Ideal Transformer

Ideal Transformer


Transformers are one of the most important components of any power system. It basically changes the level of voltages from one value to the other at constant frequency. Being a static machine the efficiency of a transformer could be as high as 99%.

Big generating stations are located at hundreds or more km away from the load center (where the power will be actually consumed). Long transmission lines carry the power to the load Centre from the generating stations. Generator is a rotating machines and the level of voltage at which it generates power is limited to several kilo volts only a typical value is 11 kV. To transmit large amount of power (several thousands of megawatts) at this voltage level means large amount of current has to flow through the transmission lines. The cross sectional area of the conductor of the lines accordingly should be large. Hence cost involved in transmitting a given amount of power rises many folds. Not only has that, the transmission lines had their own resistances. This huge amount of current will cause tremendous amount of power loss or I2R loss in the lines. This loss will simply heat the lines and becomes a wasteful energy. In other words, efficiency of transmission becomes poor and cost involved is high.

The above problems may addressed if we could transmit power at a very high voltage say, at 200 kV or 400 kV or even higher at 800 kV. But as pointed out earlier, a generator is incapable of generating voltage at these level due to its own practical limitation. The solution to this problem is to use an appropriate step-up transformer at the generating station to bring the transmission voltage level at the desired value as depicted in figure 23.1 where for simplicity single phase system is shown to understand the basic idea. Obviously when power reaches the load Centre, one has to step down the voltage to suitable and safe values by using transformers. Thus transformers are an integral part in any modern power system. Transformers are located in places called substations. In cities or towns you must have noticed transformers are installed on poles – these are called pole mounted distribution transformers. These type of transformers change voltage level typically from 3-phase, 6 kV to 3-phase 440 V line to line.

In this and the following lessons we shall study the basic principle of operation and performance evaluation based on equivalent circuit.

Principle of operation:-

A transformer in its simplest form will consist of a rectangular laminated magnetic structure on which two coils of different number of turns are wound. The winding to which A.C voltage is impressed is called the primary of the transformer and the winding across which the load is connected is called the secondary of the transformer.

Ideal Transformer:-

To understand the working of a transformer it is always instructive, to begin with the concept of an ideal transformer with the following properties.

  • Primary and secondary windings has no resistance.
  • All the flux produced by the primary links the secondary winding i,e., there is no leakage flux.
  • Permeability μr of the core is infinitely large. In other words, to establish flux in the core vanishingly small (or zero) current is required.
  • Core loss comprising of eddy current and hysteresis losses are neglected.

No load Phasor Diagram:-

A transformer is said to be under no load condition when no load is connected across the secondary i.e., the switch S is kept opened and no current is carried by the secondary windings. The phasor diagram under no load condition can be drawn starting with φ as the reference phasor.

In convention 1, phasors 1 E and 2 E are drawn 180° out of phase with respect to 1 V in order to convey that the respective power flow directions of these two are opposite. The second convention results from the fact that the quantities v1(t), e1(t) and e2(t) vary in unison then why not show them as co-phasal and keep remember the power flow business in one’s mind. Also remember vanishingly small magnetizing current is drawn from the supply creating the flux and in time phase with the flux.

Transformer under loaded condition:-

A transformer gets loaded when we try to draw power from the secondary. In practice loading can be imposed on a transformer by connecting impedance across its secondary coil. It will be explained how the primary reacts when the secondary is loaded. It will be shown that any attempt to draw current/power from the secondary, is immediately responded by the primary winding by drawing extra current/power from the source. We shall also see that mmf balance will be maintained whenever both the windings carry currents. Together with the mmf balance equation and voltage ratio equation, invariance of Volt-Ampere (VA or KVA) irrespective of the sides will be established.

We have seen in the preceding section that the secondary winding becomes a seat of emf and ready to deliver power to a load if connected across it when primary is energized. Under no load condition power drawn is zero as current drawn is zero for ideal transformer. However when loaded, the secondary will deliver power to the load and same amount of power must be sucked in by the primary from the source in order to maintain power balance. We expect the primary current to flow now. Here we shall examine in somewhat detail the mechanism of drawing extra current by the primary when the secondary is loaded. For a fruitful discussion on it let us quickly review the dot convention in mutually coupled coils.

Analysis of Ideal Transformer:-

Sinusoidally varying voltage is impressed across the primary with secondary winding open circuited. Although the current drawn Im will be practically zero, but its position will be 90° lagging with respect to the supply voltage. The flux produced will obviously be in phase with Im. In other words the supply voltage will lead the flux phasor by 90°. Since flux is common for both the primary and secondary coils, it is customary to take flux phasor as the reference.

The time varying flux φ(t) will link both the primary and secondary turns inducing in voltages e1 and e2 respectively.

Instantaneous induced voltage in primary =  N = ωNsinωt-dtφφ⎛⎞⎜⎟⎝⎠

Instantaneous induced voltage in secondary =Magnitudes of the rms induced voltages will therefore be:-

The time phase relationship between the applied voltage v1 and e1 and e2 will be same. The 180° phase relationship obtained in the mathematical expressions of the two merely indicates that the induced voltage opposes the applied voltage as per Lenz’s law. In other words if e1 were allowed to act alone it would have delivered power in a direction opposite to that of V1. By applying Kirchoff’s law in the primary one can easily say that V1 = E1 as there is no other drop existing in this ideal transformer. Thus udder no load condition,

Where, V1, V2 are the terminal voltages and E1, E2 are the rms induced voltages. In convention 1, phasors 1 E and 2 E are drawn 180° out of phase with respect to 1 V in order to convey the respective power flow directions of these two are opposite. The second convention results from the fact that the quantities v1(t), e1(t) and e2(t) vary in unison, then why not show them as co-phasal and keep remember the power flow business in one’s mind.

Equivalent circuit of an Ideal Transformer:-

The equivalent circuit of a transformer can be drawn (i) showing both the sides along with parameters, (ii) referred to the primary side and (iii) referred to the secondary side. In whichever way the equivalent circuit is drawn, it must represent the operation of the transformer correctly both under no load and load condition.

Think in terms of the supply. It supplies some current at some power factor when a load is connected in the secondary. If instead of the transformer, an impedance of value a2Z2 is connected across the supply, supply will behave identically. This corresponds to the equivalent circuit referred to the primary. Similarly from the load point of view, forgetting about the transformer, we may be interested to know what voltage source should be impressed across Z2 such that same current is supplied to the load when the transformer was present. This corresponds to the equivalent circuit referred to the secondary of the transformer. When both the windings are shown in the equivalent circuit, they are shown with chain lines instead of continuous line. Why? This is because, when primary is energized and secondary is opened no current is drawn, however current is drawn when a load is present on the secondary side. Although supply two terminals are physically joined by the primary winding, the current drawn depends upon the load on the secondary side.

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