Single phase ac circuit 01 - Webs · EEE3405 EEPII-Single Phase AC circuits_01 Representing AC...

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EEE3405 EEPII-Single Phase AC circuits_01

Example

An alternating voltage is represented by v = 170sin2450t V, find

a) its peak value

b) its frequency

c) its period

d) its values at t = 3.65ms.

Solution:

a) Vm = 170 V

b) 2πf = 2450, f = 390 Hz

c) T = 1/f = 1/390 = 2.56 ms

d) v = 170sin(2450x0.00365) = 78.85 V



Introduction to Phasors

A phasor is a rotating line whose projection on a vertical axis can be used to represent sinusoidal varying quantities.

Complex notation of a.c. quantities

b. Sinusoidal voltage waveform



Shifted Sine Wave

Phasors may be used to represent shifted waveforms.

Angle θ is the position of the phasor at t = 0

Wave MovementWave

Movement



Phase DifferenceIt refers to the angular displacement between different waveforms of the

same frequency.

Vm

Im

ω

Vm

Im

ω

θ

ω

VmIm

θ

(a) v and i in phase

(b) i leads v (c) i lags v



Example 15-19

Given v = 20sin(ωt + 30o) and i = 18sin(ωt - 40o), draw the phasor diagram, determine phase relationships, and sketch the waveforms

Solution:



Complex Number for AC calculation

A Complex number is a number of the form C = a + jb, where

a and b are real numbers and

Complex numbers may be represented geometrically, either in rectangular form or in polar form. (Fig 16-1&2)

°∠= 901j



Conversion between rectangular and polar forms

Since C = a + jb = C∠θ

To convert rectangular to polar form

a

b

baC

1

22

tan−=

+=

θ

To convert polar to rectangular form

a = C cosθ

b = C sinθ



Algebra of complex numbers

� Addition and subtraction – use rectangular form

Suppose C1 = a1 + jb1

C2 = a2 + jb2

C1 ± C2 = (a1 ± a2) + j(b1 ± b2)

� Multiplication and division – use polar form

Suppose C1 = C1∠θ1

C2 = C2 ∠θ2

C1.C2 = C1C2 ∠(θ1+ θ2)

)( 212

1

2

1 θθ −∠=C

CCCCC

CCCC



Example 16-2

Given A = 2 + j1 and B = 1 + j3. Determine their sum and difference.

Solution:

A + B = (2+1) + j (1+3) = 3 + j4

A – B = (2-1) + j(1-3) = 1 – j2

Example 16-3

Given A = 3∠35o and B = 2∠-20o. Determine the product A.B and the quotient A/B.

Solution:

A.B = (3)(2) ∠(35o – 20o) = 6∠15o

A/B = (3/2) ∠[35o – (-20o)] = 1.5∠55o



Representing AC voltages and currents by complex numbers

AC voltages and currents can be represented as phasors and can be viewed as complex numbers. (Fig 16-9)

Fig 16-7 Representation of a sinusoidal source voltage as a complex number.

θ∠=2

EE)b( m

rms

Erms



Transforming source from time domain to phasor domain

E

e(t) = 200sin(ωt + 40o) V

V404.141402

200E °∠=°∠=



Important Notes

1. When voltage and current waveforms are expressed in wave equation, Vm and Im are the peak values.

2. When represents the voltages and currents in phasor form, the magnitudes of them are in rms values.

3. Thus, the phasor V = 120∠0o V means a voltage of 120 V rms at an angle of 0o.

4. To add or subtract sinusoidal voltages and currents, follow the three steps:

• convert sine waves to phasors and express them in complex numbers forms,

• add of subtract the complex numbers,• convert back to time functions if desired.



Summing AC voltages and currents

Sinusoidal quantities can be added or subtracted as phasor sum of transformed sources.

Example 16-6

Given e1 = 10sinωt V and e2 = 15sin(ωt +60o) V. Determine

v = e1 + e2.

Solution:

°∠+°∠=+= 602

150

2

10EEV 21

V6.3641.15

V6061.10007.7

°∠=°∠+°∠=

Thus, v V6.368.21)6.36tsin()41.15(2 °∠=°+ω=

Single phase ac circuit 01 - Webs · EEE3405 EEPII-Single Phase AC circuits_01 Representing AC...

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