Transformer Performance Measures

Two important performance measures are of interest when choosing transformers. These are the voltage regulation and efficiency of the transformer.

1. Regulation

The voltage regulation is a measure of the variation in the secondary voltage when the load is varied from zero to rated value at a constant power factor.It is typically expressed as a percentage, or per unit, of the rated output voltage at rated load. A general expression for the regulation can be written as:

% regulation = { ( V_NL- V_FL )/V_FL} x 100

where V_NL is the voltage at no load and V_FL is the voltage at full load. The regulation is dependent upon the impedance characteristics of the transformer, the resistance (r), and more significantly the ac reactance (x), as well as the power factor of the load. The regulation can be calculated based on the transformer impedance characteristics and the load power factor using the following formulas:

%regulation = pr + qx + [(px – qr)²/200]

q = SQRT (1 – p²)

- Where p is the power factor of the load and r and x are expressed in terms of per unit on the transformer base. The value of q is taken to be positive for a lagging (inductive) power factor and negative for a leading (capacitive) power factor.

- It should be noted that lower impedance values, specifically ac reactance, result in lower regulation, which is generally desirable. However, this is at the expense of the fault current, which would in turn increase with a reduction in impedance, since it is primarily limited by the transformer impedance. Additionally, the regulation increases as the power factor of the load becomes more lagging (inductive).

2. Transformer losses

There are tow types of losses in transformer :

a) Copper losses (or I²R losses or ohmic losses) in the primary and secondary windings.

b) Iron losses or core losses) in the core. This again has tow component:

i) Hysteresis losses and ii) Eddy current losses
       The copper losses (Pc) also have have tow components i) The primary winding copper loss and ii) The secondary winding copper loss
Copper losses, (P_c) = I₁² R₁ + I₂² R₂
                               = I₁² R₁ + I₁² R₂^' = I₁² R₀₁
Also                   P_c= I₂² R₂ + I₂²R₁^' = I₂² R₀₂
( assume that R₀₂= R_eq2= equivalent resistance referred to secondary side)
(For correct determination of copper losses, the winding resistance should be determined at the operating temperature of windings)
       When alternating current flows through the winding, the core material under goes cycle processes of magnetisation and demagnetisation
      This process is called hysteresis.
The hysteresis losses (in watts) is given as :

P_h = K_h B_mⁿ f v

K_h= Hysteresis coefficient whose value depends up on the material (K_his 0.025 for cast steel, 0.001 for silicon steel and 0.0001 for permalloy)
B_m= Maxium flux density (in tesla)
n = a constant 1.5 ≤ n ≤ 2.5 depending upon the material
f = Frequency ( in hertz)
v = Volume of the core material (in m²)

The eddy currents are the circulating currents set up in the core. These are produced due to magnetic flux being cut by the core. The loss due these eddy current is called eddy current losses this loss (in watts) is given by,

P_e= K_e B_m² f²t²v

K_e = Constant dependent up on the material
t   = Thickness of laminations (in meter)
       A comparison of the expression of hysteresis and eddy current losses reveals that the eddy current varies as the square of the frequency. Whereas the hysteresis loss varies directly with the frequency. The hysteresis losses can be minimized by selecting suitable ferromagnetic material for the core. The eddy current losses can be minimized by using thin laminations in building the core.
The total iron losses is given as,
                                           P_i= P_h + P_i

3. Efficiency

The efficiency of a transformer, like any other device, is defined as the ratio of useful output power to input power.

Input power = P₁

Output power = P₂

- The percentage efficiency of a transformer is in the range of 95 to 99%. For large power transformers with low loss designs, the efficiency can be as high as 99.7%.

- If we deal with the transformer as referred to the secondary side, we have

P₂ = V₂I₂ cosθ2

where I₂ is the load current. The input power P₁ is the sum of the output power and power loss in the transformer. Thus

P₁ = P₂ + P_L

The power loss in the transformer is made of two parts: the I²R loss and the core loss Pc' Thus

P_L = Pc + I₂² R_eq2

As a result, the efficiency is given by

.........(1.1)

the condition for maximum efficiency, at a given load power factor, can be derived by differentiating the expression for η with respect to I₂ and equating it to zero:

...........(1.2)

Solving it further, we get

Pc = I₂² R_eq2

............(1.3)

Thus, the maximum efficiency occurs at a load at which variable load loss equals the constant core (no-load) loss. Further,

...........(1.4)

where I_2FL is the full-load (rated) current and I²_2FL R_eq2is the load loss at the rated load conditions. Therefore, the per-unit load at which the maximum efficiency occurs is

.........(1.5)

The value of maximum efficiency can be found out by substituting the value of I₂ from equation 1.1 in equation 1.5. Similarly, it can easily be shown that the maximum efficiency, for a given load, occurs at unity power factor (cosθ=1).

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