Electric Circuits II Phasor Diagram

Dr. Firas Obeidat

1 Phasor diagram for the Passive Circuit Elements The Resistor

Let 풊(풕) = 푰풎풄풐풔⁡(흎풕 + 흓)

In polar form

But Vm∟θ and Im∟ 흓 merely represent the general voltage and current phasors V and I. Thus

The angles θ and 흓 are equal, so that the current and voltage are always in phase.

The Inductor

풋(흎풕+흓) Let 풊(풕) = 푰풎풄풐 풔 흎풕 + 흓 = 푰풎풆

2 Dr. Firas Obeidat – Philadelphia University Phasor diagram for the Passive Circuit Elements

We obtain the desired phasor relationship

Note that the angle of the factor jωL is exactly +90◦ and that I must therefore lag V by 90° in an inductor.

The Capacitor

풋(흎풕+흓) Let 풗(풕) = 푽풎풄풐 풔(흎풕 + 흓) = 푽풎풆 풅풗(풕) 풊 풕 = 푪 푰 풆풋휽 = 풋흎푪푽 풆풋흓 풅풕 풎 풎 ퟏ 푽 = I 푰 = 푗휔퐶푽 푗휔퐶 Note that the angle of the factor 1/jωC is exactly -90◦ and that I must therefore lead V by 90° in an Capacitor. 3 Dr. Firas Obeidat – Philadelphia University Phasor diagram for series RL circuit Example: for the circuit shown in figure (a), draw the phasor circuit , impedance diagram and voltages phasor diagram.

V=100∟0, so the phasor circuit is shown in figure (b).

o ZT=ZR+ZL=3Ω+j4Ω =5∟53.13 . Impedance diagram is shown in figure (c).

표 푉 100∟0 o 퐼 = = o= 20∟−53.13 푍푇 5∟53.13 o o VR=IZR=(20∟-53.13 A)(3∟0Ω)=60∟-53.13 V. o o VL=IZL=(20∟-53.13 A)(4∟90Ω)=80∟36.87 V. Phasor diagram is shown in figure (d). In rectangular form

o VR=60∟-53.13 =36-j48V. o VL=80∟36.87 =64+j48V.

V=VR+VL=36-j48+64+j48=100+j0V=100∟0 V.

4 Dr. Firas Obeidat – Philadelphia University Phasor diagram for series RC circuit Example: for the circuit shown in figure (a), draw the phasor circuit , impedance diagram and voltages phasor diagram.

I=5∟0, so the phasor circuit is shown in figure (b).

o ZT=ZR+ZC=6Ω-j8Ω =10∟-53.13 . Impedance diagram is shown in figure (c).

o o 푉 = 퐼푍푇= (5∟53.13 )(10∟−53.13 )=50∟0 V

o o o VR=IZR=(5∟53.13 )(6∟0 )=30∟53.13 0 V

o o VC=IZC=(5∟53.13 A)(8∟-90Ω)=40∟-36.87 V. Phasor diagram is shown in figure (d). In rectangular form

o VR=30∟53.13 =18+j24 V o VC=40∟-36.87 =32-j24V.

V=VR+VC=18+j24+32-j24=50+j0=50∟0 V.

5 Dr. Firas Obeidat – Philadelphia University Phasor diagram for series RLC circuit Example: for the circuit shown in figure (a), draw the phasor circuit , impedance diagram and voltages phasor diagram.

V=50∟0, so the phasor circuit is shown in figure (b). o ZT=ZR+ZL+ZC=3Ω+7Ω-j3Ω =3+j4= 5∟53.13 . Impedance diagram is shown in figure (c).

표 푉 50∟0 o 퐼 = = o= 10∟−53.13 푍푇 5∟53.13 o o o VR=IZR=(10∟−53.13 )(3∟0 )=30∟−53.13 0 V o o VL=IZL=(10∟-53.13 A)(7∟90Ω)=70∟36.87 V. o o VC=IZC=(10∟-53.13 A)(3∟-90Ω)=30∟-143.13 V. Phasor diagram is shown in figure (d). In rectangular form o VR=30∟−53.13 0 V=18-j24 V o VL=70∟36.87 V=56+j42 V. o VC=30∟-143.13 V=-24-j18. V=VR+VL+VC=18-j24+56+j42-24-j18 V=50+j0=50∟0 V. 6 Dr. Firas Obeidat – Philadelphia University Phasor diagram for parallel RL circuit Example: for the circuit shown in figure (a), draw the phasor circuit , impedance diagram and currents phasor diagram.

V=20∟53.13, so the phasor circuit is shown in figure (b).

YT=YR+YL=1/3.33+1/j2.5=0.3-j0.4 =0.5∟-53.13 1 1 푍푇 = = =2 ∟53.13 푌푇 0.5∟−53.13

Impedance diagram is shown in figure (c).

표 푉 20∟53.13 o 퐼 = = o= 10∟0 푍푇 2∟53.13 표 푉 20∟53.13 o 퐼푅 = = o = 6∟53.13 푍푅 3.33∟0

표 푉 20∟53.13 o 퐼퐿 = = o = 8∟−36.87 푍퐿 2.5∟90 Currents Phasor diagram is shown in figure (d).

I=IR+IL=3.6+j4.8+6.4-j4.8=10+j0=10 ∟0. 7 Dr. Firas Obeidat – Philadelphia University Phasor diagram for parallel RC circuit Example: for the circuit shown in figure (a), draw the phasor circuit , impedance diagram and currents phasor diagram.

I=10∟0, so the phasor circuit is shown in figure (b).

YT=YR+YC=1/1.67+1/-j2.5=0.6+j0.8 =1∟53.13 1 1 푍푇 = = =1∟-53.13 푌푇 1∟53.13

Impedance diagram is shown in figure (c).

표 o 푉 = 퐼푍푇 = (10∟0 )(1∟-53.13)= 10∟−53.13

표 푉 10∟−53.13 o 퐼푅 = = o = 6∟−53.13 푍푅 1.67∟0 표 푉 10∟−53.13 o 퐼퐶 = = o = 8∟36.87 푍퐶 1.25∟−90

Currents Phasor diagram is shown in figure (d).

8 Dr. Firas Obeidat – Philadelphia University Phasor diagram for parallel RLC circuit Example: for the circuit shown in figure (a), draw the phasor circuit , impedance diagram and currents phasor diagram.

V=100∟53.13, so the phasor circuit is shown in figure (b).

YT=YR+YL+YC=1/3.33+1/j1.43+1/-j3.33 =0.3+j0.4 =0.5∟-53.13 1 1 푍 = = =2∟53.13o=1.2+j1.6 푇 푌 0.5∟−53.13 Impedance푇 diagram is shown in figure (c).

표 푉 100∟53.13 o 퐼 = = o = 50∟0 푍푇 2∟53.13 표 푉 100∟53.13 o 퐼푅 = = o = 30∟53.13 푍푅 3.33∟0

표 푉 100∟53.13 o 퐼퐿 = = o = 70∟−36.87 푍퐿 1.43∟90 표 푉 100∟53.13 o 퐼퐶 = = o= 30∟143.13 푍퐶 3.33∟−90 Currents Phasor diagram is shown in figure (d). 9 Dr. Firas Obeidat – Philadelphia University Series-parallel AC circuit

Example: for the circuit, calculate ZT, Is, VR, VC, IL and IC

퐕 ퟏퟏퟖ. ퟒퟒ∠ − ퟗ. ퟒퟔ퐨 퐋 퐈퐋 = = 퐨 = ퟑퟗ. ퟒퟖ∠ − ퟗퟗ. ퟒퟔ퐨 퐙퐋 ퟑ∠ퟗퟎ

10 Dr. Firas Obeidat – Philadelphia University Series-parallel AC circuit

Example: for the circuit, calculate Is and Vab

11 Dr. Firas Obeidat – Philadelphia University Series-parallel AC circuit

12 Dr. Firas Obeidat – Philadelphia University Series-parallel AC circuit Example: Determine the current I and

the voltage V.

13 Dr. Firas Obeidat – Philadelphia University Series-parallel AC circuit

Example: calculate I, I1, I2, I3 and ZT.

14 Dr. Firas Obeidat – Philadelphia University Series-parallel AC circuit

15 Dr. Firas Obeidat – Philadelphia University Series-parallel AC circuit

Example: calculate ZT, I, I1, andI2.

16 Dr. Firas Obeidat – Philadelphia University Series-parallel AC circuit

17 Dr. Firas Obeidat – Philadelphia University 18