Supplementary Notes on Transformers

Transcription

1 Supplementary Notes on Transformers A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors the transformer's coils. Figure 1 illustrates a typical transformer. Figure 1 A typical transformer. When a coil is wound around a core and current i passes through the coil, as depicted via Figure 2, magnetizing and leakage fluxes, Φ M and Φ L, develop. The core is usually made up of a ferromagnetic material, which enables generation of large magnetic fluxes with relatively low exciting currents and provides well a defined path for the magnetic fields. i N v Figure 2 Excited transformer. 1

2 A mathematical model for the circuit illustrated in Figure 2 can be derived as follows: Φ M : Magnetizing flux, Φ L : Leakage flux, N : Number of turns, P L : Leakage permeance P M : Magnetizing permenance i : Exciting current. The magnetizing flux is confined within the core, as, the leakage flux completes its path through the air. In this model, a linear relation between flux and current is assumed. Permeance is the measure of the ability of a magnetic circuit to conduct magnetic flux. It is analogous to the conductance term in electrical circuits. The permeability of the ferromagnetic core is much larger than that of the air; and hence, in some applications leakage flux may be negligible compared to the magnetizing flux. The total flux, Φ T, linking each turn of the coil is, The sum of fluxes linking the turns is obtained by multiplying the number of turns and the total flux,, the magnetizing inductance, the leakage inductance, the inductance 2

3 A time varying magnetic flux induces a voltage across the winding (coil): When another coil is wound on the other side of the core, a voltage is also induced across that winding. (See Figure 3) Although the winding on the left is linked by the magnetizing and leakage fluxes, the one on the right is only linked by the magnetizing flux. i 1 N 1 N 2 v 1 v 2 i 2 = 0 Figure 3 Transformer with secondary winding open circuited. The fluxes linking the primary coil, as well as the inductances, can be calculated using the formulas derived above with N = N 1. For the secondary side we have, M is referred to as the mutual inductance. 3

4 Now let us call the windings on the left and the right the primary and the secondary windings, respectively. Consider the case when the secondary side is not open circuited and i 2 is no longer zero, hence, the secondary side also introduces magnetizing and leakage fluxes, as illustrated via Figure 4. i 1 i 2 v s Figure 4 Transformer with secondary winding current non zero. and are the magnetizing and leakage fluxes generated by the primary current i 1, and and are the magnetizing and leakage fluxes generated by the secondary current i 2. For the primary side, the total flux can be written as,, 4

5 For the secondary side, the total flux can be written as,, The induced primary and secondary voltages are: The primary inductance L 1, the secondary inductance L 2 and the mutual inductance M satisfy the following relation: 0 If 0, then Finally, based on the derived model, an equivalent circuit of a transformer is given below: 5

6 i1 N1:N2 i2 v1 v2 Figure 5 An equivalent circuit model of the transformer. The equivalent circuit model depicted via Figure 5 can be modified by the inclusion of the copper losses due to the resistances of the primary and secondary windings in the model. The modified model is shown below: i1 r1 N1:N2 r2 i2 v1 v2 Figure 6 Modified equivalent circuit model of the transformer. In order to modify the equivalent circuit model further, the core losses should also be considered. There are two sources for the core losses: 1. Hysteresis effect, 2. Eddy currents. The hysteresis loss is caused by the nonlinear relation between the current and the generated magnetic flux. (Recall that for the assumed model the relation is linear.) 6

7 The eddy currents are induced in the core in order to oppose the change in the flux. Hence, in order to account for the demagnetizing effect of eddy currents, the exciting current should increase. The core losses can be modeled as a resistor in parallel with the magnetizing inductance. The equivalent circuit model accounting the core losses are illustrated in Figure 7. i1 r1 N1:N2 r2 i2 v1 rc v2 Figure 7 Further modified equivalent circuit model of the transformer. References [1] Yıldırım Üçtuğ, EE 361 Lecture Notes, Middle East Technical University, Department of Electrical and Electronics Engineering, Fall [2] Accessed 31 Nov