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- 1. An Introduction To Two – Port Networks The University of Tennessee Electrical and Computer Engineering Knoxville, TNwlg
- 2. Two Port NetworksGeneralities: The standard configuration of a two port: I1 I2 + + Input Output V1 _ Port The Network Port V2 _ The network ? The voltage and current convention ?* notes
- 3. Two Port Networks Network Equations: V1 = z11I1 + z12I2 V2 = b11V1 - b12I1 ImpedanceZ parameters V2 = z21I1 + z22I2 I2 = b21V1 – b22I1 I1 = y11V1 + y12V2 V1 = h11I1 + h12V2 Admittance HybridY parameters H parameters I2 = y21V1 + y22V2 I2 = h21I1 + h22V2Transmission V1 = AV2 - BI2 I1 = g11V1 + g12I2 A, B, C, D parameters I1 = CV2 - DI2 V2 = g21V1 + g22I2 * notes
- 4. Two Port Networks Z parameters: V z11 is the impedance seen looking into port 1 z = 1 11 I I =0 when port 2 is open. 1 2 V z12 is a transfer impedance. It is the ratio of the z = 1 12 I I =0 voltage at port 1 to the current at port 2 when 2 1 port 1 is open. V z = 2 z21 is a transfer impedance. It is the ratio of the 21 I I =0 1 2 voltage at port 2 to the current at port 1 when port 2 is open. V z = 2 z22 is the impedance seen looking into port 2 22 I I =0 2 1 when port 1 is open.* notes
- 5. Two Port Networks Y parameters: I y11 is the admittance seen looking into port 1 y = 1 11 V V =0 when port 2 is shorted. 1 2 I y12 is a transfer admittance. It is the ratio of the y = 1 12 V V =0 current at port 1 to the voltage at port 2 when 2 1 port 1 is shorted. I y = 2 y21 is a transfer impedance. It is the ratio of the 21 V V =0 1 2 current at port 2 to the voltage at port 1 when port 2 is shorted. I y = 2 y22 is the admittance seen looking into port 2 22 V V =0 2 1 when port 1 is shorted.* notes
- 6. Two Port NetworksZ parameters: Example 1 Given the following circuit. Determine the Z parameters. I1 I2 8 Ω 10 Ω + + V 1 20Ω 20 Ω V 2 _ _ Find the Z parameters for the above network.
- 7. Two Port Networks Z parameters: Example 1 (cont 1)For z11: For z22: Z11 = 8 + 20||30 = 20 Ω Z22 = 20||30 = 12 Ω I1 I2 8 Ω 10 ΩFor z12: + + V 1 20Ω 20 Ω V 2 V z = 1 _ _ 12 I I =0 2 1 20 xI 2 x 20 Therefore: 8 xI 2V1 = = 8 xI 2 z 12 = =8 Ω = z 21 20 + 30 I2
- 8. Two Port NetworksZ parameters: Example 1 (cont 2) The Z parameter equations can be expressed in matrix form as follows. V1 z 11 z 12 I 1 V = z I z 22 2 2 21 V1 20 8 I 1 V = I 2 8 12 2
- 9. Two Port NetworksZ parameters: Example 2 (problem 18.7 Alexander & Sadiku) You are given the following circuit. Find the Z parameters. I1 I2 1Ω 4 Ω + + + 2 Ω 1Ω Vx V 1 V 2 - 2Vx _ _
- 10. Two Port Networks Z parameters: Example 2 (continue p2) I1 I2 V 1Ω 4 Ω z = 1 11 I I =0 + + 1 2 + 2 Ω 1Ω Vx V 1 V 2 - V V + 2V x 6V x + V x + 2V x 2VxI1 = x + x = _ _ 1 6 6 3V xI1 = ; but V x = V1 − I 1 Other Answers 2 Z21 = -0.667 ΩSubstituting gives; 3(V1 − I 1 ) V1 5 Z12 = 0.222 Ω I1 = or = z11 = Ω 2 I1 3 Z22 = 1.111 Ω
- 11. Two Port NetworksTransmission parameters (A,B,C,D):The defining equations are: V1 A B V2 I = C D − I 1 2 V1 V1 A= B= V2 I2 = 0 − I2 V2 = 0 I1 I1 C= D= V2 I2 = 0 − I2 V2 = 0
- 12. Two Port NetworksTransmission parameters (A,B,C,D):Example Given the network below with assumed voltage polarities and Current directions compatible with the A,B,C,D parameters. I1 -I2 + + R 1 V 1 R 2 V 2 _ _ We can write the following equations. V1 = (R1 + R2)I1 + R2I2 V2 = R2I1 + R2I2 It is not always possible to write 2 equations in terms of the V’s and I’s Of the parameter set.
- 13. Two Port Networks Transmission parameters (A,B,C,D): Example (cont.) V1 = (R1 + R2)I1 + R2I2 V2 = R2I1 + R2I2 From these equations we can directly evaluate the A,B,C,D parameters. V1 R1 + R2 V1 A= = B= = R1 V2 I2 = 0 R2 − I2 V2 = 0 I1 I1 C= = 1 D= = 1 V2 I2 = 0 R2 − I2 V2 = 0Later we will see how to interconnect two of these networks together for a final answer* notes
- 14. Two Port NetworksHybrid Parameters: The equations for the hybrid parameters are: V1 h11 h12 I 1 I = h h22 V 2 2 21 V1 V1 h11 = h12 = I1 V2 = 0 V2 I1 = 0 I2 I2 h21 = h22 = I1 V2 = 0 V2 I1 = 0* notes
- 15. Two Port NetworksHybrid Parameters: The following is a popular model used to represent a particular variety of transistors. I1 K I2 1 + + + V K 2V 2 K 4 V 1 _ K 3V 1 2 _ _ We can write the following equations: V1 = AI 1 + BV 2 V2 I 2 = CI 1 + D* notes
- 16. Two Port NetworksHybrid Parameters: V1 = AI 1 + BV 2 V2 I 2 = CI 1 + D We want to evaluate the H parameters from the above set of equations. V1 V1 h11 = = K1 h12 = = K2 I1 V2 = 0 V2 I1 = 0 I2 h21 = = K3 I2 1 I1 V2 = 0 h22 = = K 4 V2 I1 = 0
- 17. Two Port NetworksHybrid Parameters: Another example with hybrid parameters. Given the circuit below. I1 -I2 The equations for the circuit are: + + R 1 V1 = (R1 + R2)I1 + R2I2 V 1 R 2 V 2 _ _ V2 = R2I1 + R2I2 The H parameters are as follows. V1 V1 h11 = h12 = I1 = R1 V2 = 1 V2=0 I1=0 I2 I2 1 h21 = = -1 h22 = = I1 V2 R2 V2=0 I1=0
- 18. Two Port Networks Modifying the two port network: Earlier we found the z parameters of the following network. I1 I2 8 Ω 10 Ω + + V 1 20Ω 20 Ω V 2 _ _ V1 20 8 I 1 V = 8 12 I 2 2* notes
- 19. Two Port NetworksModifying the two port network:We modify the network as shown be adding elements outside the two ports 6Ω I1 I2 8 Ω 10 Ω + + + 10 v V 1 20Ω 20 Ω V 2 4 Ω _ _ _ We now have: V1 = 10 - 6I1 V2 = - 4I2
- 20. Two Port Networks Modifying the two port network: We take a look at the original equations and the equations describing the new port conditions. V1 20 8 I 1 V1 = 10 - 6I1 V = 8 12 I 2 2 V2 = - 4I2 So we have, 10 – 6I1 = 20I1 + 8I2 -4I2 = 8I1 + 12I2* notes
- 21. Two Port NetworksModifying the two port network: Rearranging the equations gives, −1 I 1 26 8 10 I = 8 0 2 16 1 I 0.4545 = I 2 -0.2273
- 22. Two Port NetworksY Parameters and Beyond: Given the following network. I1 1 Ω I2 + + 1 s V 1 V s 2 _ _ 1 Ω (a) Find the Y parameters for the network. (b) From the Y parameters find the z parameters
- 23. Two Port Networks Y Parameter Example I I y = 1 y = 1 I1 = y11V1 + y12V2 11 V V =0 12 V 1 2 2 V =0 1 I2 = y21V1 + y22V2 I I I2 y = 2 y = 2 I1 1 Ω 21 V V =0 22 V V =0 1 2 2 1+ + 1 sV 1 s V 2 short_ _ We use the above equations to 1 Ω evaluate the parameters from the network. To find y11 2V1 = I 1 ( s )= I 2 so I y = 1 = s + 0.5 1 2+1 s 2s + 1 11 V 1 V =0 2
- 24. Two Port NetworksY Parameter Example I1 I2 1 Ω I y = 2 + + 21 V V =0 1 2 1 s V 1 s V 2 _ _ 1 Ω We see V1 = − 2I 2 I y = 2 = 0.5 S 21 V 1
- 25. Two Port Networks Y Parameter Example I1 I2 1 ΩTo find y12 and y21 we reverse short + + 1 s V things and short V1 1 s V 2 _ _ I y = 1 1 Ω 12 V V =0 2 1 I We have y = 2 22 V V =0 2 1 We have V2 = − 2I1 2s 1 V2 = I 2 y22 = 0.5 + ( s + 2) s I y = 1 = 0.5 S 12 V 2
- 26. Two Port NetworksY Parameter Example Summary: y11 y12 s + 0.5 − 0.5 Y = y = y22 − 0.5 0.5 + 1 s 21 Now suppose you want the Z parameters for the same network.
- 27. Two Port Networks Going From Y to Z Parameters For the Y parameters we have: For the Z parameters we have: I =Y V V =Z I −1 From above; V =Y I =Z I Therefore y −y 22 12 where z z ∆ Y ∆ Z =Y −1 = 11 12 = − y Y ∆Y = det Y z z y 21 22 21 11 ∆ ∆ Y Y
- 28. Two Port Parameter Conversions:
- 29. Two Port Parameter Conversions:To go from one set of parameters to another, locate the set of parametersyou are in, move along the vertical until you are in the row that containsthe parameters you want to convert to – then compare element for element ∆H z11 = h22
- 30. Interconnection Of Two Port NetworksThree ways that two ports are interconnected: ya Y parameters * Parallel yb [ y ] = [ ya ] [ ] + yb Z parameters [ z ] = [ za ] [ ] za * Series + zb zb ABCD parameters * Cascade Ta Tb [T ] = [Ta ] [Tb ]
- 31. Interconnection Of Two Port Networks Consider the following network: I1 I2 R 1 R 1 V2Find + + V1 V1 R 2 R V2 T1 2 T2 _ _ Referring to slide 13 we have; R1 + R2 R + R R1 1 2 R1 V V1 R R2 I = 2 2 1 1 1 1 − I 2 1 R2 R2
- 32. Interconnection Of Two Port Networks R1 + R2 R + R R1 1 2 R1 V V1 R R2 I = 2 2 1 −I 1 1 1 1 2 R2 R2 Multiply out the first row: R + R 2 R R + R V1 = 1 2 + 1 V + 1 2 R + R ( − I ) R R2 2 R2 1 1 2 2 Set I2 = 0 ( as in the diagram) V2 R2 2 Can be verified directly = V1 R1 + 3 R1 R2 R2 2 by solving the circuit 2
- 33. Basic Laws of Circuits End of Lesson Two-Port Networks

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