Solutions to Problems

Chapter l 2.6 1.1 (a)M-IC3T3/2;(b)M-IL-3y4J2; (c)ML2 r 2r 1 1.3 (a) 102.5 W; (b) 11.8 V; (c) 5900 W; (d) 6000 w 1.4 Two in series connected in parallel with two others. The combination connected in series with the fifth 1.5 6 V; 16 W 1.6 2 A; 32 W; 8 V 1.7 ~ D.;!b n 1.8 in;~ n 1.9 67.5 A, 82.5 A; No.1, 237.3 V; No.2, 236.15 V; No. 3, 235.4 V; No. 4, 235.65 V; No.5, 236.7 V Chapter 2 2.1 3il + 2i2 = 0; Constraint equation, 15vl - 6v2- 5v3- 4v4 = 0 i 1 - i2 = 5; i 1 = 2 A; i2 = - 3 A -6v1 + l6v2- v3- 2v4 = 0 2.2 va\+va!=-5;va=2i2;va=-3il; -5vl- v2 + 17v3- 3v4 = 0 -4vl - 2vz- 3v3 + l8v4 = 0 i1 = 2 A;i2 = -3 A 2.7 (a) 1 V ±and 2.4 .!?.; (b) 10 V +and 2.3 v13 + v2 2 = 0 (from supernode 1 + 2); 10 n; (c) 12 V ±and 4 n constraint equation v1- v2 = 5; v1 = 2 V; v2=-3V 2.8 3470 w 2.5 2.9 384 000 w 2.10 (a) 1 .!?., lSi W; (b) Ra = 0 (negative values of Ra not considered), 162 W 2.11 (a) 8 .!?.; (b),~ W; (c) 4 .!1. 2.12 ! n. By the principle of superposition if 1 A is fed into any junction and taken out at infinity then the currents in the four branches adjacent to the junction must, by symmetry, be i A. Similarly, feeding in 1 A at infinity and extracting it at a junction causes currents of! A in branches adjacent to the junction. If both these operations are 90. carried out simultaneously at adjacent P.2.5 junctions, the current in the branch joining The network is unique the junctions is ! A and the potential across 200 SOLUTIONS TO PROBLEMS 201

it is! V. Thus I A is passed between Junctions Otapter 4 which have a potential difference of 2 V and 4.I 36J;VtF=V2F=4V;24J the effective resistance is! n 2.I3 ! n. By similar reasoning to the above 4.2 At t = 0: ie3 = ics = ~ A; vc3 = vcs = 0 V; iL2 = iL7 = 0; VL2 = ~ V; 2.14 (a)Rt=R2=R3=~n. vL7 = ~ V; at t = ~: vL2 =vL7 = ic3 = ics = 0; (b) Gt = G2 = G3 = i S iL2 = iL7 = 5 A; vc3 = 20 V; vcs =50 V; 2.I5 .!,} n t 2.I6 (a)! M and i S; (b) 4 A~ and i S; node a: (va- Vb) +! J(va- vc) dt +Jbva = 10; (c) 3 At and! S 0 2.I7 2.1mA- d 2.18 0.67mA ~ node b: (vb - va) + 3-(vb - vct) = 0; dt 2.19 0.513 A~ 2.20 ru = ~ n; '12 = r21 =' n; r22 = ~ n; t Rt = J,fi 51; R3 =' 51; R2 = ~ H node c: ! J (vc - va) dt + ~(vc - vct) 2.21 gu = tf S;gt2 = g2t =- fl1 S;g22 = !¥ S; 0 dvc G1 = f¥ S; G3 = WS; G2 = fl S +5-=0· dt ' Otapter 3 t 3.I (a) 2~ F; (b) 3! F; (c)~ F node d: 3-d (vd- vb) +-I f(vd- ve) dt 3.2 (a) j H; (b)~ H; (c)~ H dt 7 3.3 133 pF + 556 pF = 689 pF total; the 0 same value of 689 pF 1 + -(vd- vc) = 0; 3.4 Eight possible connections: (a) coil one 4 alone-4 H; (b) coil two alone-2 H; (c) series opposition-4 H; (d) series addition-S H; t (el parallel addition-;{ H; (f) parallel reversed node e: -ve + -1 f (ve - vct) dt = 0 -g H (g) coil two short-circuited H; 6 7 -1 0 (h) coil one short-circuited-;{ H 3.5 (a) 11 H; (b) 4~ H 3.6 4.3 (a) At t = 0: iL4 = iL6 = vc2 = vc3 = 0; VL4 = 80 V; VL6 = 40 V; ic2 = ic3 = 4 A; 3H 3H 3H (b) at t = ~: ic2 = ic3 = VL4 = VL6 = 0; iL4 = iL6 = 5 A; vc2 ~50 V; vc3 = 75 V;

t (c) 5io: + 4 -(io:-d i(3) +-1 J(ia- i-y) dt = 100; $r"· dt 3 FJ:E(a) 0

t 10i(3 + 1- f (i(3) dt + IO(i(3- i-y) 0 d +4-(i(3-ia)=O; dt

di lO(i-y - i(3) + 6 ~ + 5i-y (b) t dt P.3.6 + t J (i-y - ia) dt = 0 0 202 AN INTRODUCTION TO ELECTRICAL CIRCUIT THEORY

4.4 Four meshes; three topological and t one pseudo mesh; node A:.!:. I (va - vb) dt + (va - vc) 40 d t mesh a: 10(i0 - i0 ) + 2 -(i0 - i(3) dt + ~ 1(va - vc) dt = 0; t t + t J(iQ- i-y) dt = 0; 1 [ (vb - 20) 0 node B: 4 J (vb - va) dt + 2 - 0; t 0 d d I [ vd) 0; mesh{3: 2-(i(J-i0 }+i(3+- J {i(J-i-y)dt node C: (vc- va) + 2 -(vc- = dt 3 0 dt t + 4(i(3- i")') = 0; node D: 2 -(vd-d vc) +-1 f vd dt = 0; dt 2 t 0 1 . t mesh -y: S j (i-y - i0 ) · dt + 4(i-y - i(3) node E: (ve - va) dt + (ve + 3)3 = 0 0 t I 0 di 13 8 + '::'::1.+ 6i = 4.6 vc 1 = 4(1 - e- 1 ); 36 J; 24 J; 12 J 7 dt "Y o·, 4.7 (a) v = 4(1 - e-t/4 ); (b) v = 4 e-t constraint equation; io = 10 4.8 (a} 0; (b) 8 V; (c) i = 1.6 e-t/6; (d) 3.2 V 4.5 4.9 (a)for 0 <:. t <:. 1 v = 5 (1 - e-2t); fort;;;. 1 v = 4.32 e-2(t-I); (b) 4.32 V; 0.588 V; 4.32 x e-18 V"" 0 4.10 (a} v = 6(1 - e-t/3); (b) v = 8(1 - e-t/4 ) 4.11 (a) i = 2(e-t/4 - e-t/2); (b) 2.77 s; (c) 0.5 A; (d) 1 F 4.12 (a) 0; (b) 0; (c) 0.5 V; (d) Vba = 2(e-tl2 - e-t) 4.13 v = 3 + e-t/S 4.14 (a) v = 1.03 e-t/4 sin (0.97t); (b) 2; (c) 4 S;

4.1,5 i 1 = 6t + 0.6(1 - e-2t); i2 = 1.2(1- e-2f); (a.) at t= O;i, = i2 = O;at t = oo;i2 = 1.2,i1 = oo

Chapter 5

(b) { w1o-1 3 - 10-3 w }) P.4.5 x cos + tan -I (wt 0.5 SOLUTIONS TO PROBLEMS 203

(d) w 0 = 1000 rad s-1 ; Q = 2; B = 500 rad s-1. 5.17 (a) 5 mH; (b) 20 nand 5 mH; For iL = 0.707(iL)m8 x, w = 1167 or (c) 10.6 n 625 rad s-1 5.18 G = 1.72 S; C = 6.9 mF; 0.25 W 5.3 w = 4640 rad s-1 and IL(max) = 1.75 Is 5.4 (a) 8 W; 0.895; 1.414 A; 6.32 V; 5.19 iL=2.5sin (2r-i}P=9~W (b) 25 mW; 0.707; 5 rnA; 7.07 V; (c) 5 mW; 5.20 0.414 1; 70.7 J.LA; 70.7 V w 5.5 (a) 0 V mean; 7.07 V r.m.s.; (b) 0 V mean; 5.78 V r.m.s.; (c) 0 V mean; 10 V r.m.s.; Chapter 6 (d) 3.19 V mean; 5 V r.m.s.; (e) 6.38 V mean; 6.1 (a) z 11 = z 22 = !

a parameters 168 Charge 2 ABCD parameters 170 conservation of 10 Admittance 144 stored by capacitor 62 short-circuit parameters 166 Circuit 1, 8 triangle 15 3 frequency domain 136 Alternator 127 ideal element 9, 18 Alternating current laws 1 instantaneous value 119 lumped element 18 mean power 120 non-linear 192 mean value 119 open 18 polyphase 184 short 18 r .m.s. value 119 Coefficient of coupling 73 three-phase 185 Complementary function 77 Ammeter 51 Complete solution 77 Ampere 1 Complex conjugate 100, 151 Amplidyne 196 Complex numbers 132 Amplitude frequency response 113 exponential form 132 Apparent power 151 graphical representation 132 Approximate differentiator 93 modulus 133 Approximate integrator 93 real part 13 3 Argand diagram 132 representation of sinusoid 132 Argument of 138 Complex phasor 134 Average values 119 addition 134 argument 138 B field 128 division 135 b parameters 169 multiplication 135 Bandwidth 114 polar form 134 Basic method of network analysis 21, 81 rectangular form 134 Bilateral element 51, 193 subtraction 135 Branch 18 Complex plane 132 Bridge network 182 Complex power 151 Maxwell 183 Compound sources 35 Wheatstone 183 Conductance 5, 9,144 frequency domain 136 Capacitance 4, 61 parameters 54 dimensions 62 parallel connected 12 energy stored 63 series connected 11 final condition 80 Conduction 5 frequency domain 139 electron theory 5 impedance 140 Conductors 5 initial conditions 78 metallic 4 parallel connected 64 Constant sources 9 reactive power 125 Constrained mesh current 3 7 series connected 64 Constrained node voltage 38 symbol 62 Constrained equations 37, 38 units 62 Coulomb 2 volt-ampere equations 62, 139 Coupled coils 7 3, 17 3 Capacitor 61 Cramer's rule 41 parallel-plate 62 Critical damping 99 205 206 INDEX Current Faraday 66 balanced three-phase 189 Field density 5 electric 62 divider 15 magnetic 66 ideal source 4, 10 Final conditions 80 Kirchhoff's law 10 capacitance 80 line 189 inductance 80 magnification 118 First-order network 78 phase 189 differential equations 78 physical source 34 final conditions 80 sign convention 9 initial conditions 78 transformation ratio 174 natural response 8 3 triangle 153 steady-state response 86 total response 84 Damped natural frequency 1 02 Flux linkage 67 Damping 97 Frequency domain 127,129 constant 101 analysis 136 Delta connection 43, 187 phasor 133 load 189 source 188 Graph, topological 20 Delta-star transformation 43 g parameters 54, 168 41 165 Differential equations 77 of first-order network 7 8 Half-power points 11 7 of second-order network 95 Henry 67 Differentiation of phasor 131 h parameters 167 Differentia tor, approximate 9 3 Hybrid parameters 167 Diode 193 Ideal circuit element 4, 9, 18 ideal 193 capacitance 62 semi-conductor 193 current source 10 Dimensional analysis 3 inductance 67 Dividers 15 mutual inductance 71 current 15 resistance 9 potential 15 voltage source 9 Dot convention 72 Ideal transformer 17 3-182 Driving point admittance 166 phasor diagrams 177 Driving point impedance 144, 165 volt:...ampere equations 17 3 Duality 29, 70 Impedance 144 Dual networks 29 bridge 182 Duals 29 of capacitance 140 Dynamo 33 driving-point 144, 165 Effective value 120 of inductance 138 Electrical quantities transformation 175 Electric field 4 triangle 151 Electromagnetic induction 66 three-phase 188 Electron 2 Inductance 4, 66 free 4 dimensions 67 theory of conduction 5 energy stored 68 Energy 2 final conditions 80 conservation of 11 frequency domain 137 stored.in capacitance 63 impedance 138 stored in inductance 68 initial conditions 79 stored in mutual inductance 73 mutual 70, 139 connected 69 Equivalent T and 7T circuits 52 171 parallel Equivalent resistance 12 ' reactive power 122 Euler's identity 100, 132 self 66 Exponential decay 84 series connected 68 Exponential rise 90 symbol 67 units 67 Farad 62 volt-ampere equations 67, 138 INDEX 207 Inductor, iron-cored 193 Natural behaviour 77 Initial conditions 78 Natural frequency 102 capacitance 78 damped 102 inductance 7 9 undamped 102 Instantaneous value 3, 119 Natural response 83 Insulator, 5 first-order network 83 Integration of phasor 131 second-order network 96 Integrator, approximate 9 3 Network 18 Integra-differential equations 61 analysis 18, 21 Internal resistance of source 39 characterisation 42, 164 Inverse hybrid parameters 167 graph 20 Iron-cored inductor 193 ladder 32 one- 47 j notation 132 planar 24 Joule 3 topology 18 Kirchhoffs laws 8 theorems 18, 19, 40, 83 current 10 two-port 51 frequency domain 140 variables 21 voltage 11 Neutral 185 current 189 Ladder network 32 Nodal analysis 26, 82, 147 Lagging power factor 124 frequency domain 141 Lattice structure 5 Node 10, 18 Leading power factor 127 ground 20,26 Linear differential equation 77 psuedo 38 Linearity 40 reference 20 Linearisation, piecewise 195 super 38 Line quantities topological 38 current 189 voltage 26 voltage 186 constrained 38 Load line 192, 195 Node-pair voltage 21 Loop 18 Node-to-datum voltage 26 current 21,23 Node-to-ground voltage 26 independent 19,23 Non-linear circuits 192 Lumped element 18 Non-linear devices 4, 19 3 Nonlinearity 40 Magnetic field 66 Non-linear-resistor 7, 193 :!5, 166 Norton's theorem 49, 83, 143 conductance 28 resistance 25 Ohm 5 Maximum power transfer 39, 156 Ohm, Georg Simon 5 Maxwell bridge 183 Ohm's law 5, 138 Mean value 119 One-port network 47 Mesh 24 Open circuit 18 analysis 23, 82, 141 driving-point impedance 165 current 23 impedance matrix 166 constrained 3 7 165 in frequency domain 141 transfer impedance 165 equations 37 Operating point 196 pseudo 37 Overdamping 98 topological 37 Metadyne 34 Parallel circuit Mutual inductance 70, 139 capacitance 64 dot notation 72 conductance 13 energy stored 7 3 inductance 69 equivalent voltage generator 72 resistance 13 frequency domain 139 Parallel connection 13 reactance 139 Parameters symbol 71 a 168 volt-ampere equations 71, 139 ABCD 170 208 INDEX

Parameters-continued Reactance 144 b 169 mutual 139 conductance 54 Reactive power 122 hybrid 167 Reciprocity 51 inverse hybrid 167 theorem 51 resistance 53 Reference axis 129 transmission 168 Reference node 26 y 166 Resistance 4, 144 z 165 effective 12 Particular integral 77 equivalent 12 Passive element 8 frequency domain 136 Pentode thermionic valve 34 ideal element 5, 9 Periodic time 120 parallel connected 13 Periodic waveform 110 power dissipation 7 Phase 113 series connected 12 angle 114 symbol 6 difference 113 Resistivity 6 frequency response 113 Resistor 4 lag 124 non-linear 7, 19 3 lead 125 power dissipation 7 Phasor 129, 132 Repetitive-step response addition 130, 135 CR circuit 93 angle 131 LR circuit 91 complex 132, 134 Resonance 113, 114, 157 conjugate 151 Resonant frequency 113 diagram 129,145,177 RMS value 119, 121 differentiation 131 Roots 98 division 135 complex 100 integration 131 equal 99 magnitude 131 real 98 multiplication 135 Rotating vector 127 polar form 134 rectangular form 134 Secondary winding 17 3 subtraction 130, 135 Second-order network 95 Photo-electric cell 34 differential equations 95 n network 54, 171 natural response 96 n-T equivalence 56 step response 102 Piecewise linearisation 195 Semi-conductor 5 Planar network 24 diode 193 Potential 2 Series circuit difference 2 capacitance 65 divider 15 conductance 12 Polyphase circuit 184 inductance 68 Power 2, 150 resistance 12 apparent 150 Series connection 12 complex 151 Short circuit 18 dissipated by resistor 7 admittance matrix 167 mean a.c., 119 166 polyphase 184 driving-point admittance 166 reactive 122, 125, 150 Siemen 5 real 122, 150 S.l. units 1 three-phase 191 Sign convention triangle 151 current 8 Power factor 124 current source 10 lagging 124 duality 31 leading 127 energy 9 Primary winding 1 7 3 mutual inductance 72 reactive power 125 Q-factor 104, 114 voltage 8 parallel circuit 115 voltage source 10 series circuit 114 Sinusoid 110, 121, 127 INDEX 209 Sinusoid-continued second-order network 1 02 average value 120 Transformer, ideal 17 3 complex number representation impedance transformation 175 132 phasor diagrams 177 phasor representation 129 volt-ampere equations 173 r.m.s. value 121 Transient response 77 Sinusoidal steady-state response 87, 110 first-order network 78 LR circuit 87 second-order network 95 Solenoid 66 Transistor 34, 196 Source 4, 32 Transmission line 145 compound 35 Transmission parameters 168 current 4, 34 Triangular waveform 121, 177 equivalence 35 Tungsten-filament lamp 7 ideal current 4 Turns ratio 1 7 3 ideal voltage 4 Two-port networks 51, 164 physical 33 1T equivalent circuit 53 resistanceless 36 T equivalent circuit 53 substitutions 36 Two-phase system 184 three-phase 185 voltage 4, 33 Undamped natural frequency 102 Square wave 91, 94, 110 Underdamping 100 Star connection 4 3 Unknown variables (2b), 19, 21 Star-delta transformation 43 Units 1 Star point 185 Unit vector 132 Steady-state 77 response 110 Step response 86 VAr 151 CR circuit 87 Volt 2 LR circuit 86, 89 Voltage repetitive 91, 9 3 divider 15 RLC circuit 102 ideal source 4, 9 Supernode 38 induced 127 Superposition 41,83 Kirchhofrs law 11 theorem 41 line 186 Susceptance 144 magnification 118 phase 185 Thermistor 7 physical source 3 3 Thevenin theorem 47, 83, 143, 149 reference signs 8 Three-phase network 184 sign convention 8 balanced currents 188 transformation ratio 17 4 balanced voltages 184 power 191 Watt 3 Time constant 84 Wheatstone bridge 183 CR circuit 84 LR circuit 86 Topological graph 20 Y connection 43 Topological mesh 37 voltage source 184 Topological node 38 y parameters 166 Topology 18 Total response 77 Zero damping 101 fust-order network 89 z parameters 165