TWO-PORT. C.T. Pan 1. C.T. Pan

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1 TWO-PORT CRCUTS C.T. Pan 5. Definition of Two-Port Circuits 5. Classification of Two-Port Parameters 5.3 Finding Two-Port Parameters 5.4 Analysis of the Terminated Two-Port Circuit 5.5 nterconnected Two-Port Circuits C.T. Pan

2 5. Definition of Two-Port Circuits Consider a linear two-terminal terminal circuit N consisting of no independent sources as follows : For a, b two terminals, if in = out, then it constitutes a port. C.T. Pan 3 5. Definition of Two-Port Circuits Now consider the following linear four-terminal circuit containing no independent sources. - ' ' - with = ' = ' Then terminals a, b constitute the input port and terminals c, d constitute the output port. C.T. Pan 4

3 5. Definition of Two-Port Circuits - ' ' - No external connections exist between the input and output ports. The two-port model is used to describe the performance of a circuit in terms of the voltage and current at its input and output ports. C.T. Pan 5 5. Definition of Two-Port Circuits Two-port circuits are useful in communications, control systems, power systems, and electronic systems. They are also useful for facilitating cascaded design of more complex systems. C.T. Pan 6

4 5. Classification of Two-Port Parameters There are four terminal variables, namely,,,, only two of them are independent. Hence, there are only six possible sets of two-port parameters. C = = 6 C.T. Pan 7 5. Classification of Two-Port Parameters () The impedance, or Z, parameters N z z, Zij : in = Ω z z For two-port networks, four parameters are generally required to represent the circuit. C.T. Pan 8

5 5. Classification of Two-Port Parameters () The admittance, or Y, parameters N y y = y y, Y : ij in S C.T. Pan 9 5. Classification of Two-Port Parameters (3) The hybrid, or h, parameters N h h = h h h, h : : in Ω in S h & h scalars C.T. Pan 0

6 5. Classification of Two-Port Parameters (4) The inverse hybrid, or g, parameters N g : in S g g, g : in = g g Ω g & g scalars C.T. Pan 5. Classification of Two-Port Parameters (5) The transmission, or a, parameters N a a = a a a, a : : in Ω in S a & a scalars C.T. Pan

7 5. Classification of Two-Port Parameters (6) The inverse transmission, or b, parameters N b b = b b b, b : : in Ω in S b & b scalars C.T. Pan Finding Two-Port Parameters Method : Calculate or measure by invoking appropriate short-circuit and open- circuit conditions at the input and output ports. Method : Derive the parameters from another set of two-port parameters. C.T. Pan 4

8 5.3 Finding Two-Port Parameters Method : Choose Z parameters as an illustration. =z z =z z when = 0, output port is open \ =z, z =, input impedance =0 =z, \ z =, transfer impedance =0 C.T. Pan Finding Two-Port Parameters When = 0, input port is open \ =z, z =, transfer impedance =0 \ =z, z =, output impedance =0 When the two-port does not contain any dependent source, then z =z. C.T. Pan 6

9 5.3 Finding Two-Port Parameters z & z are called driving-point impedances. z & z are called transfer impedances. When z =z, the two-port circuit is said to be symmetrical. When z =z, the two-port circuit is called a reciprocal circuit. C.T. Pan Finding Two-Port Parameters Example : Find the Z parameters of the T-networkT Assign mesh currents as shown : ØRR R øøø Øø Œ = R RR œœ 3 œ Œ œ º ߺ ß º ß \ z =RR z =z =R z =RR 3 C.T. Pan 8

10 5.3 Finding Two-Port Parameters A two-port can be replaced by the following equivalent circuit. =z z =z z C.T. Pan Finding Two-Port Parameters n case the two-port is reciprocal, z then it can also be represented by the T-equivalent circuit. =z, C.T. Pan 0

11 5.3 Finding Two-Port Parameters Example : Find the Z parameters Assign mesh currents as follows and write down the mesh equation. C.T. Pan 5.3 Finding Two-Port Parameters = A B C D = 0 3 C.T. Pan

12 5.3 Finding Two-Port Parameters A B 3= C D 3=0... ()... () From (), - 3 =-D C... (3) Substitute (3) into () - - A -BD C = A-BD C = [ ] Z = ABD C C.T. Pan Finding Two-Port Parameters Example 3: Find the Z parameters of the following circuit by definition of Z parameters. 8Ω 4Ω 8Ω C.T. Pan 4

13 5.3 Finding Two-Port Parameters Step: Let = 0 and apply = 0 8Ω 4Ω 8Ω 0 = /(4 Ω //(8 8) Ω ) = 64 8 = = z = = = Ω 0 5 z 8 = = = = Ω 5 z C.T. Pan Finding Two-Port Parameters Step: Let = 0 and apply = 8Ω 0 4Ω 8Ω 5 = /(8 Ω //(8 4) Ω ) = 4 4 = = z = = Ω = = = = Ω 3 5 z 6/5 8/5 [ Z] = Ω 8/5 4/5 C.T. Pan 6 z

14 5.3 Finding Two-Port Parameters Example 4: Containing dependent source case R R α Step: Let = 0 and apply R R = 0 α C.T. Pan Finding Two-Port Parameters R R = 0 α C.T. Pan 8 = = α R R α = α z = = R α z = = α

15 5.3 Finding Two-Port Parameters Step: Let = 0 and apply R R = 0 α = 0 = R = 0 = = z 0 R z = = R, α 0 [ Z ] = α R C.T. Pan Finding Two-Port Parameters Example 5: Find the y parameters of the following circuit. G b Ga Use nodal analysis Gc G b G a c G C.T. Pan 30

16 5.3 Finding Two-Port Parameters G b G a c G Nodal equation Ga Gb Gb Gb Gb G c = Ga Gb Gb [ y] = in S unit Gb Gb G c C.T. Pan Note that y = y =G C.T. Pan 3 b 5.3 Finding Two-Port Parameters A linear reciprocal two-port can be represented by the following equivalent Π circuit. y y y y y C.T. Pan 3

17 5.3 Finding Two-Port Parameters Similarly, a linear two-port can also be represented by the following equivalent circuit with dependent sources. y y y y = y y = y y C.T. Pan Finding Two-Port Parameters A linear two-port can be represented by the following equivalent circuit. h h h h = h h = h h C.T. Pan 34

18 5.3 Finding Two-Port Parameters Example 6 : Find the transmission parameters of the following circuits (a), (b) a a = a a C.T. Pan Finding Two-Port Parameters For circuit (a) and when =0 a = = 0 = = 0 a = = = C.T. Pan 36

19 5.3 Finding Two-Port Parameters For circuit (a) and when =0 = = R a = = = R a = = = a a R a a 0 C.T. Pan Finding Two-Port Parameters Similarly, for circuit (b) a a 0 a a = G C.T. Pan 38

20 5.3 Finding Two-Port Parameters Method () The 6 sets of parameters relate the same input and output terminal variables, hence they are interrelated. A systematical procedure for obtaining a set of parameters from another one is given as follows for reference. C.T. Pan Finding Two-Port Parameters Step: : Arrange the given two port parameters in the following standard form: k k k 3 k 4 =0 k k k 3 k 4 =0 Step: : Separate the independent variables and the dependent variables of the desired parameter set. Step3: : Find the solution of the dependent variable vector. C.T. Pan 40

21 5.3 Finding Two-Port Parameters Example 7: Given Z parameters, find h parameters. Step From =z z =z z =z z z n the standard form -z 0 -z =0 0 z - z =0 z z C.T. Pan Finding Two-Port Parameters Step z z 0 0 z = z Step3 solve z z 0 = 0 z z h h = h h C.T. Pan 4

22 5.3 Finding Two-Port Parameters A different solution approach = z z...( A) = z z...( B) From ( B )onecan obtain z =...( C) From ( A)and( C) z z = h h = z z ( h h ) = ( z z h ) z h h C.T. Pan Finding Two-Port Parameters Example 8 : Two sets of measurements are made on a two-port resistive circuit. The first set is made with port open, and the second set is made with port short-circuited. The results are as follows : Port open =0m =0μA =-40 Port short-circuited =4m C.T. Pan 44 =0μA =ma Find h parameters from there measurements.

23 5.3 Finding Two-Port Parameters = h h (A) = h h (B) When port is short circuited, =0 =4m, =0μA, =ma Hence, from (A) and (B) 4m = h (0μA) 0 ma = h (0μA) 0 h =. KΩ K, h = 50 (C) C.T. Pan Finding Two-Port Parameters When port is open, =0 =0m, =0μA, =-40 Hence, from (A), (B) and (C) 0m =. KΩ K (0μA) h (-40) 0 = 50 (0μA) h (-40) h = 5x0-5, h =.5 μs C.T. Pan 46

24 5.4 Analysis of the Terminated Two-Port Circuit Reciprocal Theorem ersion : For a reciprocal circuit, the interchange of an ideal voltage source at one port with an ideal ammeter at the other port produces the same ammeter reading. C.T. Pan Analysis of the Terminated Two-Port Circuit =z z =z z Q S A =, =-, =0 =z z (- S A 0=z z (- zs \ = z z -z A A ) ) C.T. Pan 48

25 5.4 Analysis of the Terminated Two-Port Circuit Q =0, A =-, = S \ \ 0=z (- ) z A =z (- ) z S A z S A= = A zz -z C.T. Pan Analysis of the Terminated Two-Port Circuit The effect of reciprocity on the two-port parameters is given by z =z, or y =y a a -a a =, or b b -b b =, or h =-h, or g =-g C.T. Pan 50

26 5.4 Analysis of the Terminated Two-Port Circuit Examples of symmetric two ports. C.T. Pan Analysis of the Terminated Two-Port Circuit C.T. Pan 5

27 5.4 Analysis of the Terminated Two-Port Circuit There are mainly 6 interested characteristics for a terminated two-port circuit in practical applications. C.T. Pan Analysis of the Terminated Two-Port Circuit input impedance / output current Thevenin equivalent looking into port. current gain / voltage gain / voltage gain / g C.T. Pan 54

28 5.4 Analysis of the Terminated Two-Port Circuit The derivation of any one of the desired expressions involves the algebraic manipulation of the two-port equations along with the two constraint equations imposed at input and output terminals. C.T. Pan Analysis of the Terminated Two-Port Circuit Example 9 : Use Z parameters as an illustration g - Z g [ ] Z Z L two-port equation = z z LL ( A) = z z LL ( B) C.T. Pan 56

29 5.4 Analysis of the Terminated Two-Port Circuit input port constraint ( ) g = z g LL C Output port constraint g - Z g [ ] Z Z L ( ) =z L LL D () Find Z in = / From (D) and (B), z = z z L LL ( E) C.T. Pan Analysis of the Terminated Two-Port Circuit Substitute (E) into (A) Z in zz = z z z () Find : From (A) and (C) L Z g g [ ] Z Z L g z = z z g LL ( F) Substitute (F) into (E) zg = ( z z )( z z ) z z g L LL ( G) C.T. Pan 58

30 5.4 Analysis of the Terminated Two-Port Circuit (3) Find TH and Z TH at port : With =0, from (A), (B) and (F) z = z = LL ( H) z = z = z g z From (C), (H) and () g = = LLLLLLL ( ) TH g zg z C.T. Pan 59 z 5.4 Analysis of the Terminated Two-Port Circuit With g =0, then ( ) =z g LL J From (A) and (J), = z z z z z ZTH = = z z z g g (4) Find current gain / z From (E), = z z L C.T. Pan 60

31 5.4 Analysis of the Terminated Two-Port Circuit (5) Find voltage gain / : From (B) and (D) = z z = ( ) zl z z L zl z From (A), (D), and (K) LLLLLLL (K) ( z z z z z ) L = zl z z z = z z z z z L L LLLLL( L) C.T. Pan Analysis of the Terminated Two-Port Circuit (6) Find / g : Q Zin = = ( z Z ) g g g in zzl = ( z z )( z z ) z z g L C.T. Pan 6

32 5.4 Analysis of the Terminated Two-Port Circuit Example 0 : Given the following circuit, find when R L =5KΩ b = 0 b =3000Ω b = ms b =0. C.T. Pan Analysis of the Terminated Two-Port Circuit Example 0: (cont.) = b b...( A) = b b...( B ) = ( C ) = R L...( D ) Substitute (C) and (D) into (A) and (B) to eliminate and 3 0 = 0...( E ) = ( F ) 3 3 From (E) and (F), 5000 = = C.T. Pan 64

33 5.5 nterconnected Two-Port Circuits Two-port circuits may be interconnected in five ways : () in series () in parallel (3) in series-parallel (4) in parallel-series (5) in cascade C.T. Pan nterconnected Two-Port Circuits () Series connection = a = b = a = b [ ] Z = Z a Zb = a b = a b C.T. Pan 66

34 5.5 nterconnected Two-Port Circuits () Parallel connection = a b = a b [ ] Y = Y a Yb = a = b = a = b C.T. Pan nterconnected Two-Port Circuits (3) Series-parallel connection a a - a b N a a b b N b b - = a = b = a b [ ] h = h a hb = a b = a = b C.T. Pan 68

35 5.5 nterconnected Two-Port Circuits (4) Parallel-series connection = a b = a = b g = g g a b = a = b = a b C.T. Pan nterconnected Two-Port Circuits (5) Cascade connection a a Na a a b b Nb b b = a a = b = b = a b = - a = b a a a a a a ' ' " " ' ' " " a a = a a a a C.T. Pan 70

36 5.5 nterconnected Two-Port Circuits Example : Find the [Z] and [Y] parameters of the following two port network. R 4 R R 3 R - - C.T. Pan nterconnected Two-Port Circuits Example : (cont.) For [y] parameters : a R R 3 a - R - R 4 C.T. Pan 7

37 5.5 nterconnected Two-Port Circuits Example : (cont.) y y a = a y y a a = R (R R ) = R 3 = R (R R ) = R 3 3 R R 3 y R R 3 RR y R R a R R y = y = = - y = - y R R R R C.T. Pan nterconnected Two-Port Circuits Example : (cont.) y y b = b y y b b y = = y R 4 y = y = = - R 4 y = y y a b C.T. Pan 74

38 5.5 nterconnected Two-Port Circuits Example : (cont.) For [z] parameters: C.T. Pan nterconnected Two-Port Circuits Example : (cont.) a b a b b a a b z = R (R R ) 3 4 Z = R (R R ) 3 4 Z = Z = Z Z = Z = R Z = Z = R R RR 4 C.T. Pan 76 [ ] Z = Z a Zb

39 5.5 nterconnected Two-Port Circuits Example : Find the a parameters = a a = a a C.T. Pan nterconnected Two-Port Circuits Example 3 : A common emitter circuit b hie hre b = h h c fe oe c h :base input impedence ie h :reverse voltage gain re h :forward current gain fe h :output admittance oe C.T. Pan 78

40 Summary Objective : Understand the definition of 6 sets of two port parameters. Objective : Be able to find any set of two-port parameters. Objective 3 : Be able to analyze a terminated two-port circuit. C.T. Pan 79 Summary Objective 4 : Understand the reciprocal theorem for two-port circuits. Objective 5 : Know how to analyze an interconnected two-port circuits. C.T. Pan 80

41 Summary Problem : Due within one week. C.T. Pan 8

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