Electric Circuits, 9th Edition P73 pdf

696 Two-Port Circuits If more than two units are connected in cascade, the a parameters of the equivalent two-port circuit can be found by successively reducing the original set of two-port circuits one pair at a time. Example 18.5 illustrates how to use Eqs. 18.74-18.77 to analyze a cas- cade connection with two amplifier circuits. Example 18.5 Analyzing Cascaded Two-Port Circuits Two identical amplifiers are connected in cas- cade, as shown in Fig. 18.11. Each amplifier is described in terms of its h parameters. The values are h u = 1000 H, h n = 0.0015, h n = 100, and h 22 = 100 /xS. Find the voltage gain V 2 /V g . 10.25 X 10 -6 , Figure 18.11 • The circuit for Example 18.5. Solution The first step in finding V 2 /V g is to convert from h parameters to a parameters. The amplifiers are identical, so one set of a parameters describes the amplifiers: «n = «12 = «21 = -Ah +0.05 -4 -h n 100 -1000 5 x 10 = -lo a, h 2] 100 -h 22 -100 x 10~ 6 «12 = «11 «12 + «12«22 = (5 X 10^)(-10) + (-10)(-10 -2 ) = 0.095 a, «21 = «21«11 + «22«21 = (-KT 6 )(5 X 10~ 4 ) + (-0.01)(-10 -6 ) = 9.5 x 10~ 9 S, «22 = «21«12 + «22«22 = (-10^)(-10) + (-10 -2 ) 2 = 1.1 X 10 -4 . From Table 18.2, V g («n + «2iZ<,)Zi. + a l2 + a 22 Z R 10< h 2 \ -1 100 •10 -6 S. 1 "* = % - wo = - ,0 Next we use Eqs. 18.74-18.77 to compute the a parameters of the cascaded amplifiers: «n = «ii«'n + «i2«2i = 25 x 10 -8 + (-10)(-10 -5 ) [10.25 X 10 -6 + 9.5 X 10 -9 (500)]10 4 + 0.095 + 1.1 X 10 -4 (500) = 10 4 ^ ~ 0.15 + 0.095 + 0.055 _ \tf_ 3 = 33,333.33. Thus an input signal of 150 /JLV is amplified to an output signal of 5 V. For an alternative approach to finding the voltage gain V 2 IV g , see Problem 18.41. Practical Perspective 697 I/ASSESSMENT PROBLEM Objective 3—Know how to analyze a cascade interconnection of two-port circuits 18.7 Each element in the symmetric bridged-tee circuit shown is a 15 ft resistor. Two of these bridged tees are connected in cascade between a dc voltage source and a resistive load. The dc voltage source has a no load voltage of 100 V and an internal resistance of 8 ft. The load resistor is adjusted until maximum power is delivered to the load. Calculate (a) the load resistance, (b) the load voltage, and (c) the load power. /l ^- + z a z c v, • z b —<•— z a h ~*c— + Vi • Answer: NOTE: Also try Chapter Problem 18.40. (a) 14.44 ft; (b)16V; (c) 17.73 W. Practical Perspective Characterizing an Unknown Circuit We make the following measurements to find the h parameters for our "black box" amplifier: With Port 1 open, apply 50 V at Port 2. Measure the voltage at Port 1 and the current at Port 2: V { = 50 mV; / 2 = 2.5 A. With Port 2 short-circuited, apply 2.5 mA at Port 1. Measure the volt- age at Port 1 and the current at Port 2: V, = 1.25 V; / 2 = 3.75 A. Calculate the h parameters according to Eq. 18.14: hu = —- h\ = Vi h h h 1.25 v 2=() 0.0025 3.75 v 2=0 0.0025 = 500 ft; h 12 = 77 V, = 1500; ^22 = 77 /,=o 7,=0 0.05 50 = 10 -3 -7- = 50 mS. 50 Now we use the terminated two-port equations to determine whether or not it is safe to attach a 2 V(rms) source with a 100 ft internal impedance to Port 1 and use this source together with the amplifier to drive a speaker modeled as a 32 ft resistance with a power rating of 100 W. Here we find the value of /2 fr° m TaD ^ e 18 - 2: hyVg h (1 + h 22 Z L )(h n + Z g ) - h l2 h 2l Z L 1500(2) " [1 + (0.05)(32)][500 + 100] - (1500)(10^)(32) = 1.98 A(rms) Calculate the power to the 32 ft speaker: P = Rll = (32)(1.98) 2 = 126 W. The amplifier would thus deliver 126 W to the speaker, which is rated at 100 W, so it would be better to use a different amplifier or buy a more pow- erful speaker. 698 Two-Port Circuits Summary • 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. (See page 678.) • The model is limited to circuits in which • no independent sources are inside the circuit between the ports; • no energy is stored inside the circuit between the ports; • the current into the port is equal to the current out of the port; and • no external connections exist between the input and output ports. (See page 678.) • Two of the four terminal variables (Vi, /i, V 2 , />) are independent; therefore, only two simultaneous equa- tions involving the four variables are needed to describe the circuit. (See page 680.) • The six possible sets of simultaneous equations involv- ing the four terminal variables are called the z-, y-, a-, b-, h-, and g-parameter equations. See Eqs. 18.1-18.6. (See page 680.) • The parameter equations are written in the s domain. The dc values of the parameters are obtained by setting s ~ 0, and the sinusoidal steady-state values are obtained by setting $ = jw. (See page 680.) Any set of parameters may be calculated or measured by invoking appropriate short-circuit and open-circuit con- ditions at the input and output ports. See Eqs. 18.7-18.15. (See pages 681 and 682.) The relationships among the six sets of parameters are given in Table 18.1. (See page 684.) A two-port circuit is reciprocal if the interchange of an ideal voltage source at one port with an ideal ammeter at the other port produces the same ammeter reading. The effect of reciprocity on the two-port parameters is given by Eqs. 18.28-18.33. (See page 687.) A reciprocal two-port circuit is symmetric if its ports can be interchanged without disturbing the values of the terminal currents and voltages. The added effect of symmetry on the two-port parameters is given by Eqs. 18.38-18.43. (See page 688) The performance of a two-port circuit connected to a Thevenin equivalent source and a load is summarized by the relationships given in Table 18.2. (See page 690.) Large networks can be divided into subnetworks by means of interconnected two-port models. The cas- cade connection was used in this chapter to illustrate the analysis of interconnected two-port circuits. (See page 694.) Problems Sections 18.1-18.2 18.1 Find the h and g parameters for the circuit in Example 18.1. 18.2 Find the y parameters for the circuit shown in Fig. P18.2. Figure P18.3 I + ± 1ft AAA- f WV 4 ft M 12 ft V-, Figure P18.2 + V\ 8ft 20 ft 4ft :10ft -»- v? 18.3 Find the z parameters for the circuit in Fig. PI8.3. 18.4 Use the results obtained in Problem 18.3 to calcu- late the y parameters for the circuit in Fig. PI8.3. 18.5 Find the h parameters for the circuit in Fig. PI8.5. Figure P18.5 h 10 ft -AM, Problems 699 18.6 Find the b parameters for the circuit shown in 18.11 Find the g parameters for the operational amplifier Fig. P18.6. circuit shown in Fig. PI 8.11. Figure P18.6 20 n Figure P18.ll l ion -VW # + V^ V, 50 4012 18.7 Select the values of R h R 2 , and i? 3 in the circuit in Fig. P18.7 so that h u = 4 ft, h u = 0.8, h 2] = -0.8, and/* 22 = 0.14 S. 18.12 The operational amplifier in the circuit shown in Fig. P18.12 is ideal. Find the h parameters of the circuit. Figure P18.7 + V", !± r /?i :Ri i ^3 1 / 2 + v 2 Figure P18.12 400 fi 18.8 Find the a parameters for the circuit in Fig. PI8.8. 18.13 The following direct-current measurements were made on the two-port network shown in Fig. PI8.13. Figure P18.8 'v l kn • 'Wv V, 10" 4 V 1)50/! 140 kH V 2 Port 2 Open K, = 20 mV Is -5V /, = 0.25 juA Port 2 Short-Circuited l x = 200 fiA h = 50 fiA V { = 10 V Calculate the g parameters for the network. 18.9 Use the results obtained in Problem 18.8 to calcu- late the g parameters of the circuit in Fig. PI8.8. 18.10 Find the h parameters of the two-port circuit shown inFig.P18.10. Figure P18.10 Figure P18.13 + v, •— I. ion /20a 20011 I ™ A»„ i : —< + =:-/ioon v : > • -/. + Vi g -*< h + V, 18.14 a) Use the measurements given in Problem 18.13 to find the y parameters for the network. b) Check your calculations by finding the y parameters directly from the g parameters found in Problem 18.13. 18.15 Derive the expressions for the h parameters as functions of the g parameters. 700 Two-Port Circuits 18.16 Derive the expressions for the b parameters as functions of the h parameters. 18.17 Derive the expressions for the g parameters as functions of the z parameters. 18.18 Find the ^-domain expressions for the a parameters of the two-port circuit shown in Fig. P18.18. Figure P18.18 *i 1F i „ k e : |(- 1 TVY-Y-V 0 '•' :4H 18.19 Find the ^-domain expressions for the z parameters of the two-port circuit shown in Fig. P18.19. Figure P18.19 18.20 Find the frequency-domain values of the a parame- ters for the two-port circuit shown in Fig. P18.20. Figure P18.20 I, 20 n 18.21 Find the h parameters for the two-port circuit shown in Fig. P18.20. 18.22 a) Use the defining equations to find the s-domain expressions for the h parameters for the circuit in Fig. PI 8.22. b) Show that the results obtained in (a) agree with the /i-parameter relationships for a reciprocal symmetric network. Figure P18.22 18.23 Is the two-port circuit shown in Fig. PI8.23 sym- metric? Justify your answer. Figure P18.23 Section 18.3 18.24 Derive the expression for the voltage gain V 2 /V\ of the circuit in Fig. 18.7 in terms of the y parameters. 18.25 Derive the expression for the input impedance (Z in = Vj//i) of the circuit in Fig. 18.7 in terms of the b parameters. 18.26 Derive the expression for the voltage gain V 2 /V g °f the circuit in Fig. 18.7 in terms of the h parameters. 18.27 Derive the expression for the current gain I 2 /l\ of the circuit in Fig. 18.7 in terms of the g parameters. 18.28 Find the Thevenin equivalent circuit with respect to port 2 of the circuit in Fig. 18.7 in terms of the z parameters. 18.29 The b parameters of the amplifier in the circuit shown in Fig. PI8.29 are h n = 25; b lx = -1.25 S; b l2 = 1 kO; b 22 = -40. Find the ratio of the output power to that supplied by the ideal voltage source. Figure P18.29 loo a Problems 701 18.30 The y parameters for the two-port amplifier circuit in Fig. PI 8.30 are y u = 2mS; y n = -2 /xS; y 2 \ = 100 mS; y 22 = - 50 JJLS. The internal impedance of the source is 2500 + /0 ft, and the load impedance is 70,000 + jO 0 The ideal voltage source is generating a voltage v g = 80 V2 cos 4000r mV. a) Find the rms value of V 2 . b) Find the average power delivered to Z L . c) Find the average power developed by the ideal voltage source. Figure P18.30 7 j—Z, vA 1 V 2 | L V22 i 1 1 + |^2 1 1 _ J z L 18.31 For the terminated two-port amplifier circuit in Fig. P18.30, find a) the value of Z L for maximum average power transfer to Z L b) the maximum average power delivered to Zi c) the average power developed by the ideal voltage source when maximum power is delivered to Z L . 18.32 The linear transformer in the circuit shown in Fig. P18.32 has a coefficient of coupling of 0.75. The transformer is driven by a sinusoidal voltage source whose internal voltage is v g = 260 cos 4000^ V. The internal impedance of the source is 25 + ;0 ft. a) Find the frequency-domain a parameters of the linear transformer. b) Use the a parameters to derive the Thevenin equivalent circuit with respect to the terminals of the load. c) Derive the steady-state time-domain expression for ih. Figure P18.32 25 a ^vw- '-'l 50 O o.75 400 .° "k 12.5 mH 18.33 The g parameters for the two-port circuit in Fig. PI 8.33 are 1 ! Q ^ = 6" ; 6 S ' £12 = -0.5 + /0.5; -/0.5; g 22 = 1.5 + /2.5ft. The load impedance Z L is adjusted for maximum average power transfer to Z L . The ideal voltage source is generating a sinusoidal voltage of v g = 42V2" cos 5000* V. a) Find the rms value of V 2 . b) Find the average power delivered to Z L . c) What percentage of the average power developed by the ideal voltage source is delivered by Z L ? Figure P18.33 18.34 The following dc measurements were made on the resistive network shown in Fig. P18.34. Measurement 1 V, = 4V /, - 5 raA h = -200 mA Measurement 2 V! = 20 mV /, = 20 juA V 2 = 40 V / 2 = 0A A variable resistor R 0 is connected across port 2 and adjusted for maximum power transfer to R () . Find the maximum power. Figure P18.34 1 5.25 mvf + T /. 250 O + ) ". Resistive network U + V-, i A ikn 702 Two-Port Circuits 18.35 The following measurements were made on a resis- tive two-port network: Condition 1 - create a short circuit at port 2 and apply 20 V to port 1: Measurements: I : = 1 A; h = -1 A. Condition 2 - create an open circuit at port 1 and apply 80 V to port 2: Measurements: V { = 400 V; I 2 = 3 A. Find the maximum power that this two-port circuit can deliver to a resistive load at port 2 when port 1 is driven by a 4 A dc current source with an internal resistance of 60 O. Figure P18.38 soon jf a c [h] b 1 d c d 2 e f t c»- (a) R R R •^WV f -VS-V * -•c 18.36 a) Find the s-domain expressions for the g parame- ters of the circuit in Fig. PI 8.36. b) Port 2 in Fig. P18.36 is terminated in a resistance of 400 O, and port 1 is driven by a step voltage source v x (t) = 30u(t) V. Find v 2 (t) for t > 0 if C = 0.2 /xF and L = 200 mH. R = 72 kH IR Figure P18.36 • •- sL 1/rC 1/sC + (b) 18.39 The networks A and B in the circuit in Fig. PI8.39 are reciprocal and symmetric. For network A, it is known that a' n = 5 and a\ 2 = 24 O. a) Find the a parameters of network B. b) Find V 2 when V g = 75/CT V, Z g = 1/tT n, and Z L = 10/0° a. Figure P18.39 18.37 a) Find the y parameters for the two-port network in Fig. PI8.37. b) Find v 2 for t > 0 when v 8 = 50u(t) V. Figure P18.37 Section 18.4 18.38 The h parameters of the first two-port circuit in Fig. PI8.38(a) are h n = 1000 O; h n = 5 x 10 -4 : /i 2 i = 40; h 22 = 25 fiS. The circuit in the second two-port circuit is shown in Fig. P18.38(b), where R = 72 kH. Find v a if v n = 9 mV dc. 1 r 50 j is n A j 5 a,j is a 512 ;-/ion J L 18.40 Tlie z and _y parameters for the resistive two-ports in Fig. P18.40 are given by z\\ = -r-U; ) ; n = 200/xS; 3 100 <12 ft; y 12 = 40 /x.S; z 21 = -kO; y 2l = -800/xS Z22 = v ^ ii; - y 22 = 4W A<S; Calculate t> 0 if 1?» = 30 mV dc. Problems 703 Figure P18.40 ion ? [z] [>'] + vn< Sections 18.1-18.4 18.41 a) Show that the circuit in Fig. P18.41 is an equiva- lent circuit satisfied by the //-parameter equations. b) Use the /i-parameter equivalent circuit of (a) to find the voltage gain V^Vg in the circuit in Fig. 18.11. Figure P18.41 /, 18.42 a) Show that the circuit in Fig. PI 8.42 is an equiva- lent circuit satisfied by the z-parameter equations. b) Assume that the equivalent circuit in Fig. PI 8.42 is driven by a voltage source having an internal impedance of Z ? ohms. Calculate the Thevenin equivalent circuit with respect to port 2. Check your results against the appropriate entries in Table 18.2. Figure P18.42 /, I-, + -o- h(Z[2 ~ Zl\) -•II tl\ ^22 — ^21 221 + V, 18.43 a) Show that the circuit in Fig. P18.43 is also an equivalent circuit satisfied by the z-parameter equations. b) Assume that the equivalent circuit in Fig. PI8.43 is terminated in an impedance of Z L ohms at port 2. Find the input impedance V\jl\. Check your results against the appropriate entry in Table 18.2. Figure P18.43 /. •*- + v. • ?11 - *12 z 12 —II— ^22 ~~ -¾ l 2 \^ + A(*21 _ ^12) • 18.44 a) Derive two equivalent circuits that are satisfied by the y-parameter equations. Hint: Start with Eqs. 18.2. Add and subtract y2\V 2 to the first equation of the set. Construct a circuit that satis- fies the resulting set of equations, by thinking in terms of node voltages. Derive an alternative equivalent circuit by first altering the second equation in Eq. 18.2. b) Assume that port 1 is driven by a voltage source having an internal impedance Z ? , and port 2 is loaded with an impedance Z L . Find the current gain /2//1. Check your results against the appro- priate entry in Table 18.2. 18.45 a) Derive the equivalent circuit satisfied by the ^-parameter equations. b) Use the g-parameter equivalent circuit derived in part (a) to solve for the output voltage in Problem 18.38. Hint: Use Problem 3.65 to simplify the second two-port circuit in Problem 18.38. 18.46 a) What conditions and measurements will allow you to calculate the b parameters for the "black box" amplifier described in the Practical Perspective? b) What measurements will be made if the result- ing b parameters are equivalent to the h param- eters calculated in the Practical Perspective? 18.47 Repeat Problem 18.46 for the z parameters.