chapter 14 – basic elements and phasors lecture 17 by moeen ghiyas 13/12/2015 1
TRANSCRIPT
![Page 1: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/1.jpg)
Chapter 14 – Basic Elements and Phasors
Lecture 17
by Moeen Ghiyas
21/04/23 1
![Page 2: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/2.jpg)
Chapter 14 – Basic Elements and Phasors
![Page 3: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/3.jpg)
Average Power & Power Factor
Complex Numbers
Math Operations with Complex Numbers
![Page 4: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/4.jpg)
We know for any load
v = Vm sin(ωt + θv)
i = Im sin(ωt + θi)
Then the power is defined by
Using the trigonometric identity
Thus, sine function becomes
![Page 5: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/5.jpg)
Putting above values in
We have
The average value of 2nd term is zero over one cycle, producing
no net transfer of energy in any one direction.
The first term is constant (not time dependent) is referred to as
the average power or power delivered or dissipated by the load.
![Page 6: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/6.jpg)
![Page 7: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/7.jpg)
Since cos(–α) = cos α,
the magnitude of average power delivered is independent of whether v
leads i or i leads v.
Ths, defining θ as equal to | θv – θi |, where | | indicates that only the
magnitude is important and the sign is immaterial, we have average
power or power delivered or dissipated as
![Page 8: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/8.jpg)
The above eq for average power can also be written as
But we know Vrms and Irms values as
Thus average power in terms of vrms and irms becomes,
21/04/23 8
![Page 9: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/9.jpg)
For resistive load,
We know v and i are in phase, then |θv - θi| = θ = 0°,
And cos 0° = 1, so that
becomes
or
![Page 10: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/10.jpg)
For inductive load ( or network),
We know v leads i, then |θv - θi| = θ = 90°,
And cos 90° = 0, so that
Becomes
Thus, the average power or power dissipated by the ideal
inductor (no associated resistance) is zero watts.
![Page 11: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/11.jpg)
For capacitive load ( or network),
We know v lags i, then |θv - θi| = |–θ| = 90°,
And cos 90° = 0, so that
Becomes
Thus, the average power or power dissipated by the ideal
capacitor is also zero watts.
![Page 12: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/12.jpg)
Power Factor
In the equation,
the factor that has significant control over the delivered power
level is cos θ.
No matter how large the voltage or current, if cos θ = 0, the
power is zero; if cos θ = 1, the power delivered is a maximum.
Since it has such control, the expression was given the name
power factor and is defined by
For situations where the load is a combination of resistive and
reactive elements, the power factor will vary between 0 and 1
![Page 13: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/13.jpg)
In terms of the average power, we know power factor is
The terms leading and lagging are often written in conjunction
with power factor and defined by the current through load.
If the current leads voltage across a load, the load has a leading
power factor. If the current lags voltage across the load, the load
has a lagging power factor.
In other words, capacitive networks have leading power factors,
and inductive networks have lagging power factors.
![Page 14: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/14.jpg)
EXAMPLE - Determine the average power delivered to network
having the following input voltage and current:
v = 150 sin(ωt – 70°) and i = 3 sin(ωt – 50°)
Solution
![Page 15: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/15.jpg)
EXAMPLE - Determine the power factors of the following loads,
and indicate whether they are leading or lagging:
Solution:
![Page 16: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/16.jpg)
EXAMPLE - Determine the power factors of the following loads,
and indicate whether they are leading or lagging:
Solution:
![Page 17: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/17.jpg)
Application of complex numbers result in a technique for finding
the algebraic sum of sinusoidal waveforms
A complex number represents a point in a two-dimensional
plane located with reference to two distinct axes.
The horizontal axis is called the real axis, while the vertical axis
is called the imaginary axis.
The symbol j (or sometimes i) is
used to denote the imaginary
component.
![Page 18: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/18.jpg)
Two forms are used to represent a
complex number:
rectangular and polar.
Rectangular Form
Polar Form
![Page 19: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/19.jpg)
Polar Form
θ is always measured counter-clockwise
(CCW) from the positive real axis.
Angles measured in the clockwise direction
from the positive real axis must have a
negative sign
![Page 20: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/20.jpg)
EXAMPLE - Sketch the following complex numbers in the
complex plane:
a. C = 3 + j 4 b. C = 0 - j 6
![Page 21: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/21.jpg)
EXAMPLE - Sketch the following complex numbers in the
complex plane:
c. C = -10 - j20
![Page 22: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/22.jpg)
EXAMPLE - Sketch the following complex numbers in the
complex plane:
![Page 23: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/23.jpg)
EXAMPLE - Sketch the following complex numbers in the
complex plane:
![Page 24: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/24.jpg)
Rectangular to Polar
Polar to Rectangular
Angle determined to be
associated carefully with the
magnitude of the vector as per
the quadrant in which complex
number lies
![Page 25: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/25.jpg)
EXAMPLE - Convert the following from polar to rectangular form:
Solution:
![Page 26: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/26.jpg)
EXAMPLE - Convert the following from rectangular to polar form:
C = - 6 + j 3
Solution:
![Page 27: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/27.jpg)
EXAMPLE - Convert the following from polar to rectangular form:
Solution
![Page 28: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/28.jpg)
Let us first examine the symbol j associated with imaginary
numbers. By definition,
![Page 29: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/29.jpg)
The conjugate or complex conjugate is found
by changing sign of imaginary part in rectangular form
or by using the negative of the angle of the polar form.
Rectangular form,
Polar form,
(conjugate)
(conjugate)
![Page 30: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/30.jpg)
Addition Example (Rectangular)
Add C1 = 3 + j 6 and C2 = -6 + j 3.
Solution
![Page 31: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/31.jpg)
Subtraction Example (Rectangular)
Solution
![Page 32: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/32.jpg)
Imp Note
Addition or subtraction cannot be performed in polar
form unless the complex numbers have the same
angle θ or unless they differ only by multiples of 180°.
![Page 33: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/33.jpg)
Addition Example (Polar)
![Page 34: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/34.jpg)
Subtraction Example (Polar)
![Page 35: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/35.jpg)
Average Power & Power Factor
Complex Numbers
Math Operations with Complex Numbers
![Page 36: Chapter 14 – Basic Elements and Phasors Lecture 17 by Moeen Ghiyas 13/12/2015 1](https://reader036.vdocuments.site/reader036/viewer/2022081506/5697bf941a28abf838c90411/html5/thumbnails/36.jpg)
21/04/23 36