A two-port network is an electrical network with two separate ports

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1 5.1 Introduction A two-port network is an electrical network with two separate ports for input and output. Fig(a) Single Port Network Fig(b) Two Port Network There are several reasons why we should study two-ports and the parameters that describe them. Most of the circuits which we come across have two ports. Usually an input signal is connected in one port and an output signal is obtained from the other port. The parameters of a two-port network completely describes its behaviour in terms of the voltage and current at each port. Thus knowing the parameters of a two port network permits us to describe its operation when it is connected into a larger network. Two-port networks are also important in modeling electronic devices and system components. In electronics, two-port networks are employed to model transistors and op-amps., Electrical circuits are modeled by two-ports are transformers and transmission lines. Four popular types of two-ports parameters are examined: Impedance Admittance Hybrid Transmission. In the later part of discussion, usefulness of each set of parameters, demonstration of how they are related to each other and in the end how two-port networks can be interconnected (parallel, series, series-parallel and cascade).

2 Impedence parameters: V 1 z 11 I 1 z 12 I 2 V 2 z 21 I 1 z 22 I 2 Impedance Parameters, z z 11 = Open-circuit input impedance z 12 = Open-circuit transfer impedance from port 1 to port 2 z 21 = Open-circuit transfer impedance from port 2 to port 1 z 22 = Open-circuit output impedance

3 Impedance Parameters, z Impedance Parameters, z

4 Example

5 5.3 Admittance Parameters, y I 1 Y 11 V 1 Y 12 V 2 I 2 Y 21 V 1 Y 22 V 2 Admittance Parameters Y y 11 = Short-circuit input admittance y 12 = Short-circuit transfer admittance from port 1 to port 2

6 y 21 = Short-circuit transfer admittance from port 2 to port 1 y 22 = Short-circuit output admittance Fig. Obtain the y parameters for the Π network shown in Fig. Determine the y parameters for the T network shown in Fig.

7 5.4 Hybrid Parameters, h

8 V 1 h 11 I 1 h 12 I 2 I 2 h 21 I 1 h 22 V 2 Hybrid Parameters, h h 11 = Short-circuit input impedance h 12 = Open-circuit reverse voltage gain h 21 = Short-circuit forward current gain h 22 = Open-circuit output admittance Inverse Hybrid Parameters, g I 1 g 11 V 1 g 12 I 2 V 2 g 21 V 1 g 22 I 2

9 Inverse Hybrid Parameters, g g 11 = Open-circuit input impedance g 12 = Short-circuit reverse voltage gain g 21 = Open-circuit forward current gain g 22 = Short-circuit output admittance H-Parameters applied to Common Emitter Transistor Configuration Determine the Thevenin equivalent at the output port of the circuit in Fig. Solution: To find Z TH and V TH

10

11 5.5 Transmission Parameters,T (ABCD Parameters) V 1 AV 2 BI 2 I 1 CV 2 DI 2 Transmission Parameters, T A = Open-circuit voltage ratio B = Negative short-circuit transfer impedance C = Open-circuit transfer admittance D = Negative short-circuit current ratio Inverse Transmission Parameters, t

12 a = Open-circuit voltage gain b = Negative short-circuit transfer impedance c = Open-circuit transfer admittance d = Negative short-circuit current gain AD BC=1, ad bc=1 Example 5.7 Find the transmission parameters for the two-port network in Fig.

13 Example 5.8

14 Condition for Reciprocity and Symmetry A network is said to be reciprocal, if the ratio of the response transform to the excitation transform is invariant to an interchange of the positions of the excitation and response in the network A two-port network is said to be symmetrical if the ports of the two-port network can be interchanged without changing the port voltages and currents Relationship Between Two-Port Network Parameters: Any two-port network parameter can be expressed in terms of other. For example, if the Z- parameter for given two port network is computed by solving the network, the remaining Y, h and ABCD can be established without solving the network again. This provides an opportunity for the designer to choose the required combination in interconnecting more than one two port network and forming larger systems.

15 Relationship Between Two-Port Network Parameters

16 Example Problem: Find [z] and [g] parameters of a two-port network if Solution: Example Problem: Obtain the y parameters of the op amp circuit in Fig. Show that the circuit has no z parameters. Solution: Since no current can enter the input terminals of the op amps, I 1 = 0, which can be expressed in terms of V 1 and V 2 as

17 Interconnection of Networks The series connection V 1a =z 11 I 1a +z 12 I 2a V 2a =z 21 I 1a +z 22 I 2a V 1b =z 11 I 1b +z 12 I 2b V 2b =z 21 I 1b +z 22 I 2b I 1 =I 1a =I 1b, I 2 =I 2a =I 2b V 1 =V 1a +V 1b = (z 11a +z 11b )I 1 +(z 12a +z 12b )I 2 V 2 =V 2a +V 2b = (z 21a +z 21b )I 1 +(z 22a +z 22b )I 2 [Z]=[Z a ]+ [Z b ]

18 I 1a = y 11 V 1a + y 12 V 2a I 2a = y 21 V 1a + y 22 V 2a I 1b = y 11 V 1b + y 12 V 2b I = y V + y V 2b 21 1b 22 2b V = V = V,V = V = V 1 1a 1b 2 2a 2b I = I +I 1 1a 1b =(y + y )V +(y + y )V 11a 11b 1 12a 12b 2 I2 = I2a +I2b = (y 21a + y 21b )V 1 +(y 22a + y 22b )V 2 [y]=[y a ]+ [y b ]

19 Example Problem: Evaluate V 2 /V s in the circuit in Fig. This may be regarded as two-ports in series. For N b, z 12b =z 21b =10 =z 11b =z 22b Thus, [z] =[z a ]+[z b ] But V 1 =z 11 I 1 +z 12 I 2 = 22I 1 +18I 2 V 2 =z 32 I 1 +z 22 I 2 =18I 1 +30I 2 Example Problem: : Find the y parameters of the two-port in Fig.

20 Solution: Example Problem: Find the transmission parameters for the circuit in Fig. Thus, for the total network in Fig.,

21 ApplicationsTransistor Circuits Transistor Circuits hi = h11, hr = h12, hf = h21, ho = h22 h ie = Basic input impedance h re = Reverse voltage feedback ratio h fe = Basic-collector current gain h oe = Output admittance

22 Fig: V b = h ie I b +h re V c I c = h fe I b +h oe V c Fig. Transistor Amplifiers

23 Series-parallel Connection: Useful for H-parameter interconnection For the combined network connected in series -parallel model resultant matrix is:

24 Practice problem: Evaluate the Z parameters verify symmetric and reciprocity property Solution: V 1 =Z 11 I 1 +Z 12 I 2 V 2 =Z 21 +Z 12 I 2 with I 2 =0=>Z 11 = and Z 21 = with I 1 =0=>Z 11 = and Z 21 = On applying KVL at the input port V 1 =I 1 (8+6) Z 11 = =14 Ω At output port V 2 =I 1 Z 21 = = 6 Ω On writing circuit with I 1 =0 V 2 = I 1 Z 22 = =6 Ω V 1 = I 2 Z 12 = =6 Ω Z 12 =Z 21, the given circuit is reciprocal Z 11 22the given circuit is not symmetrical

25 Find the H parameter for the same network and Verify the reciprocity and symmetry property V 1 =h 11 I 1 +h 12 V 2 I 2 =h 21 I 1 +h 22 V 2 Short Circuit the output port V 2 =0 h 11 = and h 21 = V 1 =8 I 1 h 11 = 8 Ω I 2 =-I 1, h 21 = = -I 1 Open circuit the input port I 1 = 0.7,h 21 = h 22 = V 1 =V 2 =6I 2 h 12 = = 1 and h 22 = = reciprocal h 12 =-h 21 Symmetric h 11 h 12 - h 12 h 21 = 1 8 (1) (-1) 0 [Hence the given circuit is not symmetric] Expressing h parameters in terms of Z parameters h 11 = = = 8 Ω h 12 = = = 1 h 21 = = = -1 h 22= = S ABCD Parameters expressed in terms of Z parameters: ABCD parameters eq. V 1 = AV 2 BI (1) I 1 = CV 2 DI (2) Z parameters eq. V 1 = Z 11 I 1 +Z 12 I (3) V 2 = Z 21 I 1 +Z 22 I (4) Comparing equation (4) with (2) the variables also remain same

26 Rearranging equation (4) in such a way that I 1 is on one side Z 21 I 1 = V 2 Z 22 I 2 I 1 = V 2 - I (5) On comparing eq(5) and eq(2) C= and D = Now, on substituting the value of I 1, from eq(5) in eq(3) V 1 = Z 11 [ ]+Z 12 I 2 V 1 = V 2 -[ ] I 2 V 1 = V 2 - I 2 A=, B = Now we can also express Z- parameters in terms of ABCD parameters. CV 2 = I 1 + DI 2 V 2 = I 1 + I (6) On comparing eq(6) with eq(4) Z 21 = and Z 22 = By substituting for V 2 in eq(1) V 1 = A[ ]- BI 2 = + [ ]I 2 = + [ ]I 2 = + I (7) Comparing eq(7) with eq(3) Z 11 = Z 12 = ABCD Parameters in terms of Z ; considering the following example network with evaluated values for Z 11 =60Ω, Z 12 =60Ω, Z 21 =60Ω and Z 22 =60Ω

27 Expressing ABCD parameters in terms of Z A=, B = C =, D = We know that, V 1 = AV 2 - BI 2 I 1 = CV 2 - DI 2 With output port open circuit, I 2 = 0, A = Applying KVL at input port V 1 = (20+40)I 1 V 2 = 40 I 1 A =,AC= Now, A = = C = = B = - D = - Now with output SC V 2 =0 V 1 = 20 I 1 + I = 20 I 1 + [ ] = I 1 I 2 = - = I 1 = c B= = B = = From? D = - D =

28 H parameters in terms of Z Express h parameters in terms of Z-parameters h 11 = h 12 = h 21 = h 22= we know that V 1 =h 11 I 1 +h 12 V 2 I 2 =h 21 I 1 +h 22 V 2 [ ] We have already computed Z-parameters : [ ] With output port short circuit V 2 = 0 h 11 = h 21 = V 1 =20 I V 1 =20 I [ ] V 1 =[ ] h 11 = = Now, I 2 = -I 1 = -I 1 h 21 = =- - with input port open circuit, I 1 =0 h 12 = and h 22 = on applying KVL at output port, V 2 = (40+30)I 2

29 h 22 = S but, h 22 = = S on V 1 = 40I 2 Now h 12 = = = h 21 = = Considering - network which is solved as example problem earlier to determine Y-Parameter: [ ]=[ ] [ ] Let us find the Z parameters for the same circuit With I 2 =0, i.e. open circuit the output port V 1 = Z 11 I 1 +Z 12 I 2 V 2 = Z 21 I 1 +Z 22 I 2 At the input port V 1 = I x 4 =

30 I x = I 1 V 1 = I 1 4 Similarly V 2 = I y 8 I y = I 1 = V 2 = I 1 4 With I 1 =0, open circuit the input port V 2 = I x 8 I x = I 2 = V 2 = I 2 8 = Similarly V 1 = I x 4 I y = I 2 = V 1 = I 2 4

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