Phasr Representatin Phase Phase difference Phasrs Phasr Transfrmatins
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.
Expressin fr shifted sine wave v = V sin( θ ± φ) P
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.
Illustratin f Phase shift
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
Example What are the phase angles between the tw sine waves? 45 0 phase angle between the tw wavefrms with sine wave B lagging
Phase shift Example f a wave that lags the reference Vltage (V) 40 30 20 10 0 Reference Peak vltage v = 30 V sin (θ 45 ) 0 45 90 135 180 225 270 315 360 and the equatin has a negative phase shift 405-20 -30-40 Ntice that a lagging sine wave is belw the axis at 0 Angle ( )
Phase shift Example f a wave that leads the reference Vltage (V) 40 30 20 10-45 0-10 -20-30 -40 Reference Ntice that a leading sine wave is abve the axis at 0 Peak vltage v = 30 V sin (θ + 45 ) 0 45 90 135 180 225 270 315 360 and the equatin has a psitive phase shift Angle ( )
A sinusid can be expressed in either sine r csine frm
Example #2 Determine the instantaneus value at the 90 reference pint n the hrizntal axis fr each vltage sine wave
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.
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 = 10 53.13º (plar frm)
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
Cnverting Between Frms (Summary) T cnvert frm the Cartesian frm nte: t plar frm,
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
Phasr - peratins
Example #4: Cnverting Rectangular Frm int Plar Frm
Example #5: Cnverting Plar Frm int Rectangular Frm
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
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
Example #8: Evaluate the cmplex numbers Given A =6 70 and B = 2 30. Determine A*B and A/B. A*B (6 70 ) (2 30 ) = 6 2 (70 + 30 ) = 12 100 A/B (6 70 ) 6 = (70 30 ) = 3 40 (2 30 ) 2
Example #9: Evaluate the cmplex numbers Given A =1.41 45 and B =3.61-56. Determine A*B and A/B. A*B (1.41 45 ) (3.61 56 ) = (1.41 3.61) (45 + ( 56 )) = 5.09 11 A/B (1.41 45 ) 1.41 = (45 ( 56 )) = 0.391 101 (3.61 56 ) 3.61
Example #10: Evaluate the cmplex numbers [(5 + j2)( 1+ j4) 5 60 15.5 + j13.67 ] 10 + j5 + 3 40 3 + j4 + 10 30 8.293 + j2.2
Phasr Representatin
Sinusid Phasr Transfrmatins Transfrm a sinusid t and frm the time dmain t the phasr dmain: (time dmain) (phasr dmain)
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
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
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 ) +
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 )
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 ) 100 0 ) 7 2 A 10 0
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.
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 = 0.1257 s frequency = 7.958 Hz.
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.
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 40 + 90 ) = 5sin(377t + 50 2 i = 4sin(377t + 25 ) = 4sin(377t + 180 + 25 ) = 4sin(377t + 1 ) 205 ) therefre, i 1 leads i 2 155.
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.
Example #17: Find the sinusids represented by the phasrs: a) I = 3 + j4 A i( t) = 5sin( wt + 126.87 ) A b) - 25 40 V v( t) = 35.35sin( wt + 40 ) V v( t) = 35.35sin( wt + 220 ) V
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
Example #19: