Chapter 3 Review of Basic Electrical and Magnetic Circuit Concepts • Electric Circuits • Phasors • Power, Power Factor • Fourier Analysis • Inductors and Capacitors • Magnetic Circuits • Transformers Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-1 AVERAGE POWER AND RMS CURRENT p (t ) v i T 1 Pav v i dt T 0 caso a carga seja apenasuma resist ência v R i, T 1 2 2 Pav R i dt R I RMS T 0 T I RMS 1 2 i dt T 0 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-2 Sinusoidal Steady State Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-3 Phasor Representation v(t ) V cos( t 0) utilizandoa notaçãoexponencial v(t ) V e j ( t 0 ) V cos( t 0) j V sin( t 0) notaçãofasorial v(t ) V e j 0 V0 no caso de uma carga complexa Z R j L R 2 L e j arctg ( L / R ) 2 Z Z e j Z V V 0 I I Z Z Electrónica de Potência © 2008 José Bastos Z jL R Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-4 Power, Reactive Power, Power Factor Ip Complex power S S V I* V e j 0 I e j V I e j j Iq V V 0 I I Real power is P ReS V I p V I cos Reactive power is Q ImS V I q V I sin S P 2 Power Factor 2 P V I cos cos S VI Ideally power factor should be 1.0 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-5 Example An inductive load connected to a 120V, 60Hz ac source draws 1 kW at a power factor of 0.8. Calculate the capacitance required in parallel with the load in order to bring the power factor to 0.95. for t heload P 1000W S Q P 1250VA PF S 2 P 2 750VA t hecomplexpower is S P jQ 1000 j 750 In order t oget a power fact orof 0.95one has S P j QL j QC P j (QL QC ) Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-6 Example (end) P S P (QL QC ) 0.95 2 (QL QC ) 2 P2 2 P 328.7 VA 2 0.95 t herefore QC 750 328.7 421.3VA but t hereact ivepower in a capacit ance is V2 V2 QC V 2 C Z C 1 /( C ) QC 421.3 C 2 77.6 F 2 V 120 2 60 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-7 Non-sinusoidal waveforms in steady state -current drawn from power electronic equipment is highly distorted -However -In steady state waveforms repeat with period T=1/f -f is the fundamental frequency (f1) -The current signal has many harmonics (multiples) of the fundamental -The harmonics can be calculated by Fourier Analysis Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-8 Non-sinusoidal waveforms in steady state (cont.) Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-9 Fourier Analysis -a non sinusoidal waveform f(t) repeating with angular frequency can be expressed as f (t ) F0 n 1 an bn 1 1 1 f n (t ) a0 an cos(nt ) bn sin(nt ) 2 n 1 n 1 f (t ) cos(nt )d (t ) f (t ) sin(nt )d (t ) Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-10 Fourier Analysis Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-11 Distortion in the Input Current • Voltage is assumed to be sinusoidal • Subscript “1” refers to the fundamental • The phase 1 is between the voltage and the current fundamental Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-12 Line-Current Distortion vs (t ) Vs sin(t ) is (t ) is1 (t ) isn (t ) n 1 is (t ) I s1 sin(1t 1 ) I s n sin(n1t n ) n 1 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-13 Total Harmonic Distortion -THD 1/ 2 1 T1 2 I RMS I s is (t ) T 10 because theintegralsof all thecross- product terms are zero, 1/ 2 I RMS I s21 I sn2 n 1 idist (t ) is (t ) is1 (t ) 1/ 2 I dist I s2 I I sn2 n 1 totalharmonicdistortionis defined as 2 1/ 2 s1 THD 100 I I I dist 100 2 I s1 I s1 2 s 2 s1 1/ 2 % I sn2 THD 100 2 % n 1 I s 1 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-14 POWER FACTOR - PF -starting with the definition of average power 1 P T1 T1 T 1 1 0 p(t )dt T1 0 vs (t )is (t )dt onceagain all cross product sare zero T 1 1 P Vs sin(1t ) I s1 sin(1t 1 )dt Vs I s1 cos(1 ) T1 0 t heapparentpower S is S Vs I s W e define t hepower fact oras PF P Vs I s1 cos(1 ) I s1 cos(1 ) S Vs I s Is T heDisplacement fact orDP Fis DPF cos(1 ) PF I s1 DPF Is T heP F can be expressedas PF Electrónica de Potência © 2008 José Bastos 1 1 THD2 DPF Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-15 Inductor and Capacitor Phasors (1) Capacitor vL (t ) V cos(t ) vL V e j t dvL (t ) j / 2 i (t ) C jCv L (t ) Cv L (t )e dt vL (t ) 1 j 1 j / 2 ZC e i (t ) jC C C 1 / 2 C Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-16 Inductor and Capacitor Phasors (2) Inductor i (t ) I cos(t ) i I e jt di(t ) j / 2 vL (t ) L jLi(t ) Li(t )e dt vL (t ) j / 2 ZL jL Le L / 2 i (t ) Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-17 Phasor Representation Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-18 ** Inductor and Capacitor Response Capacitor dvC iC C dt t 1 vC (t ) vC (t1 ) iC (t )dt t t1 C t1 Induct or diL vL L dt t 1 iL (t ) iL (t1 ) vL (t )dt t t1 L t1 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-19 Response of L and C Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-20 Average vL and iC in steady state -steady-state condition implies that voltage and current waveforms repeat with a time period T: v(t T ) v(t ) and i(t T ) i(t ) in case of capacitor 1 T t1 T i C dt 0 t1 averagecapacitorcurrentmust be zero in case of inductor 1 T t1 T v dt 0 L t1 averageinductor voltagemust be zero Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-21 Capacitor Voltage and Current in Steady State • Amp-seconds over T equal zero. Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-22 Inductor Voltage and Current in Steady State • Volt-seconds over T equal zero. Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-23 Ampere’s Law • Direction of magnetic field due to currents • Ampere’s Law: Magnetic field along a path Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-24 Ampere’s Law magnet icfield of int ensit yH : H dl i For most pract icalcircuit s H l N k k k i m m m densit y of magnet icflux B : B H Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-25 Direction of Magnetic Field Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-26 Flux Lines B dA A Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-27 Magnetic Flux For most practicalcases H k lk H k ( k Ak ) k k lk k Ak Bk Ak k lk k Ak k Equation H k lk N mim can be written k k lk k Ak k lk k Ak k lk k Ak m N mim m or N i m m m k lk k N i m m m Ak where is themagneticreluctance Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-28 Concept of Magnetic Reluctance Ni l /( A) Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-29 Faraday’s Law d dt v Electrónica de Potência © 2008 José Bastos d ( N ) d v N dt dt Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-30 Definition of Self-inductance L N Li d ( N ) Equationv can be written dt di vL dt Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-31 Inductance L A l N L i for a toroidalcoil Ni l/( A) 2 N L A l Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-32 Analysis of a Transformer (1) d1 v1 R1 i1 N1 dt d 2 v2 R2 i2 N 2 dt Very simple case: R1 R2 0 1 2 N2 v2 v1 N1 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-33 Analysis of a Transformer (2) 1 l1 2 l 2 being l1 , l 2 theleakagefluxes, and theflux in t hecore N1im N1i1 N 2i2 l /( A) l /( A) with N 2i2 im i1 N1 T he voltagesv1 and v2 can be written N12 di1 N12 dim v1 R1 i1 l1 /( A) dt l /( A) dt N 22 di2 N 2 N1 dim v2 R2 i2 l2 /( A) dt l /( A) dt Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-34 Analysis of a Transformer (3) notingthat N 12 2 2 N 12 N Ll1 ; Ll 2 ; Lm l1 /( A) l2 /( A) l /( A) T he voltagesv1 and v2 can be simplified dim di1 di1 v1 R1 i1 Ll1 Lm R1 i1 Ll1 e1 dt dt dt dim di2 N 2 di2 N 2 v2 R2 i2 Ll 2 Lm R2 i2 Ll 2 e1 dt N1 dt dt N1 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-35 Transformer Equivalent Circuit Lm Electrónica de Potência © 2008 José Bastos N2/N1 Lm Chapter 3 Basic Electrical and Magnetic Circuit Concepts 3-36 Ideal Transformers v1 e1 1. R1=R2=0 N2 N2 v2 e2 e1 v1 N1 N1 2. Ll1=Ll2=0 3. Core permeability = l /( A) N1i1 N 2i2 0 i2 N1 i1 N 2 v2i2 Electrónica de Potência © 2008 José Bastos Chapter 3 Basic Electrical and Magnetic Circuit Concepts N 2 N1 v1 i1 v1i1 N1 N 2 3-37