# Phasor Diagrams & Testing Transformer

The resulting equivalent circuit is known as the exact equivalent circuit. This circuit can be used for the analysis of the behavior of the transformers. As the no-load current is less than 1% of the load current a simplified circuit known as ‘approximate’ equivalent circuit is usually used, which may be further simplified to the one.

On similar lines to the ideal transformer the phasor diagram of operation can be drawn for a practical transformer also. The positions of the current and induced emf phasor are not known uniquely if we start from the phasor *V*_{1}. Hence it is assumed that the phasor*φ *is known. The *E*_{1} and *E*_{2} phasor are then uniquely known. Now, the magnetizing and loss components of the currents can be easily represented. Once *I*_{0} is known, the drop that takes place in the primary resistance and series reactance can be obtained which when added to *E*_{1} gives uniquely the position of *V*_{1} which satisfies all other parameters. This is represented in phasor diagram on no-load.

Next we proceed to draw the phasor diagram corresponding to a loaded transformer. The position of the *E*_{2} vector is known from the flux phasor. Magnitude of *I*_{2} and the load power factor angle *θ*_{2} are assumed to be known. But the angle *θ*_{2} is defined with respect to the terminal voltage *V*_{2} and not *E*_{2}. By trial and error the position of *I*_{2} and *V*_{2} are determined. *V*_{2} should also satisfy the Kirchoff’s equation for the secondary. Rest of the construction of the phasor diagram then becomes routine. The equivalent primary current *I*2 is added vectorially to *I*_{0} to yield *I*_{1}. *I*_{1}(*r*_{1} + *jx*_{l}_{1})is added to *E*_{1} to yield *V*_{1}. phasor diagram for a loaded transformer.

**Phasor diagrams of above transformer:-**

- Pure resistive,
- Resistive-inductive,
- Resistive-capacitive loads

**(a) For pure restive load**

**(b) For restive-inductive load**

**(c) For resistive-capacitive load**

**Testing of Transformers:-**

The structure of the circuit equivalent of a practical transformer is developed earlier. The performance parameters of interest can be obtained by solving that circuit for any load conditions. The equivalent circuit parameters are available to the designer of the transformers from the various expressions that he uses for designing the transformers. But for a user these are not available most of the times. Also when a transformer is rewound with different primary and secondary winding’s the equivalent circuit also changes. In order to get the equivalent circuit parameters test methods are heavily depended upon. From the analysis of the equivalent circuit one can determine the electrical parameters. But if the temperature rise of the transformer is required, then test method is the most dependable one. There are several tests that can be done on the transformer; however a few common ones are discussed here.

**Winding Resistance Test:-**

This is nothing but the resistance measurement of the windings by applying a small d.c voltage to the winding and measuring the current through the same. The ratio gives the winding resistance, more commonly feasible with high voltage windings. For low voltage windings a resistance-bridge method can be used. From the d.c resistance one can get the a.c. resistance by applying skin effect corrections.

**Polarity Test:-**

This is needed for identifying the primary and secondary phasor polarities. It is a must for poly phase connections. Both a.c. and d.c methods can be used for detecting the polarities of the induced emfs. The dot method discussed earlier is used to indicate the polarities. The transformer is connected to a low voltage a.c. source with the connections. A supply voltage *V*_{s} is applied to the primary and thereadings of the voltmeters *V*_{1}, *V*_{2} and *V*_{3} are noted. *V*_{1} : *V*_{2} gives the turns ratio. If *V*_{3} reads*V*_{1}−*V*_{2} then assumed dot locations are correct (for the connection shown). The beginning and end of the primary and secondary may then be marked by *A*_{1} − *A*_{2} and *a*_{1} − *a*_{2} respectively.

If the voltage rises from *A*_{1} to *A*_{2} in the primary, at any instant it does so from *a*_{1} to *a*_{2} in the secondary. If more secondary terminals are present due to taps taken from the windings they can be labeled as *a*_{3}*, **a*_{4}*, **a*_{5}*, **a*_{6}. It is the voltage rising from smaller number towards larger ones in each winding.

The same thing holds good if more secondaries are present the d.c. method of testing the polarity. When the switch S is closed if the secondary voltage shows a positive reading, with a moving coil meter, the assumed polarity is correct. If the meter kicks back the assumed polarity is wrong.

**Open Circuit Test:- **

the secondary is kept open circuited and nominal valueof the input voltage is applied to the primary winding and the input current and power are measured. *V, **A, W *are the voltmeter, ammeter and wattmeter respectively. Let these meters read *V*_{1}*, **I*_{0} and *W*_{0} respectively. shows the equivalent circuit of the transformer under this test.

The no load current at rated voltage is less than 1 percent of nominal current and hence the loss and drop that take place in primary impedance *r*_{1} + *jx*_{l}_{1 }due to the no load current *I*_{0} is negligible. The active component *I*_{c} of the no load current *I*_{0 }represents the core losses and reactive current *I*_{m} is the current needed for the magnetization. Thus the watt meter reading

The parameters measured already are in terms of the primary. Sometimes the primary voltage required may be in kilo-Volts and it may not be feasible to apply nominal voltage to primary from the point of safety to personnel and equipment. If the secondary voltage is low, one can perform the test with *LV *side energized keeping the *HV *side open circuited. In this case the parameters that are obtained are in terms of *LV *. These have to be referred to *HV *side if we need the equivalent circuit referred to *HV *side.

Sometimes the nominal value of high voltage itself may not be known, or in doubt, especially in a rewound transformer. In such cases an open circuit characteristics is first obtained, which is a graph showing the applied voltage as a function of the no load current. This graph is obtained by noting the current drawn by transformer at different applied voltage, keeping the secondary open circuited. The usual operating point selected for operation lies at some standard voltage around the knee point of the characteristic. After this value is chosen as the nominal value the parameters are calculated as mentioned above.

**Short Circuit Test:- **

The purpose of this test is to determine the series branch parameters of the equiv- alent circuit. As the name suggests, in this test primary applied voltage, the current and power input are measured keeping the secondary terminals short circuited. Let these values be *V*_{sc}*, I*_{sc} and *W*_{sc} respectively. The supply voltage required to circulate rated is thus assumed to be absent. Also *I*_{1} = *I*2 as *I*_{0} ≃ 0. Therefore *W*_{sc} is the sum of thecurrent through the transformer is usually very small and is of the order of a few percent of the nominal voltage. The excitation current which is only 1 percent or less even at rated voltage becomes negligibly small during this test and hence is neglected. The shunt branch copper losses in primary and secondary put together. The reactive power consumed is that absorbed by the leakage reactance of the two windings.

If the approximate equivalent circuit is required then there is no need to separate *r*_{1 }and *r*2 or *x*_{l}_{1} and *x*l2. However if the exact equivalent circuit is needed then either *r*_{1} or *r*2 is determined from the resistance measurement and the other separated from the total.As for the separation of *x*_{l}_{1} and *x*l2 is concerned, they are assumed to be equal. This is a fairly valid assumption for many types of transformer windings as the leakage flux paths are through air and are similar.

**Load Test:-**

Load Test helps to determine the total loss that takes place, when the transformer is loaded. Unlike the tests described previously, in the present case nominal voltage is applied across the primary and rated current is drown from the secondary. Load test is used mainly,

- To determine the rated load of the machine and the temperature rise.
- To determine the voltage regulation and efficiency of the Transformer.

Rated load is determined by loading the transformer on a continuous basis and observing the steady state temperature rise. The losses that are generated inside the transformer on load appear as heat. This heats the transformer and the temperature of the transformer increases. The insulation of the transformer is the one to get affected by this rise in the temperature. Both paper and oil which are used for insulation in the transformer start get- ting degenerated and get decomposed. If the flash point of the oil is reached the transformer goes up in flames. Hence to have a reasonable life expectancy the loading of the transformer must be limited to that value which gives the maximum temperature rise tolerated by the insulation. This aspect of temperature rise cannot be guessed from the electrical equivalent circuit. Further, the losses like dielectric losses and stray load losses are not modeled in the

equivalent circuit and the actual loss under load condition will be in error to that extent. Many external means of removal of heat from the transformer in the form of different cooling methods give rise to different values for temperature rise of insulation. Hence these permit different levels of loading for the same transformer. Hence the only sure way of ascertaining the rating is by conducting a load test.

It is rather easy to load a transformer of small ratings. As the rating increases it becomes difficult to find a load that can absorb the requisite power and a source to feed the necessary current. As the transformers come in varied transformation ratios, in many cases it becomes extremely difficult to get suitable load impedance.

Further, the temperature rise of the transformer is due to the losses that take place ‘inside’ the transformer. The efficiency of the transformer is above 99% even in modest sizes which means 1 percent of power handled by the transformer actually goes to heat up the machine. The remaining 99% of the power has to be dissipated in a load impedance external to the machine. This is very wasteful in terms of energy also. ( If the load is of unity power factor) Thus the actual loading of the transformer is seldom resorted to. Equivalent loss methods of loading and ‘Phantom’ loading are commonly used in the case of transformers. The load is applied and held constant till the temperature rise of transformer reaches a steady value. If the final steady temperature rise is lower than the maximum permissible value, then load can be increased else it is decreased. That load current which gives the maximum permissible temperature rise is declared as the nominal or rated load current and the volt amperes are computed using the same.

In the equivalent loss method a short circuit test is done on the transformer. The short circuit current is so chosen that the resulting loss taking place inside the transformer is equivalent to the sum of the iron losses, full load copper losses and assumed stray load losses. By this method even though one can pump in equivalent loss inside the transformer, the actual distribution of this loss vastly differs from that taking place in reality. Therefore this test comes close to a load test but does not replace one.

Suitable voltage is injected into the loop formed by the two secondaries such that full load current passes through them. An equivalent current then passes through the primary also. The voltage source V1 supplies the magnetizing current and core losses for the two transformers. The second source supplies the load component of the current and losses due to the same. There is no power wasted in a load ( as a matter of fact there is no real load at all) and hence the name Phantom or virtual loading. The power absorbed by the second transformer which acts as a load is pushed back in to the mains. The two sources put together meet the core and copper losses of the two transformers. The transformers work with full flux drawing full load currents and hence are closest to the actual loading condition with a physical load.