Lesson 1: Phasors and Complex Arithmetic

Transcription

1 //06 Lesson : Phasors an Complex Arithmetic ET 33b Ac Motors, Generators an Power Systems lesson_et33b.pptx Learning Objectives After this presentation you will be able to: Write time equations that represent sinusoial voltages an currents foun in power systems. Explain the ifference between peak an electrical quantities. Write phasor representations of sinusoial time equations. Perform calculations using both polar an rectangular forms of complex numbers. lesson_et33b.pptx

2 //06 Ac Analysis Techniques Time function representation of ac signals Time functions give representation of sign instantaneous values v(t) e(t) i(t) V E I sin( t v ) sin( t ) sin( t ) i e Voltage rops Source voltages Currents Where V = imum ( peak) value of voltage E = imum (peak) value of source voltage I = imum (peak) value of current v, e, i = phase shift of voltage or current = frequency in ra/sec Note: =pf lesson_et33b.pptx 3 Ac Signal Representations Ac power system calculations use effective values of time waveforms ( values) Therefore: V V E Where E I I So quantities can be expresse as: V E I 0.707V 0.707E 0.707I lesson_et33b.pptx 4

3 //06 Ac Signal Representations Ac power systems calculations use phasors to represent time functions Phasor use complex numbers to represent the important information from the time functions (magnitue an phase angle) in vector form. Phasor Notation V V or V V I I or I I Where: V, I = magnitue of voltages an currents = phase shift in egrees for voltages an currents lesson_et33b.pptx 5 Ac Signal Representations Time to phasor conversion examples, Note all signal must be the same frequency Time function-voltage v(t) 70sin(377 t 30 ) Time function-current Phasor i(t) 5sin(377 t 0 ) Fin magnitue V I V V 0.30 V Fin magnitue A Phasor I A Phase shift can be given in either raians or egrees. To convert, this conversion: use egree = p/80 raians. lesson_et33b.pptx 6 3

4 //06 Complex Number Representations Polar to rectangular conversion Rectangular form of complex number +j a r cos( ) z a jb -real r +real b r sin( ) -j To convert between polar form an rectangular form, use calculator P-R an R-P keys or remember concepts from trigonometry R-P conversion Magnitue: z a b P-R conversion z r (cos( ) jsin( )) Phase Angle: b tan a lesson_et33b.pptx 7 Complex Number Representations Example - Rectangular-to-polar (R-P) conversion using trigonometry. Fin the polar equivalent of the complex number z. z 30 j0 Solution a 30 b 0 z a b Magnitue Phase angle b tan a 0 tan Watch for the location of the phasor when making P-R or R-P conversions. Some calculators return tan - into st an 4th quarants only. lesson_et33b.pptx 8 4

5 //06 Rectangular-to-Polar Conversion Check the angle given from R-P conversion by geometric interpretation. If real an imaginary parts both negative, angle is in Quarant III Example -: Fin the polar form of z=-4+j5 Fin r z ( 4) Compute phase angle 5 tan This tan - function computes the angle in Quarant IV. The actual angle is 80 egrees from this value ( - real, + imaginary is Quarant II) lesson_et33b.pptx 9 Polar-to-Rectangular Using Trig Functions Conversion Equation z a jb r [cos( ) jsin( )] Example -3: Convert I to rectangular form a jb r [cos( ) jsin( )] a jb 50[cos(53. ) jsin(53. )] a jb 30 j40 Ans lesson_et33b.pptx 0 5

6 //06 Complex Number Arithmetic Properties of the imaginary operator j= - +j The operator j translates physically into a 90 o phase shift j = 90 o... a 90 egree phase lea -j = -90 o... a 90 egree phase lag 90 lea -j Also /j = -j an /-j = j j (-j) = 90 lag lesson_et33b.pptx Complex Number Arithmetic Complex Conjugate-reflection about the real axis +j -j z z * Conjugate (rectangular form) * (a jb) a jb Change sign on imaginary part b Conjugate (polar form) * (z ) z -b Change sign on angle lesson_et33b.pptx 6

7 //06 Complex Number Arithmetic Aition an Subtraction of Complex Numbers For calculators without complex number arithmetic ) convert both numbers to rectangular form ) a/subtract real parts of both numbers an imaginary parts of both numbers Multiplication an Division of Complex Numbers For calculators without complex number arithmetic ) convert both numbers to polar form ) multiply/ivie magnitues 3) a angles for multiplication, subtract angles for ivision Inverting a Complex Number ) convert number to polar form z ) perform ivision (0 o ) / (z ) = /z - lesson_et33b.pptx 3 Complex Number Arithmetic Example -4: Given the sinusoial time functions an complex numbers below: v (t) 340sin(377 t 0 ) v (t) 77sin(377 t 30 ) Z 70 j0 I 3 j Fin V +V, V -V, V /Z, I(Z) give the results in polar form for all calculations lesson_et33b.pptx 4 7

8 //06 Example -4 Solution () Convert v (t) an v (t) into phasors Fin magnitues V V V V Fin V +V Convert phasors to rectangular form V 40.4[cos(0 ) jsin( 0 )] V j4.745 V 95.9[cos( -30 ) jsin( -30 )] V j97.95 lesson_et33b.pptx 5 Example -4 Solution () A real an imaginary parts Vs V V V ( ) j(4.745 ( 97.95)) s V j56. Convert to polar form s V s ( 56.) 56. s tan V s Ans lesson_et33b.pptx 6 8

9 //06 Example -4 Solution (3) Fin V -V subtract real an imaginary parts V V V V ( ) j(4.745 ( 97.95)) V 67. j39.7 Convert to polar form V 67. j39.7 V tan V Ans lesson_et33b.pptx 7 Example -4 Solution (4) Compute the quantity V /Z an give the results in polar form Z 70 j0 Convert Z to polar form Z Z tan To compute the quotient, ivie magnitues an subtract phase angles V Z V Z Ans lesson_et33b.pptx 8 9

10 //06 Example -4 Solution (5) Compute the quantity I(Z) an give the results in polar form Convert I to polar form I 3 j I I tan Multiply magnitues an a phase angles to get result I Z ( ) ( ) I Z I Z Ans lesson_et33b.pptx 9 ET 33b Ac Motors, Generators an Power Systems END LESSON : PHASORS AND COMPLEX ARITHMETIC lesson_et33b.pptx 0 0