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:

  I RMS = 1 2T 0 2T i (2) 2 dt

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=V I *

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| 1p f 2

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= ωL R

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

comments powered by Disqus



Maximum Demand for Buildings

Estimating maximum demand is a topic frequently discussed. Working out how much power to allow for a building can be very subjective . Allowing too much...

Three Phase Current - Simple Calculation

The calculation of current in a three phase system has been brought up on our site feedback and is a discussion I seem to get involved in every now and...

What is LED?

Light Emitting Diodes (LED ) are increasing gaining favour in both the domestic and commercial sectors; due to their efficiency, sustainability and durability...

Earth Electrode Resistance

Earthing of electrical systems is essential for the correct functioning and the protecting of life and equipment in the event of faults.  The earth electrode...

Control Theory

Control theory looks at how systems work and are controlled from a mathematical view.  This note gives a brief introduction to some of the concepts – more...

Post Editing Tips

If you at all familiar with programs like office and outlook, then adding and editing posts is pretty straightforward and intuitive.  However, there are...

Electromechanical Relays

Electromechanical relays have been the traditional backbone of electrical protection systems.  While over recent years these have been replaced by microprocessor...

Voltage Drop in Installations - Concepts

Problems on achieving maximum voltage drop within an installation come up often. Depending where you live, local regulations will have different limits...

Contribute to myElectrcial

Have an opinion or something to say, want to ask or answer questions, share your knowledge then use our site to do it . As a community of people interested...

Battery Cars A to Z

Battery powered cars are a hot topic and widely debated. The pros, cons, issues and time frames can be talked about endlessly. An article by the Telegraph...

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