*This article is excerpted from "Premium-Efficiency Motors and Transformers", a CD-ROM is available from CDA through the Publications List.*

Transformer losses are produced by the electrical current flowing in the coils and the magnetic field alternating in the core. The losses associated with the coils are called the load losses, while the losses produced in the core are called no-load losses.

## What Are Load Losses?

Load losses vary according to the loading on the transformer. They include heat losses and eddy currents in the primary and secondary conductors of the transformer.

Heat losses, or I ^{2}R losses, in the winding materials contribute the largest part of the load losses. They are created by resistance of the conductor to the flow of current or electrons. The electron motion causes the conductor molecules to move and produce friction and heat. The energy generated by this motion can be calculated using the formula:

**Watts = (volts)(amperes) or VI.**

According to Ohm's law, **V=RI**, or the voltage drop across a resistor equals the amount of resistance in the resistor, R, multiplied by the current, I, flowing in the resistor. Hence, heat losses equal (I)(RI) or I ^{2}R.

Transformer designers cannot change I, or the current portion of the I ^{2}R losses, which are determined by the load requirements. They can only change the resistance or R part of the I ^{2}R by using a material that has a low resistance per cross-sectional area without adding significantly to the cost of the transformer. Most transformer designers have found copper the best conductor considering the weight, size, cost and resistance of the conductor. Designers can also reduce the resistance of the conductor by increasing the cross-sectional area of the conductor.

## What Are No-load Losses?

No-load losses are caused by the magnetizing current needed to energize the core of the transformer, and do not vary according to the loading on the transformer. They are constant and occur 24 hours a day, 365 days a year, regardless of the load, hence the term no-load losses. They can be categorized into five components: hysteresis losses in the core laminations, eddy current losses in the core laminations, I ^{2}R losses due to no-load current, stray eddy current losses in core clamps, bolts and other core components, and dielectric losses. Hysteresis losses and eddy current losses contribute over 99% of the no-load losses, while stray eddy current, dielectric losses, and I ^{2}R losses due to no-load current are small and consequently often neglected. Thinner lamination of the core steel reduces eddy current losses.

The biggest contributor to no-load losses is hysteresis losses. Hysteresis losses come from the molecules in the core laminations resisting being magnetized and demagnetized by the alternating magnetic field. This resistance by the molecules causes friction that results in heat. The Greek word, hysteresis, means "to lag" and refers to the fact that the magnetic flux lags behind the magnetic force. Choice of size and type of core material reduces hysteresis losses.

## Values of Transformer Losses (A and B Values)

The values of transformer losses are important to the purchaser of a transformer who wants to select the most cost-effective transformer for their application. The use of A and B factors is a method followed by most electric utilities and many large industrial customers to capitalize the future value of no-load losses (which relate to the cost to supply system capacity) and load losses (which relate to the cost of incremental energy). Put another way, A values provide an estimate of the equivalent present cost of future no-load losses, while B values provide an estimate of the equivalent present cost of future load losses. Most utilities regularly update their avoided cost of capacity and energy (typically on an annual basis), and use A and B values when specifying a transformer. Most smaller end users typically use life-cycle -cost evaluation methods, discussed in another article on this web site.

When evaluating various transformer designs, the assumed value of transformer losses (A and B values) will contribute to determining the efficiency of transformer to be purchased. Assuming a high value for transformer losses will generally result in purchase of a more efficient unit; assuming a lower value of losses will result in purchase of a less efficient unit. What value of losses should be assumed?

The total owning cost (TOC) method provides an effective way to evaluate various transformer initial purchase prices and cost of losses. The goal is to choose a transformer that meets specifications and simultaneously has the lowest TOC. The A and B values include the cost of no-load and load losses in the TOC formula:

**TOC = NLL x A + LL x B + C**

Where,

TOC = capitalized total owning cost,

NLL = no-load loss in watts,

A = capitalized cost per rated watt of NLL (A value),

LL = load loss in watts at the transformer's rated load,

B = capitalized cost per rated watt of LL (B value),

C = the initial cost of the transformer including transportation, sales tax, and other costs to prepare it for service.

## What Is the A Value?

The A value is an estimate of the present value of future capital cost (nonload- dependent) items at a given point in time. It can vary over time as utilities re-evaluate their costs on a periodic basis. (In other words, the A value is the answer to the question, what is a watt of no-load loss over the life of the transformer worth to me today?) Even if there is no load, there is capital that is devoted to fixed capacity to generate, transmit and distribute electricity, which contribute to the A value. The loading that may change daily on the transformer does not affect the no-load loss value. It is calculated using the following formula:

**A = [SC + (EC x 8760)] x 0.001 / [FC] = Cost of No-Load Loss in $/watt**

Where,

SC = Annual Cost of System Capacity in $/kW-year (SC is the levelized annual cost of generation, transmission and primary distribution capacity required to supply one watt of load to the distribution transformer coincident with the peak load).

EC = Energy Cost (EC is the levelized annual cost per kWh of fuel, including inflation, escalation, and any other fuel related components of operation or maintenance costs that are proportional to the energy output of the generating units).

8,760 = hours per year

FC = Fixed Charge on capital per year (FC is the levelized annual revenue required to carry and repay the transformer investment obligation and pay related taxes, all expressed as a per-unit quantity of the original).

0.001 = conversion from kilowatts to watts.

## What Is the B Value?

Similar to the way the A value is determined, the B value is an estimate of the present value of future variable, or load-dependent, cost items at a given point in time. (In other words, the B value is the answer to the question, what is a watt of load loss over the life of the transformer worth to me today?) The B value can also change over time as utilities revaluate their costs on a periodic basis, but once determined, it is a constant value for a given transformer purchase. The cost of load losses, or B value, is calculated using the following formula:

**B = [(SC x RF) + (EC x 8,760 x LF)] (PL) ^{2} (0.001) / (FC) = Cost of Load Loss Cost $/watt**

Where,

RF = Peak Loss Responsibility Factor (RF is the composite responsibility factor that reduces the system capacity requirements for load losses since the peak transformer losses do not necessarily occur at peak time).

LF = Annual Loss Factor (LF is the ratio of the annual average load loss to the peak value of the load loss in the transformer).

PL = Uniform Equivalent Annual Peak Load (PL is the levelized peak load per year over the life of the transformer. Transformer life cycle is defined as the useful life of the asset and is usually assumed to be 30-35 years).

## Specifying A and B Values

For custom-designed transformers, manufacturers optimize the design of the unit to the specified A and B values resulting in a transformer designed to the lowest total owning cost, rather than one designed for cheapest first cost.

In situations where A and B values have not been determined (or the enduser does not utilize or specify them), such as occur in commercial or small industrial applications, the suggested technique to maximize transformer efficiency is to obtain the no-load and full-load loss values of a specific transformer, in watts. This method is discussed in the article Transformer Life-Cycle Cost, elsewhere on this web site.