Most transformers are used to supply power to loads that draw current with a sinusoidal waveform (blue). However some types of load, particularly smoothed D.C. power supplies, draw the current as a pulse during a small part of the mains cycle (red).
This pulse can, when analysed, be shown to consist of a series of sine waves of varying amplitude which occur at integer multiples of the source (or fundamental) frequency. These integer multiples of the fundamental frequency are known as harmonics.
The French mathematician Fourier developed a theorem in which these current pulses ‘I’ can be described by the following equation.
Where ‘ω’ relates to the fundamental frequency of the system. ‘n’ is an integer series making ‘nω’ the harmonics and ‘An ’ is the amplitude of nth harmonic component in the series
Large power transformers are prone to a loss mechanism associated within the coils known as “Eddy Current Losses”. Eddy current losses are expressed as a percentage of the transformers normal resistive coil losses (I²R) as determined by Ohms Law. They are a phenomena that increases in severity as the frequency of the current increases hence the presence of harmonic content in the current waveform will have the effect of increasing the self heating of a transformer.
That is to say a transformer of a specific size operating with an R.M.S. current of 100A sinusoidal at the fundamental frequency would when loaded with a 100A R.M.S. current in the form of a pulse, overheat due to the harmonic effects of the eddy current losses.
The ‘K-Factor’ is a number calculated from the harmonic spectrum of the current in a load waveform such that, regardless of the actual spectrum, loads that require the same K Factor have the same heating effect on the transformer. In this way any load can be specified by its K-Factor and transformer compensation for K-Factor becomes standardised.
Typical K-Factors that are specified are 4, 10, 13 and 20. The higher the number, the worse the harmonic load and the greater the compensation required.
The effective rating KVAEFF of a transformer required to supply power to a K-Factor load is given by the equation
KVAACT is the power required by the system
K is the K factor of the load
% is the eddy current loss factor expressed as a decimal (5% = 0.05)
It should be noted that some effort can be made in the design of the transformer to reduce the required size increase by reducing the eddy current losses ‘%’
The K-Factor is calculated from the harmonic spectrum of the load current by the following equation
Where the value In is the amplitude of the nth harmonic current normalized such that
Where IRMS is the total R.M.S. load current
One particular feature of K-Factor transformers is they are subject to large currents flowing within the neutral line. With a normal transformer supplying a balanced system of purely sinusoidal loads at the fundamental frequency the total current flowing in the neutral line is given by
As with a balanced load I1 = I2 = I3 and by definition
i.e the currents cancel out so no current flows in the neutral line. In practice there are usually some small differences between the currents in each phase such that a small current amounting to a few percent of the rated load current flows.
When harmonics are present however the neutral current is given by
Here the 3rd, 9th etc harmonics do not cancel out but reinforce as
Hence the neutral current will be given by
Typically K-Factor transformers are fitted with neutral bars that are rated for double the normal rated load current of the transformer