Phasor Representation

Transcription

1 Phasr Representatin Phase Phase difference Phasrs Phasr Transfrmatins

2 Phase f sine wave An angular measurement that specifies the psitin f that sine wave relative t a reference When the sine wave is shifted left r right with respect t a reference, there is a phase shift.

3 Expressin fr shifted sine wave v = V sin( θ ± φ) P

4 Phase angle Phase angle (Θ) is the angular difference between the same pints n tw different wavefrms f the same frequency. If ϕ 0, tw wavefrms are ut f phase. If ϕ = 0, then tw wavefrms are said t be in phase Tw wavefrms that have peaks and zers at the same time are in phase and have a phase angle f 0. When ne sine wave is at its peak while anther is at zer, the tw are 90 ut f phase. When ne sine wave has just the ppsite phase f anther, they are 180 ut f phase.

5 Illustratin f Phase shift

6 Example #1 What are the phase angles between the tw sine waves? 45 0 phase angle between the tw wavefrms with sine wave B lagging 30 0 phase angle between the tw wavefrms with sine wave B leading

7 Example What are the phase angles between the tw sine waves? 45 0 phase angle between the tw wavefrms with sine wave B lagging

8 Phase shift Example f a wave that lags the reference Vltage (V) Reference Peak vltage v = 30 V sin (θ 45 ) and the equatin has a negative phase shift Ntice that a lagging sine wave is belw the axis at 0 Angle ( )

9 Phase shift Example f a wave that leads the reference Vltage (V) Reference Ntice that a leading sine wave is abve the axis at 0 Peak vltage v = 30 V sin (θ + 45 ) and the equatin has a psitive phase shift Angle ( )

10 A sinusid can be expressed in either sine r csine frm

11 Example #2 Determine the instantaneus value at the 90 reference pint n the hrizntal axis fr each vltage sine wave

12 Phasr Phasrs indicate the amplitude and phase angle f ac vltage r current. The length f the phasr represents the amplitude f the wavefrm. The angle represents the phase angle f the wavefrm. A phasr is a cmplex number that represents the amplitude and phase f a sinusid.

13 Phasr A cmplex number z can be written in rectangular frm as z = x + jy x-axis is the real axis and the y-axis is the imaginary (j) axis. C = 6 + j8 (rectangular frm) C = º (plar frm)

14 Cmplex Numbers (Engineering cnventin) A cmplex number is defined with the frm Where, are real numbers. The real part f, written is. The imaginary part f z, written, is. is written as and

15 Cnverting Between Frms (Summary) T cnvert frm the Cartesian frm nte: t plar frm,

16 Rectangular frm Plar frm Expnential frm z = x + jy = r(csφ + z = r φ z = re jφ j sinφ) where 2 r x + = φ = tan 1 y y x 2

17 Phasr - peratins

18 Example #4: Cnverting Rectangular Frm int Plar Frm

19 Example #5: Cnverting Plar Frm int Rectangular Frm

20 Example #6: Evaluate the cmplex numbers Given A =6 +j12 and B =7 + j2. Determine A+B and A-B. A+B (6 + j12) + (7 + j2) = (6 + 7) + j(12 + 2) = 13 + j14 A-B (6 + j12) (7 + j2) = (6 7) + j(12 2) = 1+ j10

21 Example #7: Evaluate the cmplex numbers Given A =1 +j1 and B =2 j3. Determine A+B and A-B. A+B (1 + j1) + (2 j3) = (1 + 2) + j(1 + ( 3)) = 3 j2 A-B (1 + j1) (2 j3) = (1 2) + j(1 ( 3)) = 1 + j4

22 Example #8: Evaluate the cmplex numbers Given A =6 70 and B = Determine A*B and A/B. A*B (6 70 ) (2 30 ) = 6 2 ( ) = A/B (6 70 ) 6 = (70 30 ) = 3 40 (2 30 ) 2

23 Example #9: Evaluate the cmplex numbers Given A = and B = Determine A*B and A/B. A*B ( ) ( ) = ( ) (45 + ( 56 )) = A/B ( ) 1.41 = (45 ( 56 )) = ( ) 3.61

24 Example #10: Evaluate the cmplex numbers [(5 + j2)( 1+ j4) j13.67 ] 10 + j j j2.2

25 Phasr Representatin

26 Sinusid Phasr Transfrmatins Transfrm a sinusid t and frm the time dmain t the phasr dmain: (time dmain) (phasr dmain)

27 Sinusid-Phasr Transfrmatins Time Dmain V m sin(ωt + φ) V m cs(ωt + φ) I m sin(ωt + θ) I m cs(ωt + θ) Phasr Dmain φ V V I I rms rms rms rms θ ( φ + 90 ) ( θ + 90 ) Assumes V m is psitive and -180 φ 180

28 Sinusid-Phasr Transfrmatins Time Dmain V m sin(ωt - φ) V m cs(ωt - φ) I m sin(ωt - θ) I m cs(ωt - θ) Phasr Dmain V φ V I I rms rms rms rms ( 90 φ) θ ( 90 θ ) Assumes V m is psitive and -180 φ 180

29 Sinusid-Phasr Transfrmatins Time Dmain -V m sin(ωt + φ) -V m cs(ωt + φ) -I m sin(ωt + θ) -I m cs(ωt + θ) Phasr Dmain φ V V I I rms rms rms rms 180 ( 90 + φ) θ ( 90 + θ ) +

30 Sinusid-Phasr Transfrmatins Time Dmain -V m sin(ωt - φ) -V m cs(ωt - φ) -I m sin(ωt - θ) -I m cs(ωt - θ) Phasr Dmain φ V V I I rms rms rms rms 180 ( 90 φ) 180 θ ( 90 θ )

31 Example #11: Sinusidal Functin : 0 3V sin(100t + 20 ) Cnverting t phasr ntatin : 3 2 V 20 0 Sinusidal Functin : 0 7Acs(350t 100 ) = 7Asin(350t + 90 = 7Asin(350t 10 Cnverting t phasr ntatin : 0 0 ) ) 7 2 A 10 0

32 Example #12: Given a sinusid vltage, 5sin(4πt 60 ), calculate its amplitude, phase, angular frequency, perid, and frequency. Draw the phasr diagram and cnvert t phasr Slutin: Vm = 5 vlts phase = 60 angular frequency = 4π rad/s Perid = 0.5 s frequency = 2 Hz.

33 Example #13: Given a sinusid vltage, 12 cs(50t + 10 ), calculate its amplitude, phase, angular frequency, perid, and frequency. Draw the phasr diagram and cnvert t phasr Slutin: Vm = 12 vlts phase = 10 angular frequency = 50 rad/s Perid = s frequency = Hz.

34 Example #14: Given a sinusid vltage, 5cs(6πt + 60 ), calculate its amplitude, phase, angular frequency, perid, and frequency. Draw the phasr diagram and cnvert t phasr Slutin: Vm = 5 vlts phase = 150 angular frequency = 4π rad/s Perid = 0.33 s frequency = 3 Hz.

35 Example #15: Find the phase angle between i 1 = 4sin(377t + and i2 = 5cs(377t 40 ), des i 1 lead r lag i 2? Draw the phasr diagram 25 ) Slutin: Since sin(ωt+90 ) = cs ωt i = 5sin(377t ) = 5sin(377t i = 4sin(377t + 25 ) = 4sin(377t ) = 4sin(377t + 1 ) 205 ) therefre, i 1 leads i

36 Example #16: Find the phase angle between v 1 = 10 cs( wt + 50 and v2 = 12sin( wt 10 ), des v 1 lead r lag v 2? Draw the phasr diagram ) Slutin: v 2 leads v 1 by 30.

37 Example #17: Find the sinusids represented by the phasrs: a) I = 3 + j4 A i( t) = 5sin( wt ) A b) V v( t) = 35.35sin( wt + 40 ) V v( t) = 35.35sin( wt ) V

38 Example #18 Draw the phasr fr the fllwing wavefrms. i 1 = 20 sin (ωt) ma. i 2 = 10 sin (ωt+90 ) ma. i 3 = 30 sin (ωt - 90 ) ma. Determine the equatin fr i T. Put it tgether in the cnversin and yu get: I = 14.1 j14.3 T i ( t) = 28.4sin ( wt 45.4 ) ma T

39 Example #19: