boy30444_formulacard.qxd 3/24/06 2:04 PM Page 1 Summary of Equations to Accompany INTRODUCTORY CIRCUIT ANALYSIS, Eleventh Edition, by Robert L. Boylestad © Copyright 2007 by Prentice Hall. All Rights Reserved. dc Introduction Capacitors Ϫ12 Conversions 1 meter ϭ 100 cm ϭ 39.37 in., 1 in. ϭ 2.54 cm, Capacitance C ϭ Q/V ϭ eA/d ϭ 8.85 ϫ 10 er A/d farads (F), 1 yd ϭ 0.914 m ϭ 3 ft, 1 mile ϭ 5280 ft, °F ϭ 9/5°C ϩ 32, °C ϭ C ϭ erCo Electric field strength Ᏹ ϭ V/d ϭ Q/eA (volts/meter) 12 Ϫt/t Ϫt/t 5/9(°F Ϫ 32), K ϭ 273.15 ϩ °C Scientific notation 10 ϭ Transients (charging) iC ϭ (E/R)e , t ϭ RC, uC ϭ E(1 Ϫ e ), ϭ 9 ϭ ϭ 6 ϭ ϭ 3 ϭ ϭ Ϫ3 ϭ ϭ Ϫt/t ϭ Ϫt/RC ϭ tera T, 10 giga G, 10 mega M, 10 kilo k, 10 (discharge) uC Ee , iC (E/R)e iC iCav C(DuC /Dt) Ϫ6 Ϫ9 Ϫ12 milli ϭ m, 10 ϭ micro ϭ m, 10 ϭ nano ϭ n, 10 ϭ pico ϭ p Series QT ϭ Q1 ϭ Q2 ϭ Q3, 1/CT ϭ (1/C1) ϩ (1/C2) ϩ (1/C3) ϩ иии ϩ n Ϫn Ϫn n n m nϩm Powers of ten 1/10 ϭ 10 , 1/10 ϭ 10 , (10 )(10 ) ϭ 10 , (1/CN), CT ϭ C1C2/(C1 ϩ C2) Parallel QT ϭ Q1 ϩ Q2 ϩ Q3, n m nϪm n m nm 2 10 /10 ϭ 10 , (10 ) ϭ 10 CT ϭ C1 ϩ C2 ϩ C3 Energy WC ϭ (1/2)CV Voltage and Current Inductors 2 9 2 2 2 Coulomb’s law F ϭ kQ1Q2/r , k ϭ 9 ϫ 10 Nиm /C , Self-inductance L ϭ N mA/l (henries), L ϭ mr Lo ϭ ϭ ϭ ϭ ϭ Q coulombs (C), r meters (m) Current I Q/t (amperes), Induced voltage eLav L(Di/Dt) Transients (storage) iL Ϫ19 Ϫt/t Ϫt/t t ϭ seconds (s), Qe ϭ 1.6 ϫ 10 C Voltage V ϭ W/Q (volts), Im(1 Ϫ e ), Im ϭ E/R, t ϭ L/R, uL ϭ Ee (decay), uL ϭ Ϫt/t ′ Ϫt/t ′ W ϭ joules (J) [1 ϩ (R2/R1)]Ee , t′ ϭ L/(R1 ϩ R2), iL ϭ Ime , Im ϭ E/R1 Series LT ϭ L1 ϩ L2 ϩ L3 ϩ иии ϩ LN Parallel 1/LT ϭ (1/L1) ϩ Resistance (1/L2) ϩ (1/L3) ϩ иии ϩ (1/LN), LT ϭ L1L2/(L1 ϩ L2) ϭ 2 Circular wire R ϭ rl/A (ohms), r ϭ resistivity, l ϭ feet, Energy WL 1/2(LI ) 2 2 ACM ϭ (dmils) , r(Cu) ϭ 10.37 Metric units l ϭ cm, A ϭ cm , Ϫ6 Magnetic Circuits r(Cu) ϭ 1.724 ϫ 10 ohm-cm Temperature (ͿTͿ ϩ T1)/R1 ϭ Ϳ Ϳ ϩ ϭ ϩ Ϫ ϭ 2 ( T T2)/R2, R1 R20[1 a20(T1 20°C)], a20(Cu) 0.00393 Flux density B ϭ F/A (webers/m ) Permeability m ϭ mrmo Color code Bands 1–3: 0 ϭ black, 1 ϭ brown, 2 ϭ red, 3 ϭ orange, (Wb/Aиm) Reluctance ᏾ ϭ l/mA (rels) Ohm’s law F ϭ Ᏺ/᏾ ϭ ϭ ϭ ϭ ϭ ϭ 4 yellow, 5 green, 6 blue, 7 violet, 8 gray, 9 white, (webers) Band 3: 0.1 ϭ gold, 0.01 ϭ silver, Band 4: 5% ϭ gold, 10% ϭ silver, Ᏺ ϭ ϭ ϭ ϭ ϭ Magnetomotive force NI (ampere-turns) Magnetizing 20% no band, Band 5: 1% brown, 0.1% red, 0.01% orange, ϭ Ᏺ ϭ ⌺ Ᏺ ϭ ϭ ϭ force H /l NI/l Ampère’s circuital law 0 0.001% yellow Conductance G 1/R siemens (S) 5 Flux ⌺ Fentering ϭ ⌺ Fleaving Air gap Hg ϭ 7.96 ϫ 10 Bg Ohm’s Law, Power, and Energy Ohm’s law I ϭ E/R, E ϭ IR, R ϭ E/I Power P ϭ W/t ϭ VI ϭ I 2R ϭ V 2/R (watts), 1 hp ϭ 746 W Greek Alphabet Efficiency h% ϭ (Po /Pi) ϫ 100%, hT ϭ h1 и h2 и h3 hn Energy W ϭ Pt, W (kWh) ϭ [P(W) t(h)]/1000 Letter Capital Lowercase Letter Capital Lowercase Series Circuits Alpha AaNu Nn RT ϭ R1 ϩ R2 ϩ R3 ϩ иии ϩ RN, RT ϭ NR, I ϭ E/RT, V ϭ IR Beta BbXi Yy Kirchhoff’s voltage law ⌺ V ϭ 0, ⌺ Vrises ϭ ⌺ Vdrops Gamma GgOmicron Oo ϭ Voltage divider rule Vx RxE/RT Delta DdPi Pp Parallel dc Circuits Epsilon EeRho Rr Zeta ZzSigma ⌺ j ϭ ϩ ϩ ϩ иии ϩ ϭ RT 1/(1/R1 1/R2 1/R3 1/RN), RT R/N, Eta HhTau Tt R ϭ R R R ϩ R I ϭ EG ϭ E/R T 1 2/( 1 2), T T Theta VvUpsilon Uu Kirchhoff’s current law ⌺ I ϭ ⌺ I entering leaving Iota IiPhi Ff Current divider rule I ϭ (R /R )I, (Two parallel elements): x T x Kappa KkChi Xx I1 ϭ R2I/(R1 ϩ R2), I2 ϭ R1I/(R1 ϩ R2) Lambda LlPsi Ww Series-Parallel Circuits Mu M m Omega ⍀ q Potentiometer loading RL >> RT Ammeter Rshunt ϭ RmICS /(Imax ICS) ϭ Ϫ Voltmeter Rseries (Vmax VVS)/ICS Prefixes Ohmmeter Rs (E/ICS) Rm zero-adjust/2 Methods of Analysis and Selected Topics (dc) Multiplication SI SI Factors Prefix Symbol Source conversions E ϭ IRp, Rs ϭ Rp, I ϭ E/Rs a b 18 exa E 10 ؍ Determinants D ϭ 1 1 ϭ a b Ϫ a b 1 000 000 000 000 000 000 ⏐⏐a b 1 2 2 1 peta P 1015 ؍ 000 000 000 000 000 1 2 2 ϭ ⌬ ′ ϭ tera T 1012 ؍ Bridge networks R1/R3 R2/R4 -Y conversions R 1 000 000 000 000 R ϩ R ϩ R , R ϭ R R /R′, R ϭ R R /R′, R ϭ R R /R′, R ϭ R /3 giga G 109 ؍ A B C 3 A B 2 A C 1 B C Y D 1 000 000 000 Y-⌬ conversions R″ ϭ R R ϩ R R ϩ R R , R ϭ R″/R , R ϭ R″/R , mega M 106 ؍ C 3 B 2 1 000 000 3 2 3 1 2 1 RA ϭ R″/R1, RD ϭ 3RY kilo k 103 ؍ 000 1 10؊3 milli m ؍ Network Theorems 0.001 ؊6 micro M 10 ؍ Superposition Voltage sources (short-circuit equivalent), current 0.000 001 ؊ nano n 9 10 ؍ sources (open-circuit equivalent) 0.000 000 001 ؊12 pico p 10 ؍ Thévenin’s Theorem RTh: (all sources to zero), ETh: (open-circuit 0.000 000 000 001 10؊15 femto f ؍ terminal voltage) 0.000 000 000 000 001 ؊18 ؍ Maximum power transfer theorem RL ϭ RTh ϭ RN, Pmax ϭ 2 2 0.000 000 000 000 000 001 10 atto a E Th /4RTh ϭ IN RN /4 boy30444_formulacard.qxd 3/24/06 2:04 PM Page 2 Summary of Equations to Accompany INTRODUCTORY CIRCUIT ANALYSIS, Eleventh Edition, by Robert L. Boylestad © Copyright 2007 by Prentice Hall. All Rights Reserved. ac 2 2 Sinusoidal Alternating Waveforms R-C filters (high-pass) fc ϭ 1/(2pRC), Vo /Vi ϭ R/͙RෆෆϩෆෆXෆC ЄtanϪ1(X /R) (low-pass) f ϭ 1/(2pRC), V /V ϭ X /͙Rෆෆ2 ϩෆෆXෆ2 Sine wave u ϭ Vm sin a, a ϭ qt ϭ 2pft, f ϭ 1/T, 1 radian ϭ 57.3°, C c o i C C ϭ ϫ ϭ ϫ Ϫ R radians (p/180°) (degrees), degrees (180°/p) (radians) ЄϪtan 1 Identities sin(qt ϩ 90°) ϭ cos qt, sin qt ϭ cos[qt Ϫ (p/2)], XC sin(Ϫa) ϭϪsin a, cos(Ϫa) ϭ cos a Average value G ϭ Octave 2 1, 6 dB/octave Decade 10 1, 20 dB/decade algebraic sum of areas/length of curve Transformers Effective (rms) value Irms ϭ 0.707Im, Im ϭ ͙2ෆIrms, 2 ϭ ͙ෆෆෆ ϭ Irms ϭ ͙arෆeaෆෆ[iෆ(tෆ)]ෆ/ෆT Mutual inductance M k LpLs Iron-core Ep 4.44fNpFm, Es ϭ 4.44fNsFm, Ep /Es ϭ Np /Ns, a ϭ Np /Ns, Ip /Is ϭ Ns/Np, The Basic Elements and Phasors 2 Zp ϭ a ZL, Ep Ip ϭ Es Is, Pi ϭ Po(ideal) ϭ ϩ 2 ϩ R: Im ϭ Vm /R, in phase L: XL ϭ qL, uL leads iL by 90° Air-core Zi Zp [qM) /(Zs ZL)] ϭ ϭ ϭ C: XC 1/qC, iC leads uC by 90° Power P (Vm Im /2) cos v Polyphase Systems V I cos v R: P ϭ V I ϭ I2 R ϭ V 2 /R Power rms rms rms rms rms rms ϭ ϭ ϭ ϭ ͙ෆ factor F ϭ cos v ϭ P/V I Rectangular form C ϭ A Ϯ jB Y-Y system Ifg IL IfL, Vf Ef, EL 3 Vf Y-D system p rms rms ϭ ϭ ͙ෆ ϭ ϭ ϭ ͙ෆ Polar form C ϭ CЄv Conversions C ϭ ͙Aෆෆ2 ϩෆෆBෆ2, v ϭ Vf EL, IL 3If D-D system Vf EL Ef, IL 3If Ϫ1 ϭ ͙ෆ ϭ ϭ ϭ tan (B/A), A ϭ C cos v, B ϭ C sin v Operations j ϭ ͙Ϫෆ1ෆ, D-Y system EL 3Vf, If IL, EL Ef Power PT 3Pf, 2 ϭ ϭ ϭ ͙ෆ ϭ j ϭϪ1, 1/j ϭϪj, C1 Ϯ C2 ϭ (ϮA1 Ϯ A2) ϩ j(ϮB1 Ϯ B2), QT 3Qf, ST 3Sf 3ELIL, Fp PT /ST и ϭ Є ϩ ϭ Є Ϫ C1 C2 C1C2 (v1 v2), C1/C2 (C1/C2) (v1 v2) Pulse Waveforms and the R-C Response Series and Parallel ac Circuits % tilt ϭ [(V1 Ϫ V2)/V] ϫ 100% with V ϭ (V1 ϩ V2)/2 ϭ Elements RЄ X Є X ЄϪ Pulse repetition frequency (prf) 1/T 0°, L 90°, C 90° ϭ ϫ ϭ ϩ ϩ ϩ иии ϩ ϭ ϭ Duty cycle (tp/T) 100% Series ZT Z1 Z2 Z3 ZN, Is E/ZT, Fp R/ZT ϭ ϩ Ϫ ϫ ϭ ϭ ϩ ϩ Vav (duty cycle)(peak value) (1 duty cycle) (Vb) Voltage divider rule Vx ZxE/ZT Parallel YT Y1 Y2 ϭ ϩ Ϫ Ϫ Ϫt/RC ϩ иии ϩ ϭ ϩ Є ЄϪ R-C circuits uC Vi (Vf Vi)(1 e ) Y3 YN, ZT Z1Z2/(Z1 Z2), G 0°, BL 90°, ϭ Є ϭ ϭ ϭ Compensated attenuator RpCp RsCs BC 90°, Fp cos vT G/YT Current divider rule I1 Z2IT/(Z1 ϩ Z2), I2 ϭ Z1IT /(Z1 ϩ Z2) Equivalent circuits Rs ϭ Nonsinusoidal Circuits 2 2 ϩ 2 ϭ 2 2 ϩ 2 ϭ 2 ϩ 2 Rp Xp /(Xp Rp ), Xs Rp Xp/(Xp Rp ), Rp (Rs Xs )/Rs, Fourier series f(a) ϭ A0 ϩ A1 sin qt ϩ A2 sin 2qt ϩ иии ϩ ϭ 2 ϩ 2 Xp (Rs Xs )/Xs An sin nqt ϩ B1 cos qt ϩ B2 cos 2qt ϩ иии ϩ Bn cos nqt Even function f(a) ϭ f(Ϫa), no B terms Odd function f(a) ϭ Series-Parallel ac Networks: n Ϫf(Ϫa), no An terms, no odd harmonics if f(t) ϭ f [(T/2) ϩ t], no even Employ block impedances and obtain general solution for reduced harmonics if f(t) ϭϪf [(T/2) ϩ t] network.