AC circuit analysis

AC Circuit Analysis Module: SE1EA5 Systems and Circuits Lecturer: James Grimbleby y URL: http://www.personal.rdg.ac.uk/~stsgrimb/ email: jj.b.grimbleby g y reading.ac.uk g Number of Lectures: 10 Recommended text book: David Irwin and Mark Nelms Basic Engineering Circuit Analysis (8th edition) John Wiley and Sons (2005) ISBN: 0-471-66158-9 James Grimbleby School of Systems Engineering - Electronic Engineering Slide AC Circuit Analysis Recommended text book: David Irwin and Mark Nelms Basic Engineering Circuit Analysis (8th edition) Paperback 816 pages J h Wil John Wiley and dS Sons (2005) ISBN: 0-471-66158-9 Price: £36 James Grimbleby School of Systems Engineering - Electronic Engineering Slide AC Circuit Analysis Syllabus This course of lectures will extend dc circuit analysis to deal with ac circuits The topics that will be covered include: AC voltages and currents Complex representation of sinusoids Phasors Complex impedances of inductors and capacitors Driving-point impedance Frequency response of circuits – Bode plots Power in ac circuits E Energy storage t in i capacitors it and d iinductors d t Three-phase power James Grimbleby School of Systems Engineering - Electronic Engineering Slide AC Circuit Analysis Prerequities You should be familiar with the following topics: SE1EA5: Electronic Circuits Ohm’s Law parallel resistances Series and p Voltage and current sources Circuit analysis y using g Kirchhoff’s Laws Thévenin and Norton's theorems The Superposition Theorem SE1EC5: Engineering Mathematics Complex numbers James Grimbleby School of Systems Engineering - Electronic Engineering Slide AC Circuit Analysis Lecture AC Voltages and Currents Reactive Components James Grimbleby School of Systems Engineering - Electronic Engineering Slide AC Waveforms Sine waveform (sinusoid) Square waveform Sawtooth waveform Audio waveform James Grimbleby School of Systems Engineering - Electronic Engineering Slide Frequency The number of cycles y per p second of an ac waveform is known as the frequency f, and is expressed in Hertz (Hz) Voltage or Current cycles y f = Hz Time James Grimbleby 1s School of Systems Engineering - Electronic Engineering Slide Frequency Examples: p Electrocardiogram: Hz Mains power: 50 Hz Aircraft power: 400 Hz Audio frequencies: 20 Hz to 20 kHz AM radio broadcasting: 0.5 MHz – 1.5 MHz FM radio di broadcasting: b d ti 80 MH MHz – 110 MH MHz g 500 MHz – 800 MHz Television broadcasting: Mobile telephones: 1.8 GHz James Grimbleby School of Systems Engineering - Electronic Engineering Slide Period The period T of an ac waveform is the time taken for a complete cycle: period = frequency Voltage or Current T = 0.167 s Time James Grimbleby 1s School of Systems Engineering - Electronic Engineering Slide Why Linear? We shall consider the steady-state steady state response of linear ac circuits to sinusoidal inputs Linear circuits contain linear components such as resistors, capacitors p and inductors A linear component has the property that doubling the voltage across it doubles the current through it Most circuits for processing signals are linear Analysis of non-linear circuits is difficult and normally requires the use of a computer James Grimbleby School of Systems Engineering - Electronic Engineering Slide 10 Example 5A Imaginary part IS = IL + IC IC IS 5A Real part IL -5A James Grimbleby School of Systems Engineering - Electronic Engineering Slide 230 Example An electric motor operating p g from the 50 Hz mains supply pp y has a lagging current with a power factor of 80 The rated motor current is A at 230 V so that the magnitude of 1/ZL is: IL = = = 0.02609 ZL VS 230 and the phase of 1/ZL is: ⎧1⎫ ∠⎨ ⎬ = cos −1 0.8 = ±0.6435 ⎩ ZL ⎭ Since the current lags g the voltage g the negative g p phase is used James Grimbleby School of Systems Engineering - Electronic Engineering Slide 231 Example = 0.02609 ∠ − 0.6435 = 0.02087 − j 0.01565 ZL = + j 0.01565 = j C ZC 0.01565 C= = 49.82 F × 50 Before correction: Pa = 230 × = 1380 P = pf × Pa = 0.8 × 1380 = 1104 James Grimbleby After correction: P = Pa = 1104 P 1104 IS = = = 230 VS School of Systems Engineering - Electronic Engineering Slide 232 Example 5A Imaginary g y part IS = IL + IC IC IS 5A Real part p IL -5A James Grimbleby School of Systems Engineering - Electronic Engineering Slide 233 Three-Phase Three Phase Electric Power Most ac power transmission systems use a three-phase system t Three-phase Three phase is also used to power large motors and other heavy industrial loads Three-phase consists of three sinusoids with phases /3 (120º)) apart (120 This allows more power to be transmitted down a given number of conductors than single phase A three-phase transmission system consists of conductors for the three p phases and sometimes a conductor for neutral James Grimbleby School of Systems Engineering - Electronic Engineering Slide 234 Three-Phase Three Phase Electric Power Three-phase generator James Grimbleby Three-phase load School of Systems Engineering - Electronic Engineering Slide 235 Three-Phase Three Phase Electric Power Phase-to-neutral Phase to ne tral voltage oltage v0 Phase-to-phase voltage vp v0 vp v p = 2v sin = 2v = v0 James Grimbleby v0 π (60o ) v0 2π (120o ) School of Systems Engineering - Electronic Engineering Slide 237 Three-Phase Three Phase Electric Power UK domestic supply pp y uses three -phase p with a p phase-toneutral voltage v0 of 230 V rms (325 V peak) This corresponds to a phase-to-phase voltage vp of 400 V rms (563 V peak) Each property is supplied with one phase and neutral If the phases are correctly balanced (similar load to neutral on each) h) then th the th overallll neutral t l currentt iis zero The UK electricity electricit distrib distribution tion net network ork operates at 275 kV rms and 400 kV rms James Grimbleby School of Systems Engineering - Electronic Engineering Slide 238 AC Circuit Analysis L t Lecture 10 Energy Storage James Grimbleby School of Systems Engineering - Electronic Engineering Slide 239 Energy Storage Reactive components (capacitors and inductors) not dissipate power when an ac voltage or current is applied Power is dissipated only in resistors Instead reactive components store energy During an ac cycle reactive components alternately store energy gy and then release it Over a complete ac cycle y there is no net change g in energy gy stored, and therefore no power dissipation James Grimbleby School of Systems Engineering - Electronic Engineering Slide 240 Energy Storage The voltage across a capacitor is increased from zero to V producing a stored energy E: T E = ∫ v (t ) i (t )dt v(t) dv = ∫ v (t ) C dt dt V V = C ∫ v dv d t T James Grimbleby v T i C dv i =C dt E = CV 2 School of Systems Engineering - Electronic Engineering Slide 241 Energy Storage Example: calculate the energy storage in an electronic flash capacitor of 1000 F charged to 400 V E = CV 2 = × 1000 × 10 −6 × 400 2 = 80 J James Grimbleby School of Systems Engineering - Electronic Engineering Slide 242 Energy Storage The current in an inductor is increase from zero to I producing a stored energy E: T E = ∫ v (t ) i (t )dt i(t) T di I = ∫L dt I t T = L ∫ i di i (t ) dt v i L di v =L dt E = LI James Grimbleby School of Systems Engineering - Electronic Engineering Slide 247 Energy Storage Example: calculate the energy storage in a mH inductor carrying a current of 10 A E = Li = × × 10 −3 × 10 2 = J James Grimbleby School of Systems Engineering - Electronic Engineering Slide 248 AC Circuit Analysis © J B Grimbleby 18 February 2009 James Grimbleby School of Systems Engineering - Electronic Engineering Slide 252 [...]... Capacitance C 1 C open circuit short circuit L short circuit i it open circuit i it Inductance L James Grimbleby School of Systems Engineering - Electronic Engineering Slide 30 AC Circuit Analysis L t Lecture 2 AC Analysis using Differential Equations Complex Numbers Complex Exponential Voltages and Currents James Grimbleby School of Systems Engineering - Electronic Engineering Slide 32 AC Circuit Analysis. .. Engineering Slide 35 AC Circuit Analysis A = v1 cos = v0 1+ 2 2 2 R C B = −v1 sin = RCv 0 1+ 2 2 2 R C Thus: v1 = v 0 1 1+ 2 2 2 tan = − RC R C At an angular frequency =1/RC: v0 =− v1 = 2 4 v0 ⎛ ⎞ cos⎜ t − ⎟ v c (t ) = 2 4⎠ ⎝ The output p voltage g lags g the input p voltage g by y /4 ((45°)) James Grimbleby School of Systems Engineering - Electronic Engineering Slide 36 AC Circuit Analysis V Voltage... Engineering Slide 32 AC Circuit Analysis The ac response p of a circuit is determined by y a differential equation: R v in (t ) C i (t ) v in (t ) = Ri (t ) + v c (t ) v c (t ) dv c (t ) i (t ) = C dt dv c (t ) v in (t ) = RC + v c (t ) dt dv c (t ) v c (t ) v in (t ) i + = dt RC RC James Grimbleby School of Systems Engineering - Electronic Engineering Slide 33 AC Circuit Analysis Now suppose that the input... Electronic Engineering Slide 22 Capacitors Does a capacitor have a “resistance”? v, i v (t ) = v 0 sin( t ) i (t ) = t1 v (t1) v 0 = =∞ i (t1) 0 Cv 0 cos( t ) t2 t v (t 2 ) 0 = =0 i (t 2 ) i 0 Thus “resistance” varies between ±∞: not a useful concept p James Grimbleby School of Systems Engineering - Electronic Engineering Slide 23 Capacitors The reactance XC of a capacitor is defined: v0 XC = i0 where... Electronic Engineering Slide 18 Capacitors i Insulating dielectric v Conducting electrodes Di l t i Dielectrics: James Grimbleby air i polymer ceramic i Al203 (electrolytic) C Capacitance it C q = Cv dv i =C dt School of Systems Engineering - Electronic Engineering Slide 20 Capacitors James Grimbleby School of Systems Engineering - Electronic Engineering Slide 21 Capacitors v i dv d i =C dt C Suppose... Engineering Slide 23 Capacitors The reactance XC of a capacitor is defined: v0 XC = i0 where v0 is the amplitude p of the voltage g across the capacitor p and i0 is the amplitude of the current flowing through it Thus: XC = v0 = C 0 Cv 1 1 = C 2 fC The reactance of a capacitor is inversely proportional to its value and to frequency James Grimbleby School of Systems Engineering - Electronic Engineering... reactance XC of an inductor is defined: v0 XC = i0 where v0 is the amplitude of the voltage across the inductor and i0 is the amplitude of the current flowing through it Thus: v0 Xc = = v0 / L L = 2 fL The reactance of an ind inductor ctor is directl directly proportional to its value and to frequency James Grimbleby School of Systems Engineering - Electronic Engineering Slide 29 Resistance and Reactance... the expression for vc: v c (t ) = v1 cos t cos − v1 sin t sin dv c (t ) = − A sin t + B cos t dt James Grimbleby = A cos t + B sin t School of Systems Engineering - Electronic Engineering Slide 34 AC Circuit Analysis The differential equation becomes: v0 A B − A sin t + B cos t + cos t + sin t = cos t RC RC RC Comparing p g the coefficients of sin t and cos t on both sides of the equation: − A RC + B... linear circuit will not change the waveform or frequency of a sinusoidal input (the amplitude and phase may be altered) Power is generated as a sinusoid by rotating electrical machinery Sinusoidal Si id l carrier i waves are modulated d l d to transmit i information (radio broadcasts) Any periodic waveform can be considered to be the sum of a fundamental pure sinusoid plus harmonics (Fourier Analysis) ... of Sinusoids The sinusoid can have a phase term : v (t ) = v 0 sin( t + A phase shift v ) is equivalent to a time shift - / = v 0 sin( t + ) t v 0 sin( t ) The phase is positive so the red trace leads the green trace James Grimbleby School of Systems Engineering - Electronic Engineering Slide 15 Resistors Ceramic tube coated with Conductive film i Metal end cap v Fil Film: carbon b metal t l oxide id ... Electronic Engineering Slide AC Circuit Analysis Syllabus This course of lectures will extend dc circuit analysis to deal with ac circuits The topics that will be covered include: AC voltages and currents.. .AC Circuit Analysis Recommended text book: David Irwin and Mark Nelms Basic Engineering Circuit Analysis (8th edition) Paperback 816 pages J h Wil John Wiley... Electronic Engineering Slide AC Circuit Analysis Lecture AC Voltages and Currents Reactive Components James Grimbleby School of Systems Engineering - Electronic Engineering Slide AC Waveforms Sine waveform