­

Alternating Current Circuits 

By on

Alternating current (a.c.) is the backbone of modern electrical power distribution.  In this article I’ll be pulling some of the more important concepts together.  It is a work in progress and I’ll be adding to it bit by bit as I start closing pages down in the Wiki.  Hopefully by the time the Wiki is fully depreciated, we should have a reasonable article here. 

Alternating Current Theory

Sinewave

Sine wave Alternating voltage or current is in the form of a sine wave and is given by:

myElectrical Equation
 

and

myElectrical Equation
 

 

Where:
V(t) - is the instantaneous voltage at time, t
I(t) - is the instantaneous voltage at time, t
Vm - is the maximum value of instantaneous voltage
Im - is the maximum value of instantaneous current
ω - is the angular frequency in radians

myElectrical Equation
- is the phase angle of the voltage
myElectrical Equation
- is the phase angle of the current

The number of full cycles per second is the frequency (f) and is measured in Hertz (Hz).  The period (T) of a sine wave is the time to complete on cycle in seconds and is related to the frequency by:

myElectrical Equation

Note: In the Europe, the frequency is 50 Hz (period to 20 mS), while in the US it is 60 Hz (period 16.67 mS).

The angular frequency ω, is given by:

image00

RMS and Average Values

The average voltage or current of a pure sine wave is zero.

Even though the average is zero the power delivered by an AC source is not zero.

The root-mean-square (rms) values of current voltage and current equate the power delivered by the AC circuit.

myElectrical Equation

The average value of a sine wave is 0.637 of the maximum.

RMS Theory

The RMS value of a function is the root of the mean of the square of the function. This is expressed mathematically as:

 IRMS=12T2T0i(2)2dt

In electrical engineering this is important as it relates to a value of voltage (V) or current (I), which when multiplied together gives power:

2d107b24f126e238f558cb2bc58045ca

The RMS value can be worked out for any periodic function. For common electrical waveforms this gives:

    RMS Value
Sine Wave
myElectrical Equation
myElectrical Equation
Square Wave
myElectrical Equation
myElectrical Equation
Saw-tooth Wave
myElectrical Equation
myElectrical Equation

Complex Power

Real, reactive power and power factor can be explained using different concepts.  Sometimes it is convenient to use the concept of complex power.  In an electrical system, if the voltage and current are treated as vectors, these can be expressed in complex numbers.  The complex power S, of the system is then given by:

 S=VI*

where:
P is the complex power in VA
V is the voltage in complex form in V
I* is the conjugate of the current in complex form (the imaginary part is multiplied by -1) in A

ComplexPower

Complex Power Complex power S, is a vector and is commonly represented as:

 S=P+jQ

where:
S is the complex power in VA
P is the real power transmitted in W
Q is the reactive power transmitted in VAr
j is the complex operator √-1

The quantity P, represents the real power transmitted in the system.  This is the power that actually does work (i.e. rotating motors, heating homes, etc.).  The reactive power Q, does not do actual real work, but is responsible for maintaining the magnetic and electric fields of the electrical system.  These fields provided the energy storage mechanisms necessary for the distribution system to work. The magnitude of the complex power S, is often referred to as apparent power.

The ratio between the real and apparent power is the power factor of a system:

 power factor=P|S|

 P=|S|×power factor

 Q=|S|1pf2

If the current is lagging the voltage in phase (for example in an inductive circuit) the power factor is said to be lagging.  Conversely if the current leads the voltage the power factor is said to be leading.  For a power factor of 1, the real and apparent power will be the same, with the reactive power being equal to zero.  As the power factor is reduce, more reactive power in the system is required.  Generation of reactive power required additional equipment,  increasing transmission system costs and operating energy loss.  By ensuring high power factors these costs and loses are reduced.

AC System Components

The three main components to an alternating current (ac) system are resistance, inductance and capacitance.

Resistance

The resistance of an alternating current circuit is denoted by the symbol R.

Inductance

The impedance of a inductor XL is given by:

12040301

where:

myElectrical Equation
- is the angular frequency
f - is the frequency
L - is the inductance in henry, H
j - is the imaginary unit

Q-factor of an Inductor

While in an ideal inductor the resistance is zero, in practice there is always some resistance. Loss in the resistance makes the inductor inefficient.  The ratio of inductive reactance to resistance is the Q-factor; the higher the value the closer to an ideal conductor:

 Q=ωLR

For more on inductance, see the Notes Post on Inductance

Capacitance

The impedance of a capacitor Xc is given by:

myElectrical Equation

where:
C - is the capacitance in Farad

See also: Capacitor Theory 

 

Impedance of a Circuit

Impedance (Z) is the resistance in ohms (Ω)  to the flow of electric current in a circuit:

image01

where:
R - is the resistance of the circuit, Ω
X - is the reactance of the circuit,  Ω

Note: Admittance (Y) is the inverse of impedance (i.e. Y=1/Z).  The unit of admittance is the siemens (S)

References

  • [1] William D.Stevenson, "Elements of Power System Analysis", McGraw-Hill, 1982
  • [2] The IDC Engineers Pocket Guide, First Edition, Volume 5


Steven McFadyen's avatar
Steven McFadyen

Steven has over twenty five years experience working on some of the largest construction projects. He has a deep technical understanding of electrical engineering and is keen to share this knowledge. About the author

myElectrical Engineering




Windows Live Writer and myElectrical

When making adding a Note to our site we have a great online WYSIWYG editor and things are pretty simple.  However, if you prefer you can write, manage...

Voltage Levels – Confused?

I was having a conversation the other day about voltage levels.  While everyone was in agreement that low voltage was 1000 V and less, there was more confusion...

Equipment Verification (to IEC Standards)

One of the requirements to ensuring that everything works is to have equipment selected, manufactured and verified [tested] to IEC standards. Not all equipment...

Getting Started with Patents

If you have a great idea or invent something the last thing you want is someone to steal the idea. One of the things you can do is protect the intellectual...

EU Code of Conduct on Data Centres - Best Practices

The European Union is implementing a voluntary code of practice for participants with the aim of improving the overall efficiency of data centres. As part...

Periodic Electrical Installation Inspection – What to Inspect?

This is the second post in a series of two on periodic electrical inspections. In the first post, I discussed how often inspections should be carried out...

ABB Technical Guides - Motor Operation

ABB has produced a range of technical guides that offer concise explanations of the major technologies and technical issues in low voltage AC drives. ...

Motor Starting - Introduction

Motor starting and its associated problems are well-known to many people who have worked on large industrial processes. However, these things are, of course...

Lead us, Warleader

Delum, who had watched all in silence, his face empty of expression, now spoke in turn. ' "Lead us, Warleader, into glory."' Reading is something I do...

Load Flow Study – how they work

A load flow study is the analysis of an electrical network carried out by an electrical engineer. The purpose is to understand how power flows around...

Have some knowledge to share

If you have some expert knowledge or experience, why not consider sharing this with our community.  

By writing an electrical note, you will be educating our users and at the same time promoting your expertise within the engineering community.

To get started and understand our policy, you can read our How to Write an Electrical Note