Transformer as a Matching Device Example and Formula

Transformer as a Matching Device

We recall that for maximum power transfer, the load resistance RL must be matched with the source resistance Rs. In most cases, the two resistances are not matched; both are fixed and cannot be altered.

However, an iron-core transformer can be used to match the load resistance to the source resistance. This is called impedance matching.

For example, to connect a loudspeaker to an audio power amplifier requires a transformer, because the speaker’s resistance is only a few ohms while the internal resistance of the amplifier is several thousand ohms.

Transformer as a Matching Device

Consider the circuit shown in Figure.(1).

transformer as a matching device
Figure 1. Transformer used as a matching device

We recall from the ideal transformer, that the ideal transformer reflects its load back to the primary with a scaling factor of n2. To match this reflected load RL/n2 with the source resistance Rs, we set them equal,

transformer as a matching device

Equation.(1) can be satisfied by proper selection of the turns ratio n. From Equation.(1), we notice that a step-down transformer (n < 1) is needed as the matching device when Rs > RL, and a step-up (n > 1) is required when Rs < RL.

Transformer as a Matching Device Example

The ideal transformer in Figure.(2) is used to match the amplifier circuit to the loudspeaker to achieve maximum power transfer. The Thevenin (or output) impedance of the amplifier is 192 Ω, and the internal impedance of the speaker is 12 Ω. Determine the required turns ratio.

transformer as a matching device
Figure 2

Solution:
We replace the amplifier circuit with the Thevenin equivalent and reflect the impedance ZL = 12 Ω of the speaker to the primary side of the ideal transformer.

transformer as a matching deviceFigure.(3) shows the result. For maximum power transfer,

transformer as a matching device
Figure 3

Using P = I2R, we can show that indeed the power delivered to the speaker is much larger than without the ideal transformer. Without the ideal transformer, the amplifier is directly connected to the speaker.

The power delivered to the speaker is

transformer as a matching device With the transformer in place, the primary and secondary currents are

transformer as a matching device

Hence,

transformer as a matching deviceconfirming what was said earlier.