PowerPoint Presentation Fundamentals of Electric Circuits AC Circuits Chapter 9 Sinusoids and Phasors 9 1 Introduction 9 2 Sinusoids 9 3 Phasors 9 4 Impedance and admittance 9 5 Kirchhoff’s law in the[.]
Fundamentals of Electric Circuits AC Circuits Chapter Sinusoids and Phasors 9.1 Introduction 9.2 Sinusoids 9.3 Phasors 9.4 Impedance and admittance 9.5 Kirchhoff’s law in the frequency domain 9.6 Impedance combinations FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.1 Introduction + DC sources: the main means of providing electric power up until the late 1800s + Comparing direct current (DC) and alternating current (AC): AC is more efficient and economical to transmit over long distances + Begin the analysis of circuits: source voltage or current is time varying (sinusoidal time varying excitation (sinusoid)) o A sinusoid is a signal that has the form of the sine or cosine funcions o A sinusoidal current is usually referred to as alternating current o Circuits driven by sinusoidal current or voltage sources are called AC circuits FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.2 Sinusoids v(t) Vm sin(t ) Consider the sinusoidal voltage: where: Vm: the amplitude of the sinusoid [V] ω: the angular frequency [radians/s] ωt + φ: the argument of the sinusoid φ: the phase of the sinusoid Period of the sinusoid: T Cyclic frequency f: f 2 v(t T) v(t) Hz 2f T A periodic function is one that satisfies f(t) = f(t + nT), for all t and all integers n FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.3 Phasors Sinusoids: easily expressed in terms of phasors (convenient to work with than sine and cosine functions) A phasor: a complex number that represents the amplitude and phase of a sinusoid A complex number Z can be written as Rectangular form Z x jy x: the real part of Z Polar form Z r r x y Exponential form Z re j x r cos y: the imaginary part of z ar tan y r sin Z x jy r r cos j sin y x FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.3 Phasors Mathematical functions with complex numbers: Z1 x1 jy1 r11 + Addition, subtraction: Z1 Z x1 x2 j y1 y2 + Multiplication: Z1.Z r1r2 1 + Division: Z1 r1 1 Z r2 Z1 r1 + Square root: + Complex conjugate: Note: 1 Z x1 jy1 r1 1 r1e j1 Z x2 jy2 r2 1 Z r j j FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.3 Phasors j e cos Re(e j ) v(t) Vm cos(t ) Re Vme j (t ) j sin Im(e ) v(t) Re Vmej ejt Re Vejt cos j sin V is the phasor representation of the sinusoid v(t) Time domain representation v(t) Vm cos t Phasor domain representation V Vm dv(t) dt j V v(t)dt V j Note: sinusoidal signals are of the same frequency FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.3 Phasors Phasor relationships for circuit elements Resistor R i R(t) I m cos t vR(t) R I m cos t V RI m RI Inductor L i L (t) I m cos t vL (t) L di LI m cos t 900 dt V LI m e j 90 jLI Capacitor C vC (t) Vm cos t dv iC (t) C CV m cos t 900 dt I CVm e j 90 jCV V I j C j I C FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.4 Impedance and Admittance + Impedance Z of a circuit: the ratio of the phasor voltage V to the phasor current I, measured in Ohms (Ω) Z V V ZI I Y I I YV Z V Impedances & admittances of passive elements Element Impedance Z [Ω] R ZR L Z j L C Z j C Admittance Y [S] R Y j L Y Y j C R Re Z : resistance Z R jX X Im Z : reactance G Re Y : conductance Y G jB B Im Y : susceptance FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.5 Kirchhoff’s law in the frequency domain + Both KVL and KCL hold in the frequency domain o KCL (for any node or section): n I I I n I k k0 o KVL (for any loop): n V1 V2 Vn Vk k0 FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.6 Impedance Combinations + All equivalent transformations are still correct: Z1 o N series connected impedances: Z2 Zn V V1 V2 Vn I (Z Z Z N ) V Z eq Z1 Z Z n Z1 o Voltage division: Z1 V ; V1 Z1 Z Z2 V2 V Z1 Z Z2 V FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.6 Impedance Combinations o N parallel connected impedances: 1 I I I I n V ZN Z1 Z V Z eq Z1 Z2 Yeq Y1 Y2 Yn o Current division: Z1 Z2 I I1 Z1 Z Z1 I2 I Z1 Z ; Z2 Zn FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.6 Impedance Combinations o Y∆ conversion Z12 Z1 Z13 Z Z3 Z1 Z Z12 Z1 Z Z3 Z1 Z Z13 Z1 Z Z2 Z 23 Z Z Z 2Z3 Z1 Z 23 (A delta or wye circuit is said to be balanced if it has equal impedances in all three braches) o ∆Y conversion Z1 Z12 Z1 Z Z3 Z 23 Z13 Z12 Z13 Z1 Z Z Z12 Z 23 Z2 Z1 Z Z Z13 Z 23 Z3 Z1 Z Z FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.6 Impedance Combinations + Example 1: Determine v0(t) in the circuit if v(t) = 20cos(4t - 150) R o Transform the time-domain circuit to the equivalent phasor domain vS(t) 20 cos(4t 15 ) Vs 20 150 V Z Z ZC j 25 , ZL j L j 20 Z LC C L j100 ZC ZL j C o Applying the voltage division law: V0 Z LC j100 Vs 20 150 17.1515.960 V Z R Z LC 60 j100 o Convert it to the time domain: v (t) 17.15cos(4t 15,.96 )V 60Ω v(t) C 10mF L + 5H v - FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 9.6 Impedance Combinations + Example 2: Find the current i(t) in the given circuit R2 2Ω o Convert the delta network to Y network Z1 Z L1 R2 Z C1 1.6 j 0.8 Z L1 R2 Z C1 R3 i(t) L1 R1 Z L1.R3 Z2 j 3.2 Z L1 R2 Z C1 R3 R3 R2 Z C1 Z3 1.6 j 3.2 Z L1 R2 Z C1 R3 -j4Ω 12Ω 500 V v(t) C1 R3 8Ω j4Ω L2 C2 -j3Ω j6Ω R4 8Ω o Total impedance of circuit Z R1 Z1 Z Z C // R4 Z Z L 13.6 j1 13.644.20 o The current through R1 is V 5000 0 I 66 A i t 66 cos t A Z 13.644.2 FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits Appendix: Complex numbers with calculator a jb A Type shift + = Fx500A →A=5 shift → φ = 36.870 [( - shift A a jb - 36.87 shift = →a=4 →b=3 [( - Select complex mode Fx570 shift Abs ( + i ) = shift (-) 36.87 = →a=4 →A=5 Move the cursor to the Arg position shift Arg → φ = 36.870 shift Re - Im →b=3 FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits Appendix: Complex numbers with calculator A a jb a jb A Type Pol ( Fx500MS RCL = tan = , shift →A=5 Pol ( → φ = 36.870 , RCL 36.87 = →a=4 tan = →b=3 Select complex mode Fx570MS + shift i shift + = shift ∟ 36.87 = →a=4 shift = →b=3 →A=5 shift = → φ = 36.870 Select complex mode Fx570ES + shift ENG shift A shift (-) 36.87 = a jb ... Vejt cos j sin V is the phasor representation of the sinusoid v(t) Time domain representation v(t) Vm cos t Phasor domain representation V Vm dv(t) dt j V v(t)dt