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Electrical Power and Machines

(B38EI)

(Mech Eng students’ stream)

Chapter 1

Lecture 2: AC in R, L and C components

0. Introduction

1. ac Voltage and Current in the Resistor

1.1 ac Power in a Resistor

2. ac voltage and current in an Inductor

2.1 Power in an Inductor

3. ac Voltage and Current in a Capacitor

3.1 Power in a Capacitor

4. What about this (j) operator thing?

4.1 The Complex Planes

4.2 Aide-memoire to

voltage/currents/lead/lag/Inductor/Capacitor

Contents:

ac voltage and current in the Resistor

Ohm's Law (V=iR) can be applied to an ac circuit containing a resistance to

determine the ac current in the resistance when an ac voltage is connected.

i(t) = V/R sin t

v(t) = V sint

R

i (t) = V/R sin ωt

In a resistor voltage and current are in phase

4

Resistor

Power calculation with AC voltage and current in the Resistor

You will recall that in a dc circuit power can be calculated using any of the three

relationships:

VI I

2 R V

2 /R Watts.

In ac circuits both voltage and current are time-varying quantities, and so

therefore is power.

The power at any instant, the instantaneous power , can be computed using

instantaneous values of voltage and/or current.

ac Power, p(t) = v(t).i(t) = (V sin t)(I sin t) = VI sin

2 t Watts

ac Power in a Resistor

The Passive components: Resistor D

ac voltage and current in the Resistor

ac Power, p(t) = v(t).i(t) = (V sin t)(I sin t) = VI sin

2 t Watts

Now the periodicity of the sin

2 t term can be understood in two ways:

Secondly, we can use standard trigonometric formulae to expand sin

2 t:

sin

2 t = 0.5 (1 - cos 2t)

Thus ac power = VI sin

2 t = 0.5VI (1 - cos 2t) Watts.

Remember V and I are peak values!

ac voltage and current in the Resistor

ac Power, p(t) = v(t).i(t) = (V sin t)(I sin t) = VI sin

2 t Watts

Now the sin

2 t term can be viewed in two ways:

Secondly, we can use standard trigonometric formulae to expand sin

2 t:

sin

2 t = 0.5 (1 - cos 2t)

Thus ac power = VI sin

2 t = 0.5VI (1 - cos 2t) Watts.

Now, consider the cos 2t term_._

The average value of cos 2  t over some period of time (t>> a period) is zero

(equal positive and negative areas).

Thus average ac power = 0.5VI

Remember V and I are peak values!

RMS measurements

Vertical Voltmeter for 400-10.000 Volt

End XIX Century, The Hunterian

Museum and Art Gallery, University of

Glasgow

An inductor has the ability to store magnetic field.

Basically an inductor can be considered, at a most fundamental

level, as a coil of conducting wire.

The ability of an inductor to store magnetic field is measured in

Henrys , or more commonly as mH , which is a thousandth of a

Henry. The equivalent definition to the capacitor is:

Inductance [Henrys] =

Or L = Henrys

Current[Amp]

FluxLinkages[Webers]

I

N

Inductors

Inductors and Time, in DC

For the inductor, = LI and as induced emf is

then

This in words says that the voltage across an inductor is proportional

to the rate of change of the current in the inductor.

N

dt

d N



dt

d N



= dt

dI L or

dt

dI V  L

 V dt

L

I

In Integral terms:

The current in an inductor cannot change instantaneously unless the

voltage is infinite as voltage is proportional to the rate of change of the

current.

Inductors and Time, in DC

Time Constants

dt

di

V  Ri  L

The current at the inductor will be:

close at time t = 0

R

L

i

V

( 1 )

L

Rt

i I e



 

(for a full demonstration see the Hughes, but it is not necessary)

ac voltage and current in the Inductor

Whereas in the resistor the Voltage and Current are related by the linear

relationship we call Resistance (), for the Inductor (and the Capacitor) the

situation is not so straightforward.

For an Inductor the voltage to current relationship is:

V = L

dt

di

where V is the voltage across the inductor, L is the inductance and i is the current

through the inductor

dt

di

L = V

ac voltage and current in the Inductor

V = L

So let us return to our basic circuit with a voltage v(t) = V sin t volts applied

across an inductor (L).

i(t)

v(t) = V sint

L

dt

di

ac voltage and current in the Inductor

If the voltage is sinusoidal then the current must be I sin (t - /2),

V sin t = L {-I cos t}

In an Inductor the voltage LEADS the current by 90

dt

d

If the voltage is sinusoidal then the current must be I sin (t - /2),

V sin t = L {-I cos t}

Now {-I cos t} =  I sin t

so substituting in the above equation gives:

V sin t = L {-I cos t} = L  (I sin t)

So the fundamental relationship between Voltage and Current for an inductor is

L (which naturally has units of .)

L is known as the Inductive Reactance and is denoted XL.

dt

d

dt

d

dt

d