K-factor & Transformers

Transformers serving heavy nonlinear loads are subject to increased

winding temperatures due to the harmonic currents generated by those
loads. This overheating can result in a shortened service life for the
transformer. For example, operating a transformer at 10 degrees C
above its insulation-rated class will cause approximately a 50%
reduction in the transformer's life. If the over-temperature gets high
enough or lasts long enough, the insulation will fail, resulting in a
transformer failure. K-factor-rated transformers are designed to
compensate for harmonic loading, thereby preventing excess heating.

The IEE Std provides the definition for the K-factor. C57.110-2008 (IEEE
Recommended Practice for Establishing Transformer Capacity When
Supplying Non-sinusoidal Load Currents) states:

“A rating optionally applied to a transformer indicating its suitability for

use with loads that draw non-sinusoidal currents.

Where:

Ih(pu) = the rms current at harmonic “h” (per unit of rated rms load
current);
h = the harmonic order.”
This paper aims to demystify the subject and provide a basic
understanding of the impact of the K-factor on transformer design and
operation.

Although transformers are inherently very efficient, there are losses

associated with their design and loading. These losses comprise core,
winding (I2R), and eddy currents. Although core loss and winding loss
values are largely constant and directly dependent on the quality and
amount of material used, eddy currents can vary depending on load
profiles.

In a transformer, the primary windings induce a voltage in the secondary

windings through an expanding and contracting magnetic field. Eddy
currents are stray electricity currents created by induction in conductors.
These counter-electromotive forces (emf) are induced in opposition to
the original field, thereby creating opposition to current flow
(resistance), which translates to losses.

Eddy current losses are expressed as a percentage of the transformer’s

normal winding (I2R) as determined by Ohms law. They are a
phenomena that increase in severity as the frequency of the current
increases.

In an electrical power system, harmonics are current and voltage with

frequencies that are integer multiples of the fundamental power
frequency. That is, in a power system with a fundamental frequency of
60Hz, the second harmonic is 120Hz, the third harmonic is 180Hz, and
so on. Harmonics have no useful purpose yet contribute to losses and
lower system efficiency. Harmonics return over the neutral and are
dissipated as heat in connecting cables and transformers. These
frequencies are referred to as non-sinusoidal loads. The presence of
non-sinusoidal harmonic content in the current waveform will increase
eddy current losses in the transformer, leading to “harmonic distortion”
of the fundamental power frequency waveform.

In the image to the right “A” depicts a single 60Hz cycle waveform
(fundamental power frequency). “B” represents a 3rd harmonic (180Hz)
waveform. The resultant waveform as shown in “C” provides an example
as to the impact that the harmonic load may have on the fundamental
waveform. Although the example depicts an unlikely 3rd harmonic
magnitude, it provides a graphic depiction as to the impact that non-
sinusoidal content can have on the fundamental waveform. The
magnitude of the distortion is dependent on the number and
magnitude of the total harmonic load profile.
The table provides an example of environments in which various K-
factor rated transformers would be used. A transformer “K-factor” rating
conveys its ability to manage varying degrees of nonlinear loads
without exceeding the rated temperature rise limits. For any given
nonlinear load, if the harmonic current components are known, the K-
factor can be calculated and compared to the transformer’s nameplate
K-factor. As long as the load K-factor is equal to or less than the
transformer’s rated K-factor, the transformer does not need to be de-
rated. The higher the K-factor, the more non-linear loads the
transformer can handle. The actual formula to determine K-factor takes
into account the frequency and current intensity of each individual
harmonic.

The identification, measurement, and determination of the presence of

non-sinusoidal frequency loads is essential in determining the impact
on a transformer load. ANSI/IEEE C57.110, is the guide for determining
the heating effects of nonlinear loads. It developed an equation for
calculating these heating effects. By squaring the frequency and the
per-unit current and multiplying them together, the guide arrived at a
number without a designation. Originally it was going to be called C for
“constant”, but was decided against because of possible confusion with
“centigrade”. The letter K for “Konstant” was selected and Underwriters
Laboratory used this designation in the original submission of a low
voltage dry type transformer. K since became the standard measure of
the ability of a transformer to withstand nonlinear loads.

The K-factor is a number derived from a numerical calculation based on

the summation of harmonic currents generated by the non-linear load.
The higher the K-factor, the more significant the harmonic current
content.

Standard K-factor transformers come in K-factors of 4, 9, 13, 20, 30, 40,

and 50. After K-factor load calculations are made, a transformer rated
equal to or higher than the result is specified. It is more economical to
purchase a K- factor transformer than to use a de-rated oversized
transformer.
 As a “rule of thumb”:

 0% electronic, 100% electrical – standard (K-1 rated) transformer

 25% electronic, 75% electrical – K-4 rated transformer
 50% electronic, 50% electrical – K-9 rated transformer
 75% electronic, 25% electrical – K-13 rated transformer
 100% electronic, 0% electrical – K-20 rated transformer

“electronic” = Nonlinear Loads

“electrical” = Inductive and Resistive Loads

K-factor rated transformers are preferred over oversized (de-rated)

conventional transformers because they are designed to supply
nonlinear loads, are equipped with 200% rated neutral bus, and are
likely to be smaller and less expensive. Disadvantages of an over-sized
standard transformer may include the requirement for a higher short-
circuit rating on circuit breakers and a higher inrush current. De-rating
a standard transformer is only a temporary fix that often translates into
lower efficiency operation.

To calculate the K-factor, multiply the square of the percentage of

harmonic current by the square of the harmonic order and add the
results. For example, if a load is 100% of the fundamental, 65% of the
third harmonic, 30% of the fifth harmonic, and 35% of the seventh
harmonic, the resulting K-factor would be 12.93

(1.00² * 1²) + (.65² * 3²) + (.30² * 5²) + (.35² * 7²) = 1.00 * 1 + (.42 * 9) +
(.09 * 25) + (.12 * 49) = 1.00 + 3.8 + 2.25 + 5.88 = 12.93

In this example, a transformer with a K-factor of 13 should be specified.

The K-factor rating defines the transformer’s ability to withstand odd-
harmonic currents while operating within its insulation class. When the
K-factor is unknown, a transformer may be selected by using the above
“Examples…” table as a guide.

For existing installations, one can validate the K-factor load by using a
3 phase analyzer such as pictured below.

Such a device can also be used to check the impact of adding additional
load devices on K-factor loading so as to insure that the existing
transformer can be used.

So, what changes must be made to a transformer design in order to

accommodate the additional losses caused by non-sinusoidal loads?
The most notable is that the capacity of the transformer neutral is
increased a minimum of 200% of the transformer kva rating. This is to
accommodate the presence of triplen harmonics. The triplen harmonics
are defined as the odd multiples of the 3rd harmonic (ex. 3rd, 9th, 15th,
21st etc.). Triplen harmonics are of particular concern because they are
zero sequence harmonics, unlike the fundamental, which is positive
sequence. The consequence of this fact is that the magnitude of these
currents on the 3 phases are additive in the neutral which if not
accommodated for, can lead to significant heating.
For K-factor rated transformers, it is utilized round coil construction with
cruciform cores which allow for 360 degree cooling ducts. This approach
minimizes the potential for localized heating within the coils since the
cooling fluid flows freely throughout the core/coil assembly.

With the additional loss requirements identified via the K-factor rating,
additional cooling radiators may be added to insure that while
delivering nameplate capacity, the temperature limit will not be
exceeded. In extreme cases (K-20 and above), winding conductor
current densities and/or core material flux densities may be adjusted.

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