Impeadance Of Autotransformer

 Impeadance Of Autotransformer

The impedance of a transformer can be determined by measuring the impedance across the primary terminals with the secondary terminals short-circuited. Consider a two-winding transformer having a turns ratio n connected as a boosting autotransformer. The transformer impedance, which is mainly leakage reactance, is split between the common winding and the series winding as Zc and Zs, respectively. The magnetizing impedance is neglected. To determine the impedance of the autotransformer, the secondary (low-voltage) terminals are short-circuited and a voltage source Ep is applied to the primary (highvoltage) terminals, as shown in Figure 1.

The current through the common winding Ic is equal to the current through the series winding divided by the turns ratio:

Ic = Is/ n                                            eq 1

The voltage across the common winding Ec is equal to the voltage drop across Zc:

Ec = Ic x Zc = Is x Zc/ n                   eq 2

FIGURE 1 A boosting autotransformer with a short circuit applied at the low-voltage

output.

The voltage across the series winding Es is equal to Ec divided by the turns ratio:

Es= Ec/ n = Is x Zc /n2                                     eq  3

The current through the series winding Is is equal to the difference between the input voltage Ep and the  voltage across the series winding Es divided by Zs:

Is= (Ep - Es) / Zs                                               eq 4

Multiplying both sides of Eq. (.4) by Zs and substituting Eq. (.3) for Es:

Is x Zs =Ep - Is x Zc/n^2                             eq 5

Solving for Ep by rearranging Eq. (5):

Ep = Is x (Zs +Zc/n^2)                                      eq 6

The impedance as seen at the primary or high-voltage side of the transformer Zp is equal to Ep/Is. This is obtained by dividing both sides of Eq. (6) by Is:

Zp = Zs +( Zc/n^2)                                                eq 7

Notice that the value of Zp in Eq. (7) is equal to the series impedance as seen at the secondary or low-voltage side when the transformer is connected as a conventional two-winding transformer with the high-voltage windings short circuited. For a large ratio n this impedance is much smaller than the series impedance seen at the primary side.

In general, the series impedance of a small two-winding transformer is greater than the series impedance of a large two-winding transformer if the two impedances expressed as percent of the winding KVA bases are equal. However, when a small transformer is connected as an autotransformer the series impedance seen at the transformer terminals can be substantially smaller than the impedance of a much larger equivalent two-winding transformer. This can be quantified as follows. Let:

% ZAT = impedance of a unit connected as autotransformer expressed in percent of the winding KVA base
KVAAT = winding KVA base a unit connected as an autotransformer
% ZTW =impedance of a two-winding transformer expressed in percent of the winding KVA base
KVATW = winding KVA base of a two-winding transformer
ZAT = impedance, in ohms, of an autotransformer across the primary terminals
ZTW = impedance, in ohms, of a two-winding transformer across the primary terminals
r = ratio of the primary terminal voltage to the secondary terminal voltage 

n = turns ratio of the common/series windings of the autotransformer
Ep = primary voltage
Es = autotransformer series winding voltage
η = ratio of ZAT/ZTW

The impedance of a transformer in ohms is equal to the per unit impedance times the quantity E2 base/KVAbase. Recall from Eq. (7) that the impedance of an autotransformer seen at the primary terminals is equal to the series impedance as seen at the secondary or low-voltage side when the transformer
is connected as a conventional two-winding transformer. Therefore, Ebase = Es, and

Zat = 0.01 x %Zat x Es^2 /KVAat  ohms                              eq 8

For a two-winding transformer, Ebase = Ep, and

Ztw = 0.01 x % Ztw x(Ep^2/KVAtw)  ohms                            eq 9

The ratio ZAT/ZTW is found by dividing Eq. (8) by Eq. (9).

Zat /Ztw = (Es^2 x KVAtw x %ZAT)/(Ep^2 x KVAat x %Ztw)                        eq 10

If the two-winding transformer is replaced by an autotransformer having the same KVA capacity, the KVA rating of the autotransformer windings is smaller than the KVA rating of the two-winding transformer windings. Recalling the capacity multiplication factor of a boosting autotransformer in Eq. (1),

KVAtw/KVAat = Fc = r /r - 1                                      eq 11

For the boosting autotransformer connection, the secondary voltage is equal to the voltage across the common winding. The voltage across the series winding Es divided by the secondary voltage is therefore equal to 1/n:

Es/secondary voltage= 1/ n                                             eq 12

The ratio r was defined as the primary terminal voltage divided by the secondary terminal voltage:

Ep/secondary voltage=  r                                               eq 13

For a boosting autotransformer,

1/ n = r - 1,                                                                   eq 14

Es/Ep=(r - 1)/r                                                              eq 15

(Es/Ep)^2=(r - 1)^2/r^2                                                eq 16

Combining Eqs. (11) and (16) with Eq. (10), the following result is obtained.

η =( (r - 1) x %Zat)/(r %Ztw)                                     eq 17

Since %ZAT and %Ztw are expressed on the KVA bases of the transformer windings, the values of these values should be comparable even though the physical sizes and KVA ratings of the transformers may be very different.
For the special case where %Zat and %Ztw are equal,

η = (r - 1)/r < 1                                                           eq 18

The meaning of Eq. (18) is that the engineer has the opportunity to design an autotransformer with much lower conductor losses and regulation than would be practical with a two-winding transformer with the same KVA capacity. However, in order to limit short-circuit currents, the reactance component of the impedance must still be maintained above some minimum design limit. In Section 4.4, we will see that an autotransformer behaves very differently from a two-winding transformer when a short-circuit is applied to the output.

reference :

http://en.wikipedia.org/wiki/Autotransformer