13.1 Introduction

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13.1 Introduction

The circuits we have considered so far may be regarded as conductively coupled, because one loop affects the neighboring loop through current conduction. When tw o loops with or without contacts between them affect each other through the magnetic field generated by one of them, they are said to be magnetically coupled.

The transformer is an electrical de vice designed on the basis of the concept of magnetic coupling. It uses magnetically coupled coils to transfer energy from one circuit to another. Transformers are key circuit elements. They are used in power systems for stepping up or stepping down ac voltages or currents. They are used in electronic circuits such as radio and television receivers for such purposes as impedance matching, isolating one part of a circuit from another, and again for stepping up or down ac voltages and currents.

We will begin with the concept of mutual inductance and introduce the dot convention used for determining the v oltage polarities of inductively coupled components. Based on the notion of mutual inductance,

we then introduce the circuit element known as the transformer. We will consider the linear transformer, the ideal transformer, the ideal autotransformer, and the three-phase transformer. Finally, among their important applications, we look at transformers as isolating and matching devices and their use in power distribution.

13.2 Mutual Inductance

When two inductors (or coils) are in a close proximity to each other, the magnetic flux caused by current in one coil links with the other coil, thereby inducing v oltage in the latter . This phenomenon is kno wn as mutual inductance.

Let us first consider a single inductor, a coil with N turns. When current i flows through the coil, a magnetic flux ϕ is produced around it (Fig. 13.1). According to Faraday’s law, the voltage v induced in the coil is proportional to the number of turns N and the time rate of change of the magnetic flux ϕ; that is,

$$v = N \frac{d\phi}{dt} \tag{13.1}$$

But the flux ϕ is produced by current i so that any change in ϕ is caused by a change in the current. Hence, Eq. (13.1) can be written as

$$v = N \frac{d\phi}{di} \frac{di}{dt}$$

(13.2)

or

$$v = L \frac{di}{dt}$$

(13.3)

which is the voltage-current relationship for the inductor. From Eqs. (13.2) and (13.3), the inductance L of the inductor is thus given by

$$L = N \frac{d\phi}{di}$$

(13.4)

This inductance is commonly called self-inductance, because it relates the voltage induced in a coil by a time-varying current in the same coil.

Now consider two coils with self-inductances L1 and L2 that are in close proximity with each other (Fig. 13.2). Coil 1 has N1 turns, while coil 2 has N2 turns. F or the sak e of simplicity, assume that the second inductor carries no current. The magnetic flux ϕ1 emanating from coil 1 has two components: One component ϕ11 links only coil 1, and another component ϕ12 links both coils. Hence,

$$\phi_1 = \phi_{11} + \phi_{12} \tag{13.5}$$

Although the tw o coils are ph ysically separated, the y are said to be magnetically coupled. Since the entire flux ϕ1 links coil 1, the v oltage induced in coil 1 is

$$v_1 = N_1 \frac{d\phi_1}{dt} \tag{13.6}$$

Only flux ϕ12 links coil 2, so the voltage induced in coil 2 is

$$v_2 = N_2 \frac{d\phi_{12}}{dt}$$

(13.7)

Figure 13.1 Magnetic flux produced by a single coil with N turns.

Again, as the fluxes are caused by the current i1 flowing in coil 1, Eq. (13.6) can be written as

$$$$