346 Sinusoidal Steady-State Analysis in Fig. 9.57 depicts these observations. The dotted phasors represent the pertinent currents and volt- ages before the addition of the capacitor. Thus, comparing the dotted phasors of I, RJL, ja>L-[l, and V v with their solid counterparts clearly shows the effect of adding C to the circuit. In par- ticular, note that this reduces the amplitude of the source voltage and still maintains the amplitude of the load voltage. Practically, this result means that, as the load increases (i.e., as I a and I h increase), we can add capacitors to the system (i.e., increase I c ) so that under heavy load conditions we can main- tain V L without increasing the amplitude of the source voltage. Figure 9.56 • The effect of the capacitor current I c on the line current I. Figure 9.55 • The addition of a capacitor to the circuit shown in Fig. 9.53. Figure 9.57 A The effect of adding a load-shunting capacitor to the circuit shown in Fig. 9.53 if V L is held constant. NOTE: Assess your understanding of this material by trying Chapter Problems 9.84 and 9.85. Practical Perspective A Household Distribution Circuit Let us return to the household distribution circuit introduced at the begin- ning of the chapter. We will modify the circuit slightly by adding resistance to each conductor on the secondary side of the transformer to simulate more accurately the residential wiring conductors. The modified circuit is shown in Fig. 9.58. In Problem 9.88 you will calculate the six branch currents on the secondary side of the distribution transformer and then show how to calculate the current in the primary winding. NOTE: Assess your understanding of this Practical Perspective by trying Chapter Problems 9.88 and 9.89. — h 13.2/0! kV • + 1(1 -AAA- • + 120/0! l|20O _v 2 n h AW [3 — r, 120/0! j i40 0 -VW * 10 nf iii Figure 9.58 • Distribution circuit. Summary Summary 347 The general equation for a sinusoidal source is v = V m cos(a)t + 4>) (voltage source), or i = I m cos(a»r + (j>) (current source), where V m (or I m ) is the maximum amplitude, a> is the frequency, and <f) is the phase angle. (See page 308.) The frequency, a>, of a sinusoidal response is the same as the frequency of the sinusoidal source driving the circuit. The amplitude and phase angle of the response are usu- ally different from those of the source. (See page 311.) The best way to find the steady-state voltages and cur- rents in a circuit driven by sinusoidal sources is to per- form the analysis in the frequency domain. The following mathematical transforms allow us to move between the time and frequency domains. • The phasor transform (from the time domain to the frequency domain): V = V m e^ = &{V m cos(tot + <£)}. • The inverse phasor transform (from the frequency domain to the time domain): (See pages 312-313.) When working with sinusoidally varying signals, remember that voltage leads current by 90° at the ter- minals of an inductor, and current leads voltage by 90° at the terminals of a capacitor. (See pages 317-320.) Impedance (Z) plays the same role in the frequency domain as resistance, inductance, and capacitance play in the time domain. Specifically, the relationship between phasor current and phasor voltage for resis- tors, inductors, and capacitors is V = 21, where the reference direction for I obeys the passive sign convention. The reciprocal of impedance is admittance (Y), so another way to express the current- voltage relationship for resistors, inductors, and capaci- tors in the frequency domain is v = i/y. (See pages 320 and 324.) All of the circuit analysis techniques developed in Chapters 2-4 for resistive circuits also apply to sinu- soidal steady-state circuits in the frequency domain. These techniques include KVL, KCL, series, and paral- lel combinations of impedances, voltage and current division, node voltage and mesh current methods, source transformations and Thevenin and Norton equivalents. The two-winding linear transformer is a coupling device made up of two coils wound on the same nonmagnetic core. Reflected impedance is the impedance of the sec- ondary circuit as seen from the terminals of the primary circuit or vice versa. The reflected impedance of a linear transformer seen from the primary side is the conjugate of the self-impedance of the secondary circuit scaled by the factor (a)M/\Z 22 \) 2 . (See pages 335 and 336.) The two-winding ideal transformer is a linear trans- former with the following special properties: perfect coupling (k = 1), infinite self-inductance in each coil (Z-! = L 2 = oo), and lossless coils (R { = R 2 = 0). The circuit behavior is governed by the turns ratio a = N 2 /N]_. In particular, the volts per turn is the same for each winding, or Afc* and the ampere turns are the same for each winding, or JVili = ± N 2 l 2 - (See page 338.) TABLE 9.3 Impedance and Related Values Element Impedance (Z) Resistor R (resistance) Capacitor ;(—l/a»C) Inductor jwL Reactance - 1/wC coL Admittance (Y) G (conductance) j(oC }{-\/a>L) Susceptance -1/coL 348 Sinusoidal Steady-State Analysis Problems Section 9.1 9.1 Consider the sinusoidal voltage v(t) = 80 cos (lOOOirt - 30°) V. a) What is the maximum amplitude of the voltage? b) What is the frequency in hertz? c) What is the frequency in radians per second? d) What is the phase angle in radians? e) What is the phase angle in degrees? f) What is the period in milliseconds? g) What is the first time after t = 0 that v = 80 V? h) The sinusoidal function is shifted 2/3 ms to the left along the time axis. What is the expression for v(t)l i) What is the minimum number of milliseconds that the function must be shifted to the right if the expression for v(t) is 80 sin 100()7r/ V? j) What is the minimum number of milliseconds that the function must be shifted to the left if the expression for v(t) is 80 cos IOOO77-? V? 9.2 At t = — 2 ms, a sinusoidal voltage is known to be zero and going positive. The voltage is next zero at t = 8 ms. It is also known that the voltage is 80.9 V at t = 0. a) What is the frequency of v in hertz? b) What is the expression for vl 9.3 A sinusoidal current is zero at t = — 625/xs and increasing at a rate of 800077 A/s. The maximum amplitude of the current is 20 A. a) What is the frequency of i in radians per second? b) What is the expression for /? 9.4 A sinusoidal voltage is given by the expression v = 10 cos (3769.911 - 53.13°) V. Find (a) / in hertz; (b) T in milliseconds; (c) V m ; (d) v(0); (e) <£ in degrees and radians; (f) the smallest positive value of t at which v = 0; and (g) the small- est positive value of t at which dv/dt = 0. 9.5 In a single graph, sketch v = 100 cos (cot + 4>) ver- sus cot for 4> = -60°, -30°, 0°, 30°, and 60°. a) State whether the voltage function is shifting to the right or left as <f> becomes more positive. b) What is the direction of shift if 4> changes from 0to30°? 9.6 Show that / ' 9.7 The rms value of the sinusoidal voltage supplied to the convenience outlet of a home in Scotland is 240 V. What is the maximum value of the voltage at the outlet? 9.8 Find the rms value of the half-wave rectified sinu- soidal voltage shown. Figure P9.8 K„,sin^r£,0 t« 7/2 37/2 Section 9.2 Vj, cos 2 (wr -I- (j>)dt = VlT 9.9 The voltage applied to the circuit shown in Fig. 9.5 at t = 0 is 20 cos (800* + 25°) V. The circuit resist- ance is 80 fl and the initial current in the 75 mH inductor is zero. a) Find i(t) for t > 0. b) Write the expressions for the transient and steady-state components of i(t). c) Find the numerical value of i after the switch has been closed for 1.875 ms. d) What are the maximum amplitude, frequency (in radians per second), and phase angle of the steady-state current? e) By how many degrees are the voltage and the steady-state current out of phase? 9.10 a) Verify that Eq. 9.9 is the solution of Eq. 9.8. This can be done by substituting Eq. 9.9 into the left- hand side of Eq. 9.8 and then noting that it equals the right-hand side for all values of t > 0. At t = 0, Eq. 9.9 should reduce to the initial value of the current. b) Because the transient component vanishes as time elapses and because our solution must sat- isfy the differential equation for all values of /, the steady-state component, by itself, must also satisfy the differential equation. Verify this observation by showing that the steady-state component of Eq. 9.9 satisfies Eq. 9.8. Sections 9.3-9.4 9.11 Use the concept of the phasor to combine the fol- lowing sinusoidal functions into a single trigono- metric expression: a) y = 50 cos(500/ + 60°) + 100 cos(500* - 30°), b) y = 200 cos(377/ + 50°) - 100 sin(377/ + 150°), Problems 349 c) y = 80 cos(100f + 30°) - 100 sin(100f - 135°) + 50 cos(100r - 90°), and d) v = 250 cos oit + 250 cos(wr + 120°) + 250 cos(o* - 120°). 9.12 The expressions for the steady-state voltage and current at the terminals of the circuit seen in Fig. P9.12 are v g = 300 cos (5000^ + 78°) V, i g = 6sin(50007rf + 123°) A a) What is the impedance seen by the source? b) By how many microseconds is the current out of phase with the voltage? Figure P9.12 1» Circuit 9.13 A 80 kHz sinusoidal voltage has zero phase angle and a maximum amplitude of 25 mV. When this voltage is applied across the terminals of a capaci- tor, the resulting steady-state current has a maxi- mum amplitude of 628.32 /xA. a) What is the frequency of the current in radians per second? b) What is the phase angle of the current? c) What is the capacitive reactance of the capacitor? d) What is the capacitance of the capacitor in microfarads? e) What is the impedance of the capacitor? 9.14 A 400 Hz sinusoidal voltage with a maximum amplitude of 100 V at t = 0 is applied across the terminals of an inductor. The maximum amplitude of the steady-state current in the inductor is 20 A. a) What is the frequency of the inductor current? b) If the phase angle of the voltage is zero, what is the phase angle of the current? c) What is the inductive reactance of the inductor? d) What is the inductance of the inductor in millihenrys? e) What is the impedance of the inductor? Sections 9.5 and 9.6 9.15 A 40 H resistor, a 5 mH inductor, and a 1.25/xF PSPICE capacitor are connected in series. The series-connected elements are energized by a sinusoidal voltage source whose voltage is 600 cos (8000^ + 20°)V. a) Draw the frequency-domain equivalent circuit. b) Reference the current in the direction of the voltage rise across the source, and find the pha- sor current. c) Find the steady-state expression for i(t). 9.16 A 10 O resistor and a 5 /xF capacitor are connected PSPICE i n parallel. This parallel combination is also in par- 1 allel with the series combination of an 8 O resistor and a 300 /xH inductor. These three parallel branches are driven by a sinusoidal current source whose current is 922 cos(20,000r + 30°) A. a) Draw the frequency-domain equivalent circuit. b) Reference the voltage across the current source as a rise in the direction of the source current, and find the phasor voltage. c) Find the steady-state expression for v{t). 9.17 a) Show that, at a given frequency w, the circuits in Fig. P9.17(a) and (b) will have the same imped- ance between the terminals a,b if 2 J 2 /e, - a> z LjR 2 R\ + co 2 L 2 2 U Rl^i R\ + (o 2 L\ b) Find the values of resistance and inductance that when connected in series will have the same impedance at 4 krad/s as that of a 5 kH resistor connected in parallel with a 1.25 H inductor. Figure P9.17 'a Ri \U L, (a) (b) 9.18 a) Show that at a given frequency eo, the circuits in Fig. P9.17(a) and (b) will have the same imped- ance between the terminals a,b if Ro = R] + a?L\ u = R\ + o) 2 L\ 2 f (Hint: The two circuits will have the same impedance if they have the same admittance.) b) Find the values of resistance and inductance that when connected in parallel will have the same impedance at 1 krad/s as an 8 kft resistor con- nected in series with a 4 H inductor. 350 Sinusoidal Steady-State Analysis 9.19 a) Show that at a given frequency to, the circuits in Fig. P9.19(a) and (b) will have the same imped- ance between the terminals a,b if /?i = Q = R 2 1 + o?R\c\ 1 + arRJCJ w 2 RJC 2 b) Find the values of resistance and capacitance that when connected in series will have the same impedance at 40 krad/s as that of a 1000 ft resis- tor connected in parallel with a 50 nF capacitor. Figure P9.19 'a A', c,; c, (a) 9.20 a) Show that at a given frequency QJ, the circuits in Fig 9.19(a) and (b) will have the same imped- ance between the terminals a,b if R 2 C\ 1 + (D 2 R]C\ orR { C 2 1 + io 2 R]C\ (Hint: The two circuits will have the same impedance if they have the same admittance.) b) Find the values of resistance and capacitance that when connected in parallel will give the same impedance at 50 krad/s as that of a 1 kft resistor connected in series with a capacitance of 40 nF. 9.21 a) Using component values from Appendix H, combine at least one resistor, inductor, and capacitor in series to create an impedance of 300 - /400 ft at a frequency of 10,000 rad/s. b) At what frequency does the circuit from part (a) have an impedance that is purely resistive? 9.22 a) Using component values from Appendix H, combine at least one resistor and one inductor in parallel to create an impedance of 40 + /20 ft at a frequency of 5000 rad/s. (Hint- Use the results of Problem 9.18.) b) Using component values from Appendix H, combine at least one resistor and one capacitor in parallel to create an impedance of 40 - /20 ft at a frequency of 5000 rad/s. (Hint: Use the result of Problem 9.20.) 9.23 a) Using component values from Appendix H, find a single capacitor or a network of capacitors that, when combined in parallel with the RL cir- cuit from Problem 9.22(a), gives an equivalent impedance that is purely resistive at a frequency of 5000 rad/s. b) Using component values from Appendix H, find a single inductor or a network of inductors that, when combined in parallel with the RC circuit from Problem 9.22(b), gives an equivalent impedance that is purely resistive at a frequency of 5000 rad/s. 9.24 Three branches having impedances of 3 + /4 O, 16 - /12 ft, and -/4 ft, respectively, are connected in parallel. What are the equivalent (a) admittance, (b) conductance, and (c) susceptance of the parallel connection in millisiemens? (d) If the parallel branches are excited from a sinusoidal current source where i = 8 cos w/ A, what is the maximum amplitude of the current in the purely capacitive branch? 9.25 a) For the circuit shown in Fig. P9.25, find the fre- PSPICE quency (in radians per second) at which the impedance Z ab is purely resistive, b) Find the value of Z ab at the frequency of (a). Figure P9.25 160/tH a« TVYV^. 25 nF 9.26 Find the admittance K, b in the circuit seen in Fig. P9.26. Express K (lb in both polar and rectangu- lar form. Give the value of Y ab in millisiemens. Figure P9.26 -/12.8 ft a* >V 6 ft -{/12 ft 5ft |/10 ft 13.6 ft Problems 351 9.27 Find the impedance Z ab in the circuit seen in 9.31 Find the steady-state expression for /„(/) in the cir- Fig. P9.27. Express Z ab in both polar and rectangular PSPICE cu it in Fig. P9.31 if v s = 100 sin 50/ mV. form. MULTISIM Figure P9.27 1 Q a« 'vw- Zau -/8 n 40 n :10O ^ -/20 n !/20 0 Figure P9.31 412 240 mH VW orwx. 2.5 mF 9.32 Find the steady-state expression for v (> in the circuit of Fig. P9.32 if i g = 500 cos 2000/ m A. 9.28 The circuit shown in Fig. P9.28 is operating in the sinusoidal steady state. Find the value of co if i (} = 40 sin (a)/ + 21.87°) mA. v g = 40cos(o>/ - 15°) V. Figure P9.28 600 tt 3.2 H ^YYYV 2.5 /xF 9.29 The circuit in Fig. P9.29 is operating in the sinu- PSPICE soidal steady state. Find the steady-state expression M"LTISIM f orVw ( f ) ifu = 40 cos 50,000/ V. Figure P9.29 1/u.F 30 n vAl.2mU Figure P9.32 <0 120 a 12.5/tF 40 a + 60 mH iv. 9.33 The phasor current I a in the circuit shown in PSPICE Fig. P9.33 is 2/0° A. lumsiM a) Find I b , I c , and V g . b) If a) = 800 rad/s, write the expressions for i b (t), / c (/), and vM). Figure P9.33 120 n "4 -/40 a- 6 +/3.5 AI © 9.30 a) For the circuit shown in Fig. P9.30, find the steady- PSPICE state expression for v () if /„ = 2 cos (16 X 10^/) A. MULTISIM b) By how many nanoseconds does v a lag /',,? Figure P9.30 9.34 The circuit in Fig. P9.34 is operating in the sinusoidal PSPICE steady state. Find v 0 (t) if /,(/) = 3 cos 200/ mA. MULTISIM Figure P9.34 6ft r^ .©J : 22 n ^ i i "12.5mFj2mH : i • 1 :5 n i + vjt) • 352 Sinusoidal Steady-State Analysis 9.35 Find the value of Z in the circuit seen in Fig. P9.35 if V g = 100 - /50 V, I g = 30 + /20 A, and \i = 140 + /30 V. Figure P9.35 7, 20 a 12 a /16 a -A/W /5 ft Vj 9.39 The frequency of the sinusoidal voltage source in PSPICE the circuit in Fig. P9.39 is adjusted until the current "1ULTISIM • • • i ;.«_ i a is in phase with v g . a) Find the frequency in hertz. b) Find the steady-state expression for i g (at the frequency found in [a]) if v g = 30 cos wt V. 9.36 Find I b and Z in the circuit shown in Fig. P9.36 if VJJ = 25/(TV and I a = 5 /90° A. Figure P9.36 ••© !/3 a in -/2 ft -/5 n 4n K ^ -/3 n 9.37 Find Z ab for the circuit shown in Fig P9.37. Figure P9.37 PSPICE MULTISIM -/in 9.38 a) The frequency of the source voltage in the circuit in Fig. P9.38 is adjusted until i g is in phase with V r What is the value of co in radians per second? b) If v g = 20 cos a)t V (where a> is the frequency found in [a]), what is the steady-state expression for v n l PSPICE MULTISIM Figure P9.38 500 n {1? \ 500 mH v„ 11 kn Figure P9.39 (50/3) kil 1.2 kn >vw- 200 mH 9.40 The circuit shown in Fig. P9.40 is operating in the PSPICE sinusoidal steady state. The capacitor is adjusted ' until the current L is in phase with the sinusoidal voltage Vg a) Specify the capacitance in microfarads if Vg = 80 cos 5000f V. b) Give the steady-state expression for L when C has the value found in (a). Figure P9.40 800 mH 9.41 a) The source voltage in the circuit in Fig. P9.41 is Vg - 50 cos 50,000f V. Find the values of L such that ig is in phase with v g when the circuit is operating in the steady state. b) For the values of L found in (a), find the steady- state expressions for ig. Figure P9.41 5nF 10 kn 9.42 The frequency of the sinusoidal current source in PSPICE the circuit in Fig. P9.42 is adjusted until v a is in mTISIM phase with i r a) What is the value of a) in radians per second? b) If ig = 2.5 cos oit mA (where to is the frequency found in [a]), what is the steady-state expression for u,? Problems 353 Figure P9.42 50 nF Section 9.7 9.43 The device in Fig. P9.43 is represented in the fre- quency domain by a Norton equivalent. When a resistor having an impedance of 5 kft is connected across the device, the value of V 0 is 5 — /15 V. When a capacitor having an impedance of -/3 kft is connected across the device, the value of I () is 4.5 - /6 mA. Find the Norton current I N and the Norton impedance Z N . Figure P9.43 I A f + Device 9.44 The sinusoidal voltage source in the circuit in Fig. P9.44 is developing a voltage equal to 247.49 cos (lOOOf+ 45°) V. a) Find the Thevenin voltage with respect to the terminals a,b. b) Find the Thevenin impedance with respect to the terminals a,b. c) Draw the Thevenin equivalent. Figure P9.44 < D 100 mH iioon JlOOmH ( ^10 A iF » ob 9.45 Use source transformations to find the Thevenin equivalent circuit with respect to the terminals a,b for the circuit shown in Fig. P9.45. Figure P9.45 240/0° V /60 ft 36 a 9.46 Use source transformations to find the Norton equivalent circuit with respect to the terminals a,b for the circuit shown in Fie. P9.46. Figure P9.46 /60 ft 4/0! A f \ 50 O 30 ft -AM/ * -/100 ft 9.47 Find the Thevenin equivalent circuit with respect to the terminals a,b for the circuit shown in Fig. P9.47. Figure P9.47 /4 0 4 ft: 4 ft: lft •AAA- 60/0° V x 4ft 4ft -/4 ft -•b 9.48 Find the Thevenin equivalent circuit with respect to the terminals a,b of the circuit shown in Fig. P9.48. Figure P9.48 2504)° V 20 ft /10 ft 9.49 Find the Norton equivalent with respect to termi- nals a,b in the circuit of Fig. P9.49. Figure P9.49 6½ 1 ()/-45°A( f ) 2ft| /lft IK 9.50 Find Z ab in the circuit shown in Fig. P9.50 when the circuit is operating at a frequency of 100 krad/s. Figure P9.50 400 nF 5 i 4 600 u,H /'A 130 ft 354 Sinusoidal Steady-State Analysis 9.51 Find the Thevenin impedance seen looking into the terminals a,b of the circuit in Fig. P9.51 if the fre- quency of operation is (25/TT) kHz. Figure P9.51 2.5 nF am— 2.4 kfl >s 39/ A 5nF : 9on :3.3 kO 9.52 Find the Norton equivalent circuit with respect to the terminals a,b for the circuit shown in Fig. P9.52 whenV y = 5/0° V. 9.53 The circuit shown in Fig. P9.53 is operating at a fre- quency of 10 rad/s. Assume a is real and lies between -10 and +10, that is, -10 < a < 10. a) Find the value of a so that the Thevenin imped- ance looking into the terminals a,b is purely resistive. b) What is the value of the Thevenin impedance for the a found in (a)? c) Can a be adjusted so that the Thevenin impedance equals 500 — /500 O? If so, what is the value of a? d) For what values of a will the Thevenin imped- ance be inductive? Figure P9.53 100/uF a«- »A S1 kfl '«% Section 9.8 9.54 Use the node-voltage method to find the steady- PSPICE state expression for v () (t) in the circuit in Fig. P9.54 if MULTISIM % = 20cos(2000r - 36.87°) V, Figure P9.54 1 mH 9.55 Use the node-voltage method to find \ (> in the cir- cuit in Fig. P9.55. Figure P9.55 240/0° V /ion /io a 50 n + v., 30 il 9.56 Use the node-voltage method to find the phasor voltage V« in the circuit shown in Fig. P9.56. Figure P9.56 -/4 n -/812 + V„ 1211 -'VW- 5/Q°A( f /4 0 I., \(Z) 20/90 ° V 9.57 Use the node voltage method to find the steady-state PSPICE expressions for the branch currents / a , i b , and / c in the MULTISIM circuit seen in Fig. P9.57 if v & = 50sinl0 f VV and V b = 25 cos (10 6 / + 90°) V. Figure P9.57 i ?"» 100 nF 1/ K lOfxH i lion i IO a 1 ^v b v g2 = 50sin(2000r - 16.26°) V. Problems 355 9.58 Use the node-voltage method to find V f) and I„ in the circuit seen in Fig. P9.58. Figure P9.58 Figure P9.63 f )6+yl3 mA |50O ( 9.59 Use the node-voltage method to find the phasor voltage V„ in the circuit shown in Fig. P9.59. Express the voltage in both polar and rectangular form. Figure P9.59 10+/10 Section 9.9 9.60 Use the mesh-current method to find the steady- state expression for v a (t) in the circuit in Fig. P9.54. 9.61 Use the mesh-current method to find the steady- state expression for i () {t) in the circuit in Fig. P9.61 if v a = 60 cos 40,000/ V, v h = 90 sin (40,000* + 180°) V. Figure P9.61 25 fiF 9.62 Use the mesh-current method to find the phasor current l g in the circuit in Fig. P9.56. 9.63 Use the mesh-current method to find the branch currents I.„ I h , I c , and I d in the circuit shown in Fig. P9.63. i dJ c A 5 a ^vw- -/1 n /in 1U/0°V **J i l n 15/0° V 9.64 Use the mesh-current method to find the steady- PSPICE s tate expression for v a in the circuit seen in «"»" Fi g< p9. 6 4 if v equa | s \ 30 CO s 10,000/ V. Figure P9.64 5mH *© 40 a 30i A + 100 O 5 »„ Sections 9.5-9.9 9.65 Use the concept of current division to find the PSPICE steady-state expression for i (> in the circuit in Mum™ pig. P9.65 if/^ = 125 cos 12,500* mA. Figure P9.65 9.66 Use the concept of voltage division to find the PSPICE steady-state expression for v () (t) in the circuit in ™LTISIM Fig p9 66 if v = 75 cos 20,000/ V. Figure P9.66 12 kO —VA>- 3.125 nF