Chapter 18 Two-Port Circuits

18.1 The Terminal Equations 18.2 The Two-Port Parameters 18.3 Analysis of the Terminated Two-Port Circuit 18.4 Interconnected Two-Port Circuits

1 Motivation

 Thévenin and Norton equivalent circuits are used in representing the contribution of a circuit to one specific pair of terminals.

 Usually, a signal is fed into one pair of terminals (input port), processed by the system, then extracted at a second pair of terminals (output port). It would be convenient to relate the v/i at one port to the v/i at the other port without knowing the element values and how they are connected inside the “black box”.

2 How to model the “black box”?

Source Load (e.g. (e.g. CD speaker) player)

 We will see that a two-port circuit can be modeled by a 22 matrix to relate the v/i variables, where the four matrix elements can be obtained by performing 2 experiments.

3 Restrictions of the model

 No energy stored within the circuit.

 No independent source.

 Each port is not a current source or sink, i.e.

 ,  iiii 2211 .  No inter-port connection, i.e. between ac, ad, bc, bd.

4 Key points

 How to calculate the 6 possible 22 matrices of a two-port circuit?

 How to find the 4 simultaneous equations in solving a terminated two-port circuit?

 How to find the total 22 matrix of a circuit consisting of interconnected two-port circuits?

5 Section 18.1 The Terminal Equations

6 s-domain model

 The most general description of a two-port circuit is carried out in the s-domain.

 Any 2 out of the 4 variables {V1, I1 , V2 , I2 } can be determined by the other 2 variables and 2 simultaneous equations. 7 Six possible sets of terminal equations (1)

V1   zz 1211  I1         ; Z theis impedance matrix; V2   zz 2221  I2   I   yy  V   1  1211  1 ;  ZY 1- theis admittance matrix;       I2   yy 2221  V2 

V1  11  aa 12  V2         ; A is a transmis sion matrix; I1   21  aa 22  I2   V    bb  V   2  11 12  1 ;  AB 1- is a transmis sion matrix;       I2   21  bb 22  I1 

8 Six possible sets of terminal equations (2)

V1   hh 1211   I1         ; H is hybrida matrix; I2   hh 2221  V2    I   gg  V   1  1211  1 ;  HG 1- is  hybrida matrix;       V2   gg 2221  I2 

 Which set is chosen depends on which variables are given. E.g. If the source voltage

and current {V1 , I1} are given, choosing transmission matrix [B] in the analysis.

9 Section 18.2 The Two-Port Parameters

1. Calculation of matrix [Z] 2. Relations among 6 matrixes

10 Example 18.1: Finding [Z] (1)

 Q: Find the impedance matrix [Z] for a given resistive circuit (not a “black box”):

V1   zz 1211  I1          V2   zz 2221  I2 

 By definition, z11 =(V1 /I1) when I2 =0, i.e. the

input impedance when port 2 is open.  z11 = (20 )//(20 )=10 .

11 Example 18.1: (2)

 By definition, z21 =(V2 /I1) when I2 =0, i.e. the transfer impedance when port 2 is open.

 When port 2 is open:

 15  V   VV ,75.0  2 5 15  1 1  V1 V1  z11 10 , I1  ,  I1 10 

V2 75.0 V1 z21  5.7  . I1 V1 10( )

12 Example 18.1: (3)

 By definition, z22 = (V2 /I2 ) when I1 =0, i.e. the

output impedance when port 1 is open.  z22 = (15 )//(25 ) =9.375 .

 z12 =(V1 /I2) when I1 =0, 

 02  V   VV ,8.0  1 02 5  2 2  V2 V2  z22 375.9 , I2  ,  I2 375.9 

V1 8.0 V2 z12  5.7  . I2 V2 375.9( )

13 Comments

 When the circuit is well known, calculation of [Z] by circuit analysis methods shows the physical meaning of each matrix element.

 When the circuit is a “black box”, we can perform 2 test experiments to get [Z]: (1) Open

port 2, apply a current I1 to port 1, measure the

input voltage V1 and output voltage V2 . (2) Open

port 1, apply a current I2 to port 2, measure the

terminal voltages V1 and V2 .

14 Relations among the 6 matrixes

 If we know one matrix, we can derive all the others analytically (Table 18.1).

 [Y]=[Z]-1, [B]=[A]-1, [G]=[H]-1, elements between mutually inverse matrixes can be easily related.

 E.g. 1  zz 1211   yy 1211  1  22  yy 12         ,  zz 2221   yy 2221  y  21 yy 11 

det where det   yyyyYy 21122211 .

15 Represent [Z] by elements of [A] (1)

 [Z] and [A] are not mutually inverse, relation between their elements are less explicit.

 By definitions of [Z] and [A],

V1   zz 1211  I1  V1  11  aa 12  V2         ,        , V2   zz 2221  I2  I1   21  aa 22  I2 

the independent variables of [Z] and [A] are {I1 ,

I2 } and {V2 , I2} , respectively.

 Key of matrix transformation: Representing the

distinct independent variable V2 by {I1 , I2 }. 16 Represent [Z] by elements of [A] (2)

 By definitions of [A] and [Z],

  IaVaV 2122111  )1(    IaVaI 2222211  )2(

1 a22 )2( V2 I1  IzIzI 2221212  ),3( a21 a21  1 a   22  )3(),1(  aV 111  I1    IaI 2122  a21 a21  a  aa  11  2211  I1     IzIzIa 212111212  )4( a21  a21   zz  1  aa   1211  11 where,  Aa .det  zz  a  1 a   2221  21  22  17 Section 18.3 Analysis of the Terminated Two-Port Circuit

1. Analysis in terms of [Z] 2. Analysis in terms of [T][Z]

18 Model of the terminated two-port circuit

 A two-port circuit is typically driven at port 1 and loaded at port 2, which can be modeled as:

 The goal is to solve {V1 , I1 , V2 , I2} as functions

of given parameters Vg , Zg , ZL, and matrix elements of the two-port circuit. 19 Analysis in terms of [Z]

 Four equations are needed to solve the four

unknowns {V1 , I1 , V2 , I2} .

  IzIzV 2121111  )1(   -two port equations   IzIzV 2221212  )2(

 1 g  1ZIVV g  )3(  constraint equations todue terminati ons   22 ZIV L  )4(

 01 zz 1211  V1   0        {V1 , I1 , V2, I2 } are 10 zz 2221 V2 0         , derived by inverse  Z g 001   I1  Vg        matrix method.  010 ZL  I2   0 

20 Thévenin equivalent circuit with respect to port 2

 Once {V1 , I1 , V2 , I2} are solved, {VTh , ZTh } can

be determined by ZL and {V2 , I2} :

 Z V  L V )1(  2 Th    ZZ LTh   2 VV Th I2   )2(  ZTh

1  L VZ 2  VTh   2ZV L  VTh   L VZ 2   2ZV L          ;        .  1 I2  ZTh   V2  ZTh   1 I2   V2 

21 Terminal behavior (1)

 The terminal behavior of the circuit can be

described by manipulations of {V1 , I1, V2 , I2 }: V zz  Input impedance: Z 1 z  2112 ; in I 11  Zz 1 22 L  21Vz g  Output current: I  ; 2 ))((  zzZzZz 11 g 22 L 2112 I z  Current gain: 2  21 ; I  Zz 1 22 L

V2 21Zz L  Voltage gains:   ; V1 11 L  zZz  V Zz  2  21 L ; Vg 11 g 22 L ))((  zzZzZz 2112  22 Terminal behavior (2)

z  Thévenin voltage: V  21 V ; Th  Zz g 11 g zz  Thévenin impedance: zZ  2112 ; Th 22  Zz 11 g

23 Analysis in term of a two-port matrix [T][Z]

 If the two-port circuit is modeled by [T][Z], T={Y , A, B, H, G}, the terminal behavior can be determined by two methods:

 Use the 2 two-port equations of [T] to get a new

44 matrix in solving {V1 , I1, V2 , I2 } (Table 18.2);  Transform [T] into [Z] by Table 18.1, borrow the formulas derived by analysis in terms of [Z].

24 Example 18.4: Analysis in terms of [B] (1)

 Q: Find (1) output voltage V2 , (2,3) average

powers delivered to the load P2 and input port

P1 , for a terminated two-port circuit with known [B]. Rg

V g RL

  320 k  -b12 B     2 2.0mS -b22   25 Example 18.4 (2)

 Use the voltage gain formula of Table 18.2:

V bZ 2  L ; Vg 1112 g 22 L  21 ZZbZbZbb Lg

bbbbb 21122211 3()2.0)(20( 2)(k  ,264)mS V 5)(2(  )k 10 2   , Vg 3( 5.0)(20()k 5)(2.0()k  ...)k 19 10 V   016.2630500  .V 2 19

26 Example 18.4 (3)

 The average power of the load is formulated by 2 2  1 V2 1 016.263 V P2   93.6 .W 2 RL 2 5 k   The average power delivered to port 1 is 1 2 formulated by  ZIP .Re 1 2 1 in

V1 22 L  bZb 12  5)(2.0(   3()k )k Zin   33.133  ; I1 21 L  bZb 11 2( 5)(mS  20)k V  g 0500 V  I1    0789.0 A,  ZZ ing 500( 33.133()  )

1 2 P1   55.41)33.133()789.0( W. 2 27 Section 18.4 Interconnected Two-Port Circuits

28 Why interconnected?

 Design of a large system is simplified by first designing subsections (usually modeled by two-port circuits), then interconnecting these units to complete the system.

29 Five types of interconnections of two-port circuits

a. Cascade: Better use [A]. b. Series: [Z] c. Parallel: [Y] d. Series-parallel: [H]. e. Parallel-series: [G].

30 Analysis of cascade connection (1)

 Goal: Derive the overall matrix [A] of two cascaded two-port circuits with known transmission matrixes [A'] and [A"].

A A

Overall two-port circuit [A]=? 31 Analysis of cascade connection (2)

V1  V2  V1     A     A    )1( I1  I2   I1

V1 V2   11  aa 12  V2     A         , I1 I2   21  aa 22  I2 

 V1   11  aa 12  V2  V2           A    )2(  I1  21 aa 22  I2  I2 

V1  V2  V2  ),2(),1(By ),2(),1(By      AA     A   , I1  I2  I2 

 11  aa 12     21121111   (   aaaaaaaa 22121211  )    AAA  ,      .          21  aa 22    21221121  (  aaaaaaaa 22221221 ) 32 Key points

 How to calculate the 6 possible 22 matrices of a two-port circuit?

 How to find the 4 simultaneous equations in solving a terminated two-port circuit?

 How to find the total 22 matrix of a circuit consisting of interconnected two-port circuits?

33 Practical Perspective Audio Amplifier

34 Application of two-port circuits

 Q: Whether it would be safe to use a given audio amplifier to connect a music player modeled by

{Vg =2 V (rms), Zg =100 } to a speaker modeled

by a load resistor ZL =32  with a power rating of 100 W?

35 Find the [H] by 2 test experiments (1)

V1   hh 1211   I1   Definition of hybrid matrix [H]:        ; I2   hh 2221  V2 

 Test 1:

I1 =2.5 mA (rms) V2 =0 (short)

V1=1.25 V (rms) I2 =3.75 A (rms)

V 25.1 V Input , hIhV 1 500  . 1111 11 impedance I1 5.2 mA V2 0 I 75.3 A Current , hIhI 2  .1500 1212 21 gain I1 5.2 mA V2 0 36 Find the [H] by 2 test experiments (2)

V1   hh 1211   I1   Definition of hybrid matrix [H]:        ; I2   hh 2221  V2 

 Test 2:

I1 =0 (open) V2 =50 V (rms)

V1 =50 mV I2 =2.5 A (rms) (rms)

V 50 mV Voltage , hVhV 1  3.10 2121 12 gain V2 05 V I10 I 5.2 A Output , hVhI 2 20(  1- .) 2222 22 admittance V2 05 V I10 37 Find the power dissipation on the load

 For a terminated two-port circuit:

the power dissipated on ZL is

* * 2 L ReIVP 22  L )(Re 22  2 ZIIZI L,Re

where I2 is the rms output current phasor. 38 Method 1: Use terminated 2-port eqs for [H]

 By looking at Table 18.2:

21Vh g I2   98.1 A (rms),  22 L 11 g ))(1(  2112 ZhhZhZh L

 hh  500  10 3  where 1211  ;    1-   hh 2221   20(1500 ) 

Vg  2 V ,(rms) Z g 100 , ZL 32  . Not safe! 2 2 L  2 ZIP L 126)32()98.1(Re  100W .W

39 Method 2: Use system of terminated eqs of [Z]

 Transform [H] to [Z] (Table 18.1):

 zz 1211  1   hh 12   02.0470          .  zz 2221  h22  h21 1   20000,30 

 By system of terminated equations:

1 V1   01 zz 1211   0   66.1 V  V   10 zz   0   63.5 V   2    2221       .  I1   Z g 001  Vg   .43 mA          I2   010 ZL   0   98.1 A  2 2 L  2 ZIP L 126)32()98.1(Re  100W .W 40