Friday, May 6, 2011

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 = { ( VNL- VFL )/VFL } x 100

where VNL is the voltage at no load and VFL 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, (Pc) =  I12 R1 + I22 R
                               = I12 R1 + I12 R2' =  I12 R01
 Also                   Pc = I22 R2 + I22R1' = I22 R02
( assume that R02  = Req2 = 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 :
Ph = Kh Bmn f v
Kh = Hysteresis coefficient whose value depends up on the material (Kh is 0.025 for cast steel, 0.001 for silicon steel and 0.0001 for permalloy)
Bm = 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,
Pe = Ke Bm2 f2 t2 v
Ke = 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,
                                           Pi = Ph + Pi

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

PL = Pc + I22  Req2
As a result, the efficiency is given by
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:
Solving it further, we get
 Pc = I22  Req2
Thus, the maximum efficiency occurs at a load at which variable load loss equals the constant core (no-load) loss. Further,
where I2FL is the full-load (rated) current and I22FL  Req2 is the load loss at the rated load conditions. Therefore, the per-unit load at which the maximum efficiency occurs is
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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