Chapter 18 Two-Port Circuits

1 1 Chapter 18 Two-Port The Terminal The Two-Port Analysis of the Terminated Two-Port circuit Interconnected Two-Port Circuits2 Motivation Th venin 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 . 3 How to model the black box ? 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 ( CD player)Load ( speaker)4 Restrictions of the model No energy stored within the circuit .

2 No independent source. Each port is not a current source or sink, No inter-port connection, between ac, ad, bc, ,2211iiii 5 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 ? 6 Section The Terminal Equations7s-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 possible sets of terminal equations (1) matrix; admittance theis ;matrix; impedance theis ;1-212221121121212221121121 ZYVV yyyyIIZII zzzzVV matrix; siona transmis is ;matrix; siona transmis is ;1-112221121122222221121111 ABIV bbbbIVAIV aaaaIV9 Six possible sets of terminal equations (2) matrix; hybrida is ;matrix; hybrida is ;1-212221121121212221121121 HGIV ggggVIHVI hhhhIV Which set is chosen depends on which variables are given.

3 If the source voltage and current {V1 , I1 } are given, choosing transmission matrix [B] in the The Two-Port Parameters1. Calculation of matrix [Z]2. Relations among 6 matrixes11 Example : Finding [Z] (1) Q: Find the impedance matrix [Z] for a given resistive circuit (not a black box ): By definition, z11 =(V1 /I1 ) when I2 =0, the input impedance when port 2 is open. z11 = (20 )//(20 )=10 . 212221121121 IIzzzzVV12 Example : (2) By definition, z21 =(V2 /I1 ) when I2 =0, the transfer impedance when port 2 is open. When port 2 is open:. ) 10( , 10 , 10, 15 5 15111221111111112 VVIVzVIzIVVVV13. ) ( , , , 5 02 02222112222222221 VVIVzVIzIVVVVE xample : (3) By definition, z22 = (V2 /I2 ) when I1 =0, the output impedance when port 1 is open. z22 = (15 )//(25 ) = . z12 =(V1 /I2 ) when I1 =0, 14 Comments When the circuit is well known, calculation of [Z] by circuit analysis methods shows the physical meaning of each matrix element.

4 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 .15 Relations among the 6 matrixes If we know one matrix, we can derive all the others analytically (Table ). [Y]=[Z]-1, [B]=[A]-1, [G]=[H]-1, elements between mutually inverse matrixes can be easily related..det where,1211222111121122212221121122211211 yyyyYyyyyyyyyyyzzzz 16 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],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 }.

5 , ,222221121111212221121121 IVaaaaIVII zzzzVV17 Represent [Z] by elements of [A] (2) By definitions of [A] and [Z], )2()1(22222112122111 IaVaIIaVaV .det where,1122112122211211 Aaaaaazzzz ),3(1)2(222121221221212 IzIzIaaIaV )4(1)3(),1(21211121221221112111212221221 21111 IzIzIaaaaIaaIaIaaIaaV 18 Section Analysis of the Terminated Two-Port Circuit1. Analysis in terms of [Z]2. Analysis in terms of [T] [Z]19 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 in terms of [Z] Four equations are needed to solve the four unknowns {V1 , I1 , V2 , I2 }.ons terminati todue equations constraint)4()3(equationsport -two)2()1(221122212122121111 LggZIVZIVVIzIzVIzIzV ,0000100011001212122211211 gLgVIIVVZZ zzzz{V1 , I1 , V2 , I2 } are derived by inverse matrix venin equivalent circuit with respect to port 2 Once {V1 , I1 , V2 , I2 } are solved, {VTh , ZTh } can be determined by ZL and {V2 , I2 }: )2()1(222 ;1221222222 VZVIVZZVVZVZVIVZLLThThLThThL22 Terminal behavior (1) The terminal behavior of the circuit can be described by manipulations of {V1 , I1 , V2 , I2 }: Input impedance: Output current: Current gain: Voltage gains:;2221121111 LinZzzzzIVZ ;))((21122211212zzZzZzVzILgg ;222112 LZzzII ;))((;21122211212112112zzZzZzZzVVzZzZzVV LgLgLL23 Terminal behavior (2) Th venin voltage: Th venin impedance:;1121ggThVZzzV.

6 11211222gThZzzzzZ 24 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 ); Transform [T] into [Z] by Table , borrow the formulas derived by analysis in terms of [Z].25 Example : 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]. 2k 320B-b12-b22 RLVgRg26 Example (2) Use the voltage gain formula of Table :.V , )k 5)( ()k )(20()k 3()k 5)(2(,264)mS 2)(k 3() )(20(;2221122211212211122 VVVbbbbbZZbZbZbbbZVVgLgLgLg27 Example (3) The average power of the load is formulated by.

7 W 5V LRVP The average power delivered to port 1 is formulated by .Re21211inZIP W. ) () (21A, ) () 500(V 0500; )k 5)(mS 2()k 3()k 5)( (2111121122211 PZZVIbZbbZbIVZinggLLin 28 Section Interconnected Two-Port Circuits29 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 types of interconnections of Two-Port circuitsa. Cascade: Better use [A].b. Series: [Z]c. Parallel: [Y]d. Series-parallel: [H].e. Parallel-series: [G].31 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]=? )2(,2222222112111122222112112211 IVAIV aaaaIVIV aaaaIVAIV32 Analysis of cascade connection (2) )1(112211 IVAIVAIV.

8 ()( ,, ),2(),1(By 2222122121221121221212112112111122211211 222211 aaaaaaaaaaaaaaaaaaaaAAAIVAIVAAIV33 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 ? 34 Practical Perspective Audio Amplifier35 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?36 Find the [H] by 2 test experiments (1);212221121121 VIhhhhIV Definition of hybrid matrix [H]: Test 1:I1 = mA (rms)V2 = 0 (short). 500mA ,0111111112 ,0122112122 VIIhIhIV1 = V (rms)I2 = A (rms)Input impedanceCurrent gain37 Find the [H] by 2 test experiments (2);212221121121 VIhhhhIV Definition of hybrid matrix [H]: Test 2:I1 = 0 (open)V2 = 50 V (rms).)

9 10V 05mV 50 ,30211221211 IVVhVhV.) 20(V 05A ,1-0222222221 IVIhVhIV1 =50 mV (rms)I2 = A (rms)Voltage gainOutput admittance38 Find the power dissipation on the load For a terminated Two-Port circuit : ,Re)(ReRe22*22*22 LLLZIIZIIVP the power dissipated on ZL iswhere I2 is the rms output current 1: Use terminated 2-port eqs for [H] By looking at Table :(rms), A ))(1(21121122212 LgLgZhhZhZhVhI. 32 , 100 ,(rms) V 2;) 20(150010 500 where1-322211211 LggZZVhhhh .W 100W 126)32() (Re222 LLZIPNot safe!40 Method 2: Use system of terminated eqs of [Z] Transform [H] to [Z] (Table ): .W 100W 126)32() (Re222 LLZIP. 20000, .43V gLgVZZzzzzIIVV By system of terminated equations.