 ∆y ∆y y11 y12 y21 y22
(43-13)
(43-14)
comparing (43-13) and (43-14) with (42-1) shows:
z11 =
y22 ∆y
(43-15)
z12 = −
y12 (43-16) ∆y
z21 = −
y21 (43-17) ∆y
z22 =
y11 ∆y
(43-18)
To find z-parameters as a function of the a-parameters, we rearrange Eqs. (43-7) in the form of Eqs. (42-1) and then compare the coefficients. From the second equation in (42-5):
V2 =
1 a I1 + 22 I 2 a21 a21
(43-19)
therefore, substituting Eq. (43-19) into the first equation of Eqs. (42-5) yields
V1 =
⎛a a ⎞ a11 I1 + ⎜ 11 22 − a12 ⎟ I 2 a21 ⎝ a21 ⎠
a11 (43-21) a21 1 From Eq. (43-19) z21 = (43-23) a21
From Eq. (43-20)
z11 =
∆a a21 a z22 = 22 a21 z12 =
(43-20)
(43-22) (43-24)
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EE 205 Dr. A. Zidouri Example 43-3 Two sets of measurements are made on a two-port resistive circuit. The first set is made with port 2 open, and the second set is made with port 2 short-circuited. The results are as follows: Port 2 Open Port 2 Short Circuited
V1 = 10mV
V1 = 24mV
I1 = 10 µ A
I1 = 20 µ A
V2 = −40V
I 2 = 1mA
Find the h parameters of the circuit. Solution:
V1 24 ×10−3 I2 10−3 From the short circuit test: h11 = = = 1.2k Ω and h21 = = = 50 I1 V =0 20 × 10−6 I1 V =0 20 × 10−6 2 2 h12 and h22 cannot be obtained directly from the open circuit test. However we can obtain them from aparameters either from a conversion table or through a similar procedure as seen in previous slides in this lecture. We get
h12 =
∆a a , h22 = 21 The a-parameters are a22 a22
V1 10 × 10−3 a11 = = = −0.25 × 10−3 V2 I =0 −40 2
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EE 205 Dr. A. Zidouri
10 × 10−6 I1 = = −0.25 × 10−6 S a21 = −40 V2 I =0 2
V1 24 ×10−3 =− = −24Ω a12 = − −3 I 2 V =0 10 2 20 ×10−6 I1 =− = −20 × 10−3 which give ∆a = a11a22 − a12 a21 = 5 × 10−6 − 6 × 10−6 = −10−6 a22 = − −3 10 I 2 V =0 2 ∆a −10−6 a21 −0.25 × 10−6 −5 = = 5 × 10 and h22 = = = 12.5µ S Therefore h12 = −3 −3 a22 −20 × 10 a22 −20 × 10
Reciprocal Two-Port Circuits If a two-port circuit is reciprocal the following relationships exist among the port parameters: z12 = z21 (43-23) y12 = y21 (43-24)
a11a22 − a12 a21 = ∆a = 1 h12 = − h21
(43-25)
b11b22 − b12b21 = ∆b = 1 g12 = − g 21
(43-26)
(43-27) (43-28) A two-port circuit is reciprocal if the interchange of an ideal voltage source at one port with an ideal ammeter at the other port produces the same ammeter reading.
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EE 205 Dr. A. Zidouri Example 43-4
b
a
c
Ammeter 15V
d d Fig. 42-5 The Circuit in Fig. 42-4 with the Voltage and Ammeter Interchanged
Consider the circuit of Fig. 43-5. When a voltage source of 15 V is applied to port ad, it produces a current of 1.75 A in the ammeter at port cd. The ammeter current can be determined easily from Vbd
Vbd Vbd − 15 Vbd + + =0 60 30 20
(43-29) and Vbd
= 5V therefore I =
5 15 + = 1.75 A 20 10
(43-30)
If the voltage source and ammeter are interchanged the ammeter will still read 1.75 A. We verify this by solving the circuit of Fig. 43-6:
Vbd Vbd Vbd − 15 + + =0 60 30 20
(43-31) therefore Vbd
= 7.5V the current Iad is
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EE 205 Dr. A. Zidouri
I ad =
7.5 15 + = 1.75 A 30 10
(43-32)
A reciprocal circuit is also symmetric, if its port can be interchanged without disturbing the values of the terminal currents and voltages. The following additional relationships will hold: z11 = z22 (43-33) y11 = y22 (43-34)
h11h22 − h12 h21 = ∆h = 1
(43-35) g11 g 22 − g12 g 21 = ∆g = 1 (43-36) For a symmetric reciprocal circuit only two calculations or measurements are necessary to determine all the two-port parameters. Self Test 43: Two sets of measurements are made on a two-port network that is symmetric and reciprocal. The first set is made with port 2 open, and the second set is made with port 2 short-circuited as follows: Port 2 Open Port 2 Short Circuited
V1 = 95V
V1 = 11.52V
I1 = 5 A
I 2 = −2.72 A
Calculate the z parameters of the two-port network. Answer:
z11 = z22 = 19Ω
z12 = z21 = 17Ω
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