chapter 1b - power in ac circuits

8/2/2019 Chapter 1b - Power in AC Circuits

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Electrical Power and Machines EPE491

Chapter 1b Power in AC Circuits

Introduction

There are 2 fundamentals laws that are important in AC circuits

o Ohms Law

States that the currentthrough a conductor btwn 2 points is directly proportionalto the

potential difference across the 2 points, and is inversely proportional to the resistancebetween them.

Mathematical equation: V IR

Where V - potential difference in volts (V)

I - current in amperes (A)

R - resistance in ohms ()

o Kirchoffs Laws

Kirchoffs Voltage Law

States that the algebraic sum of all voltages around any closed loop is zero

Kirchoffs Current Law

braic sum of currents at a node is zero States that the alge

D. Johari, FKE UiTM 1


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AC Circuit

Means voltage signal is in alternating currentform (AC) that has positive & negative portions.

Normally referred as sinusoidal form.

General equation: V(t) = Vm sin t

Where Vm - the maximum voltage

- angular frequency in rad/s = 2f

f - supply frequency in Hz

T - period = 1/f

AC Signal

AC Signal

Two AC Signals with Different Phases

Vm sin (t + ) is a signal that leads the original signal by an angle of



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Phase

o When capacitors or inductors are involved in an AC cct, the current & voltage do notpeak

at the same time.

o The period diff btwn the peaks expressed in degrees is said to be thephase difference .

o This phase relation is often represented graphically in aphasor diagram.

Phasor diagram between V and I

is the angle between V & I

= v - i

RMS Voltage

o Stands forroot-mean-square.

o In AC cct, current & voltages are generally stated as rms values instead of maximum

values.

o It is the effective voltage that is utilized in practical or theoretical analysis given by:

2

m

rms

VV = where Vm - the maximum voltage

Example 1

An AC signal is given as V (t) = 141.4 sin 314t. Determine the following:

a) Maximum voltage

b) RMS voltage

c) Frequency

d) Period to complete 1 cyclee) Phase shift



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Components of AC Circuit

Ohms Law in AC Circuits

o Modified to the form: V = I Z where

V - the effective or rms voltage values

I - the effective or rms current values

Z - the impedance

3 main components

o Resistor R in Ohms ()

o Inductor L in Henry (H)

o Capacitor C in Farad (F)

Pure Resistive Load

o The impedance, Z consists only of a resistor i.e. Z = R

Pure Resistive Circuit

o Phasor diagram

Current is in phase with voltage

= v - i = 0 Therefore, v= i

With v taken as reference point, the phasor diagram is shown below

o Power Factor

p.f. = cos

Power factor is 1.0 or unity since = 0

Inductive Load

o The impedance, Z consists of a resistor in series with an inductor i.e. Z = R + j XL

Inductive Circuit

Where XL = inductive reactance ()

= 2fL = L

L = inductance (H)

f = supply frequency



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o Phasor diagram

Current lags voltage

o When a voltage is applied to an inductor, it resists the change in current. Current

builds up more slowly than the voltage, lagging it in time and phase.

= v - i > 0

o Power Factor

p.f. = cos

Power factor is lagging

Capacitive Load

o The impedance, Z consists of a resistor in series with a capacitor i.e. Z = R jXC

Capacitive Circuit

Where XC = capacitive reactance ()

=CfC

1

2

1=

C = capacitance (F)

o Phasor diagram

Current leads voltage

o Since the voltage on a capacitor is directly proportional to the charge on it, the

current must lead the voltage in time and phase to conduct charge to the capacitor

plates and raise the voltage.

= v - i < 0

o Power Factor

p.f. = cos

Power factor is leading



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Example 2

A resistance of 7.0 is connected in series with a pure inductance of 31.4mH and the circuit is

connected to a 100V, 50Hz, sinusoidal supply. Calculate the circuit current & the phase angle.



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Power in AC Circuits

The behavior of AC machines & systems are often easier to understand by working with power,

rather than working with voltages and currents

Active, reactive & apparent power apply to steady-state AC circuits with sinusoidal waveforms

only

o Cannot be used to describe transient (temporary) behaviour

o Cannot be used to describe DC circuits

Instantaneous Power in AC circuits

o The product of instantaneous voltage & current, unit in Watts (W)

o Given by: P = V I

Active Power, P

o The average value of the instantaneous power over one cycle of the voltage

o Also known as real power @ true power @ actual power. The effective power that does

real work, unit in Watts (W)

o Given by: P = V I cos where = angle btwn V & Io Since V = IZ, P can also be written as P = I2Z cos

Reactive Power, Q

o The circulating power in the circuit.

o Power which does no real work.Also known as the imaginary power, unit in volt-amperes-

reactive [var]

o Given by: Q = V I sin

o Q can also be written as Q = I2Z sin

Sources and Loadso Generator

Active source, delivers active power

o Resistor

Active load, absorbs active power

o Capacitor

Reactive source, delivers reactive power

o Inductor

Reactive load, absorbs reactive power

Complex Power, So The product of the voltage across the load and the current through the load, unit in VA

(volts-amperes)

o Given by: S= V I*

S= (Vv)(I-i)* = (Vv)(I+ i )

= P + j Q = VI cos + j VI sin

o Apparent power the power that supplied to the load if phase angle diff btwn V & I are

ignored

o

Given by S = V Io S can also be written as S = I2Z



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Power Triangle

o The relationship btwn S, P and Q can be represented graphically by a power triangle.

Where S2

= P2

+ Q2

For inductive load, since > 0, Q = VI sin + = +ve Q

For capacitive load, since < 0, Q = VI sin - = -ve Q

Based on the discussion earlier, the summary is as follows:

Resistive Load Inductive Load Capacitive Load

I is in phase with V I lags V I leads V

= v - i = 0 = v - i > 0 = v - i < 0

Q is zero Q is +ve Q is -ve

pf is unity pf is lagging pf is leading

Example 3

Figure below shows an AC voltage source supplying power to a load with impedance Z = 20-30

. Calculate the current, I supplied to the load, the power factor of the load, and the real, reactive,

apparent, and complex power supplied to the load.



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From phasor diagram for inductive load, the power factor is lagging.

When capacitor is connected in parallel with the load, 2 < 1

Therefore, cos 2 > cos 1 i.e. p.f.2 > p.f.1

Calculating the Capacitance Value

o Consider the power triangle

o From diagram:

21QQQC =

o The reactive power is given by:

CVX

VQ C

C

C

C 2

2

==

o

The capacitance value can then be calculated as below:

2

C

C

V

QC=

Example 4

A single phase circuit is depicted in the following figure. The supply rms mode is 120V with 60Hz

frequency. Calculate the corresponding capacitance value that is needed to improve the circuit

power factor to 0.95 lagging. Show the answer through the aid of phasor diagram.


chapter 1b - power in ac circuits

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