AC Network Analysis Learning objectives 1. Compute current, voltage and energy stored in capasitors and inductors 2. Calculate the average and rms value of periodic signals 3. Write the differential equation for circuits containing inductors and capasitors 4. Convert time domain sinusoidal voltages and currents to phasors and vice versa 5. Represent circuits using impedances 6. Apply known circuit analysis methods to AC circuit in phasor form 1 2 Structure of parallel-plate capacitor Ideell Kondensator (Capasitor) Kretsparameter: Capasitans C Kan både ta opp og avgi elektrisk energi. Lagrer energi i form av elektrisk felt 3 4 Viktig observasjon for Ideell Kondensator (Capacitor) i Serie og Parallel kobling av Kapasitanser v C Tillater ikke sprang (diskontinuitet) i spenningen Tillater imidlertid sprang i strømmen gjennom kapasitansen Representeres som en åpen krets ved konstant spenning 5 6 1
Calculation of energy stored in a capasitor Iron-core inductor Magnetic flux lines Iron core inductor i (t) L _ Circuit symbol v L (t) = L di dt 7 8 Inductanse and practical inductor Ideell Spole (Inductor) Kretsparameter: Induktans L Kan både ta opp og avgi elektrisk energi. Lagrer energi i form av magnetisk felt 9 10 Viktig observasjon for Ideell Spole (Inductor) Serie og Parallel kobling av Induktorer C v _ 1 2 L Tillater ikke sprang (diskontinuitet) i strømmen Tillater imidlertid sprang i spenningen over induktansen Representeres som en kortsluttning ved konstant strøm 11 12 2
Energy stored in an Inductor 13 14 Analogy between electrical and fluid resistance R v 1 v 2 i Rf p 1 q f p 2 p 1 p 2 q f Analogy between fluid capacitance and electrical capacitance i C v1 v _ v 2 p2 q f gas P 1 q f Cf p2 p _ p 1 15 16 Analogy between fluid inertance and electrical inertance Analogy between electrical and fluid circuits v 1 i v L v 2 p 2 q f p 1 I f 17 18 3
Time-dependent signal sources Sinusoidal waveform v(t) _ i (t) v ( t), i( t) _ Generalized time-dependent sources Sinusoidal source 19 20 Phase angle / phase shift Periodic signal waveforms The sine wave V m sin(ωt θ) leads V m sinωt by θ rad. 21 22 Average and RMS values RMS Value of Sinusoidal Waveform The rms, or effective, value of a current (or voltage) is the DC (or DC voltage) that causes the same average power to be dissipated by the resistor. 23 24 4
Exercise Analysis of circuits containing dynamic elements Given the sinusoidal voltage: o vt ( ) = 325cos(100π t 30 ) V 1. What is the period of the voltage? 2. Calculate the frequency? 3. What is the value of the voltage at t=3.333ms? 4. Calculate the rms value of the voltage? 25 26 The steady-state response of circuits excited by sinusoidal sources The steady-state response of circuits excited by sinusoidal sources cont. 27 28 The steady-state sinusoidal response In a sinusoidally excited linear circuit, all branch voltages and currents are sinusoids at the same frequency as the excitation signal. The amplitudes of these voltages and currents are a scaled version of the excitation amplitude, and the voltages and currents may be shifted in phase with respect to the excitation signal. Relationship between polar and rectangular coordinates of a complex number jθ ρe = ρcosθ jρsinθ ρ jθ e = a jb b θ = tg, ρ = a b a 2 2 29 30 5
The steady state response The steady state response Complex Forcing Function From Eulers identity and and the superposition theorem we find: A complex forcing function may be considered as the sum of a real and an imaginary forcing function The real part of the complex response is produced by the real part of the forcing function. The imaginary part of the response is produced by the imaginary part of the complex forcing function The sinusoidal forcing function V m cos (ωt θ) produces the steady-state response I m cos (ωt θ). The imaginary sinusoidal forcing function j V m sin (ωt θ) produces the imaginary sinusoidal response j I m sin (ωt θ). The complex forcing function V m e j(ωt θ) produces the complex response I m e j(ωt θ). 31 32 Definition of a Phasor Phasor Transformation En prosess hvor en sinusformet strøm eller spenninger blir konvertert fra en størrelser i tidsplanet til en kompleks størrelse i frekvensplanet Time domain vt () = V cos( ωt φ ) m j( t ) { m } vt () = Re Ve ω φ 33 34 Vi merker oss at frekvensplan representasjonen ikke eksplisitt inneholder informasjon om den aktuelle frekvensen til sinussignalet. Frekvensen er kjent på forhånd og er derfor unødvendig i representasjonen j V = Ve φ Frequency domain m V = V φ m Phasor diagram Complex eksponetial function Ve jωt 35 36 6
A graphical representation of two sinusoids v 1 and v 2 The magnitude of each sine function is represented by the length of the corresponding arrow, and the phase angle by the orientation with respect to the positive x axis. In this diagram, v 1 leads v 2 by 100 o 30 o = 130 o, although it could also be argued that v 2 leads v 1 by 230 o. Summasjon av visere 37 It is customary, however, to express the phase difference by an angle less than or equal to 180 o in magnitude. 38 Diagrammet viser at summasjon av sinus størrelser kan illustreres geometrisk ved hjelp av visere. Motstand representasjon i frekvensplanet Spole representasjon i frekvensplanet i(t) v(t) - R Spenningen over resistansen blir i frekvensplanet : V= RI 39 40 Kondensator representasjon i frekvensplanet Definisjon Impedans Resistans og Reaktans 41 42 7
Definisjon Admittans Konduktans og Suseptans Enkelte ganger (f.eks i forbindelse med paralellkobling av impedanser) kan det være hensiktsmessig å innføre størrelsen admittans Resistor, Capasitor, and inductor in time and in the phasor domain (a) (b) Admittansen til et element defineres som den inverse av impedansen: 43 Måleenheten for admittans (konduktans og suseptans) er Siemens (S) 44 (c) In the phasor domain, (a) a resistor R is represented by an impedance of the same value; (b) a capacitor C is represented by an impedance 1/jωC; (c) an inductor L is represented by an impedance jωl. Impedances R, L and C in the complex plain Bestemmelse av impedansen For å bestemme impedansen til et ukjent kretselement eller en toport må vi først transformere støm og spenning til frekvensplanet. Dvs.utrykke støm og spenning som komplekse visere 45 46 Impedansen beregnes som forholdet mellom spenningsviseren og strømviseren. Impedansen blir generelt en kompleks størrelse. Måleenheten er ohm Merk: Impedansen er ikke en viser Den imaginære delen til impedansen kalles reaktansen Numerical example Calculation of impedans Kirchhoffs laws in the frequency domain This circuit is operating in the sinusoidal steady state with v(t) = 50 cos(500t) V and i(t) = 4 cos(500t < 60 ) A. Find the impedance of the elements in the box. 47 48 8
Example Kirchhoffs laws in the frequency domain Flow diagram for Phasor circuit Analysis. 49 50 Series connection of impedances Numerical example Series connection of impedances 51 52 Numerical example Series connection of impedances 53 54 Design the voltage divider so that an input v S =15cos2000t V produces a steady-state output v 0 (t) = 2sin2000t V. 9
Parallel connection of impedances Figure 15-15 (p. 682) 55 56 Parallell connection of two impedances Numerical Example Steady-state currents 57 58 Find the steady-state currents i(t), i C (t), and i R (t) for v S = 100 cos 2000t V, L = 250 mh, C = 0.5 µf, and R = 3 kω. Numerical Example steady-state currents Find the i(t), i C (t), and i R (t) for v S = 100 cos2000t V, L = 250 mh, C = 0.5 µf, and R = 3 kω. Figure 4.37 59 60 10
Phasor diagram (a) A phasor diagram showing the sum of V 1 = 6 j8 V and V 2 = 3 j4 V, V 1 V 2 = 9 j4 V = 9.85 24.0 o V. (b) The phasor diagram shows V 1 and I 1, where I 1 = YV 1 and Y = 1 j S = 1.4 45 o S. The current and voltage amplitude scales are different. 61 62 An AC circuit Figure 4.41 63 11