revisionlecture2: phasors - imperial college london · phasor diagram revisionlecture2: phasors...
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![Page 1: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/1.jpg)
Revision Lecture 2: Phasors
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 1 / 7
![Page 2: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/2.jpg)
Basic Concepts
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 2 / 7
• Phasors and Complex impedances are only relevant to sinusoidalsources.
A DC source is a special case of a cosine wave with ω = 0.
![Page 3: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/3.jpg)
Basic Concepts
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 2 / 7
• Phasors and Complex impedances are only relevant to sinusoidalsources.
A DC source is a special case of a cosine wave with ω = 0.
• For two sine waves, the leading one reaches its peak first, the laggingone reaches its peak second. So sinωt lags cosωt.
![Page 4: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/4.jpg)
Basic Concepts
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 2 / 7
• Phasors and Complex impedances are only relevant to sinusoidalsources.
A DC source is a special case of a cosine wave with ω = 0.
• For two sine waves, the leading one reaches its peak first, the laggingone reaches its peak second. So sinωt lags cosωt.
• If A cos (ωt+ θ) = F cosωt−G sinωt, then
A =√F 2 +G2, θ = tan−1 G
F.
![Page 5: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/5.jpg)
Basic Concepts
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 2 / 7
• Phasors and Complex impedances are only relevant to sinusoidalsources.
A DC source is a special case of a cosine wave with ω = 0.
• For two sine waves, the leading one reaches its peak first, the laggingone reaches its peak second. So sinωt lags cosωt.
• If A cos (ωt+ θ) = F cosωt−G sinωt, then
A =√F 2 +G2, θ = tan−1 G
F.
F = A cos θ, G = A sin θ.
![Page 6: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/6.jpg)
Basic Concepts
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 2 / 7
• Phasors and Complex impedances are only relevant to sinusoidalsources.
A DC source is a special case of a cosine wave with ω = 0.
• For two sine waves, the leading one reaches its peak first, the laggingone reaches its peak second. So sinωt lags cosωt.
• If A cos (ωt+ θ) = F cosωt−G sinωt, then
A =√F 2 +G2, θ = tan−1 G
F.
F = A cos θ, G = A sin θ.
In CMPLX mode, Casio fx-991ES can do complex arithmetic andcan switch between the two forms with SHIFT,CMPLX,3 orSHIFT,CMPLX,4
![Page 7: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/7.jpg)
Reactive Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 3 / 7
• Impedances: R, jωL, 1
jωC= −j
ωC.
Admittances: 1
R, 1
jωL= −j
ωL, jωC
![Page 8: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/8.jpg)
Reactive Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 3 / 7
• Impedances: R, jωL, 1
jωC= −j
ωC.
Admittances: 1
R, 1
jωL= −j
ωL, jωC
• In a capacitor or inductor, the Current and Voltage are 90 apart :
CIVIL: Capacitor - current leads voltage; Inductor - current lagsvoltage
![Page 9: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/9.jpg)
Reactive Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 3 / 7
• Impedances: R, jωL, 1
jωC= −j
ωC.
Admittances: 1
R, 1
jωL= −j
ωL, jωC
• In a capacitor or inductor, the Current and Voltage are 90 apart :
CIVIL: Capacitor - current leads voltage; Inductor - current lagsvoltage
• Average current (or DC current) through a capacitor is always zero
• Average voltage across an inductor is always zero
![Page 10: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/10.jpg)
Reactive Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 3 / 7
• Impedances: R, jωL, 1
jωC= −j
ωC.
Admittances: 1
R, 1
jωL= −j
ωL, jωC
• In a capacitor or inductor, the Current and Voltage are 90 apart :
CIVIL: Capacitor - current leads voltage; Inductor - current lagsvoltage
• Average current (or DC current) through a capacitor is always zero
• Average voltage across an inductor is always zero
• Average power absorbed by a capacitor or inductor is always zero
![Page 11: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/11.jpg)
Phasors
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 4 / 7
A phasor represents a time-varying sinusoidal waveform by a fixed complexnumber.
Waveform Phasor
x(t) = F cosωt−G sinωt X = F + jG
![Page 12: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/12.jpg)
Phasors
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 4 / 7
A phasor represents a time-varying sinusoidal waveform by a fixed complexnumber.
Waveform Phasor
x(t) = F cosωt−G sinωt X = F + jG [Note minus sign]
![Page 13: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/13.jpg)
Phasors
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 4 / 7
A phasor represents a time-varying sinusoidal waveform by a fixed complexnumber.
Waveform Phasor
x(t) = F cosωt−G sinωt X = F + jG [Note minus sign]
x(t) = A cos (ωt+ θ) X = Aejθ = A∠θ
![Page 14: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/14.jpg)
Phasors
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 4 / 7
A phasor represents a time-varying sinusoidal waveform by a fixed complexnumber.
Waveform Phasor
x(t) = F cosωt−G sinωt X = F + jG [Note minus sign]
x(t) = A cos (ωt+ θ) X = Aejθ = A∠θ
max (x(t)) = A |X| = A
![Page 15: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/15.jpg)
Phasors
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 4 / 7
A phasor represents a time-varying sinusoidal waveform by a fixed complexnumber.
Waveform Phasor
x(t) = F cosωt−G sinωt X = F + jG [Note minus sign]
x(t) = A cos (ωt+ θ) X = Aejθ = A∠θ
max (x(t)) = A |X| = A
x(t) is the projection of a rotating rod ontothe real (horizontal) axis.
X = F + jG is its starting position at t = 0.
![Page 16: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/16.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
![Page 17: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/17.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
![Page 18: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/18.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
= 1
1+3j
![Page 19: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/19.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
= 1
1+3j
= 0.1− 0.3j = 0.32∠− 72
![Page 20: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/20.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
= 1
1+3j
= 0.1− 0.3j = 0.32∠− 72
x(t) = cos 300t⇒ X = 1
![Page 21: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/21.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
= 1
1+3j
= 0.1− 0.3j = 0.32∠− 72
x(t) = cos 300t⇒ X = 1
Y = X × YX
![Page 22: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/22.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
= 1
1+3j
= 0.1− 0.3j = 0.32∠− 72
x(t) = cos 300t⇒ X = 1
Y = X × YX
= 0.1− 0.3j = 0.32∠− 72
![Page 23: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/23.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
= 1
1+3j
= 0.1− 0.3j = 0.32∠− 72
x(t) = cos 300t⇒ X = 1
Y = X × YX
= 0.1− 0.3j = 0.32∠− 72
0 0.5 1
-0.4
-0.2
0
Real
![Page 24: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/24.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
= 1
1+3j
= 0.1− 0.3j = 0.32∠− 72
x(t) = cos 300t⇒ X = 1
Y = X × YX
= 0.1− 0.3j = 0.32∠− 72
y(t) = 0.1 cos 300t+ 0.3 sin 300t
0 0.5 1
-0.4
-0.2
0
Real
![Page 25: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/25.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
= 1
1+3j
= 0.1− 0.3j = 0.32∠− 72
x(t) = cos 300t⇒ X = 1
Y = X × YX
= 0.1− 0.3j = 0.32∠− 72
y(t) = 0.1 cos 300t+ 0.3 sin 300t
= 0.32 cos (300t− 1.25)
= 0.32 cos (300 (t− 4.2ms))
0 0.5 1
-0.4
-0.2
0
Real
![Page 26: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/26.jpg)
Phasor Diagram
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 5 / 7
Draw phasors as vectors. Join vectors end-to-end to show how voltages inseries add up (or currents in parallel).
Find y(t) if x(t) = cos 300t .
YX
=1
jωC
R+ 1
jωC
= 1
jωRC+1
= 1
1+3j
= 0.1− 0.3j = 0.32∠− 72
x(t) = cos 300t⇒ X = 1
Y = X × YX
= 0.1− 0.3j = 0.32∠− 72
y(t) = 0.1 cos 300t+ 0.3 sin 300t
= 0.32 cos (300t− 1.25)
= 0.32 cos (300 (t− 4.2ms))
0 0.5 1
-0.4
-0.2
0
Real
0 20 40 60 80 100 120-1
-0.5
0
0.5
1x
y
time (ms)
x=bl
ue, y
=re
d
w = 300 rad/s, Gain = 0.32, Phase = -72°
![Page 27: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/27.jpg)
Complex Power
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 6 / 7
If V = |V |∠θV is a phasor, we define V = 1√2× V to be the
corresponding r.m.s. phasor. The r.m.s. voltage is∣∣∣V
∣∣∣ = 1√2× |V |.
![Page 28: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/28.jpg)
Complex Power
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 6 / 7
If V = |V |∠θV is a phasor, we define V = 1√2× V to be the
corresponding r.m.s. phasor. The r.m.s. voltage is∣∣∣V
∣∣∣ = 1√2× |V |.
Power Factor cosφ = cos (θV − θI)
![Page 29: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/29.jpg)
Complex Power
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 6 / 7
If V = |V |∠θV is a phasor, we define V = 1√2× V to be the
corresponding r.m.s. phasor. The r.m.s. voltage is∣∣∣V
∣∣∣ = 1√2× |V |.
Power Factor cosφ = cos (θV − θI)
Complex Power S = V × I∗ = P + jQ [∗ = complex conjugate]
![Page 30: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/30.jpg)
Complex Power
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 6 / 7
If V = |V |∠θV is a phasor, we define V = 1√2× V to be the
corresponding r.m.s. phasor. The r.m.s. voltage is∣∣∣V
∣∣∣ = 1√2× |V |.
Power Factor cosφ = cos (θV − θI)
Complex Power S = V × I∗ = P + jQ [∗ = complex conjugate]
Apparent Power |S| =∣∣∣V
∣∣∣×∣∣∣I∣∣∣
![Page 31: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/31.jpg)
Complex Power
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 6 / 7
If V = |V |∠θV is a phasor, we define V = 1√2× V to be the
corresponding r.m.s. phasor. The r.m.s. voltage is∣∣∣V
∣∣∣ = 1√2× |V |.
Power Factor cosφ = cos (θV − θI)
Complex Power S = V × I∗ = P + jQ [∗ = complex conjugate]
Apparent Power |S| =∣∣∣V
∣∣∣×∣∣∣I∣∣∣
Average Power P = ℜ (S) = |S| cosφ unit = Watts → heat
![Page 32: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/32.jpg)
Complex Power
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 6 / 7
If V = |V |∠θV is a phasor, we define V = 1√2× V to be the
corresponding r.m.s. phasor. The r.m.s. voltage is∣∣∣V
∣∣∣ = 1√2× |V |.
Power Factor cosφ = cos (θV − θI)
Complex Power S = V × I∗ = P + jQ [∗ = complex conjugate]
Apparent Power |S| =∣∣∣V
∣∣∣×∣∣∣I∣∣∣ unit = VAs
Average Power P = ℜ (S) = |S| cosφ unit = Watts → heatReactive Power Q = ℑ (S) = |S| sinφ unit = VARs
![Page 33: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/33.jpg)
Complex Power
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 6 / 7
If V = |V |∠θV is a phasor, we define V = 1√2× V to be the
corresponding r.m.s. phasor. The r.m.s. voltage is∣∣∣V
∣∣∣ = 1√2× |V |.
Power Factor cosφ = cos (θV − θI)
Complex Power S = V × I∗ = P + jQ [∗ = complex conjugate]
Apparent Power |S| =∣∣∣V
∣∣∣×∣∣∣I∣∣∣ unit = VAs
Average Power P = ℜ (S) = |S| cosφ unit = Watts → heatReactive Power Q = ℑ (S) = |S| sinφ unit = VARs
Conservation of power (Tellegen’s theorem): in any circuit the total complexpower absorbed by all components sums to zero⇒ P = average power and Q = reactive power sum separately to zero.
![Page 34: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/34.jpg)
Complex Power
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 6 / 7
If V = |V |∠θV is a phasor, we define V = 1√2× V to be the
corresponding r.m.s. phasor. The r.m.s. voltage is∣∣∣V
∣∣∣ = 1√2× |V |.
Power Factor cosφ = cos (θV − θI)
Complex Power S = V × I∗ = P + jQ [∗ = complex conjugate]
Apparent Power |S| =∣∣∣V
∣∣∣×∣∣∣I∣∣∣ unit = VAs
Average Power P = ℜ (S) = |S| cosφ unit = Watts → heatReactive Power Q = ℑ (S) = |S| sinφ unit = VARs
Conservation of power (Tellegen’s theorem): in any circuit the total complexpower absorbed by all components sums to zero⇒ P = average power and Q = reactive power sum separately to zero.
VARs are generated by capacitors and absorbed by inductors.
φ > 0 for inductive impedance.φ < 0 for capacitive impedance.
![Page 35: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/35.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z
![Page 36: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/36.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
![Page 37: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/37.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
![Page 38: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/38.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
![Page 39: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/39.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
V = 230. Motor is 5||7j Ω.
![Page 40: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/40.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
V = 230. Motor is 5||7j Ω.
I = 46− 33j = 56.5∠− 36
![Page 41: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/41.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
V = 230. Motor is 5||7j Ω.
I = 46− 33j = 56.5∠− 36
S = V I∗ = 13∠36 kVA
![Page 42: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/42.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
V = 230. Motor is 5||7j Ω.
I = 46− 33j = 56.5∠− 36
S = V I∗ = 13∠36 kVAcosφ = P
|S|= cos 36 = 0.81
![Page 43: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/43.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
V = 230. Motor is 5||7j Ω.
I = 46− 33j = 56.5∠− 36
S = V I∗ = 13∠36 kVAcosφ = P
|S|= cos 36 = 0.81
Add parallel capacitor: 300µF
![Page 44: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/44.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
V = 230. Motor is 5||7j Ω.
I = 46− 33j = 56.5∠− 36
S = V I∗ = 13∠36 kVAcosφ = P
|S|= cos 36 = 0.81
Add parallel capacitor: 300µFI = 46− 11j = 47∠− 14
![Page 45: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/45.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
V = 230. Motor is 5||7j Ω.
I = 46− 33j = 56.5∠− 36
S = V I∗ = 13∠36 kVAcosφ = P
|S|= cos 36 = 0.81
Add parallel capacitor: 300µFI = 46− 11j = 47∠− 14
S = V I∗ = 10.9∠14 kVA
![Page 46: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/46.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
V = 230. Motor is 5||7j Ω.
I = 46− 33j = 56.5∠− 36
S = V I∗ = 13∠36 kVAcosφ = P
|S|= cos 36 = 0.81
Add parallel capacitor: 300µFI = 46− 11j = 47∠− 14
S = V I∗ = 10.9∠14 kVAcosφ = P
|S| = cos 14 = 0.97
![Page 47: RevisionLecture2: Phasors - Imperial College London · Phasor Diagram RevisionLecture2: Phasors •Basic Concepts •Reactive Components •Phasors •Phasor Diagram •Complex Power](https://reader031.vdocuments.net/reader031/viewer/2022020319/5d1f5b7b88c9934c378d892b/html5/thumbnails/47.jpg)
Complex Power in Components
RevisionLecture 2: Phasors
• Basic Concepts
• Reactive Components
• Phasors
• Phasor Diagram
• Complex Power
• Complex Power inComponents
E1.1 Analysis of Circuits (2014-4472) Revision Lecture 2 – 7 / 7
S = V I∗ =|V |2Z∗
=∣∣∣I∣∣∣2
Z ⇒ ∠S = ∠Z
Resistor: positive real (absorbs watts)Inductor: positive imaginary (absorbs VARs)Capacitor: negative imaginary (generates VARs)
Power Factor Correction
V = 230. Motor is 5||7j Ω.
I = 46− 33j = 56.5∠− 36
S = V I∗ = 13∠36 kVAcosφ = P
|S|= cos 36 = 0.81
Add parallel capacitor: 300µFI = 46− 11j = 47∠− 14
S = V I∗ = 10.9∠14 kVAcosφ = P
|S| = cos 14 = 0.97
Current decreases by factor of 0.810.97
. Lower power transmission losses.