Transformer as a Matching Device Example and Formula

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

After we learn about the linear transformer, we will have some important explanation such as:

1. Mutual inductance and dot convention
2. What is ideal transformer
3. Three phase transformer
4. Transformer as isolation device
5. Transformer as matching device

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).

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 R_L/n² with the source resistance R_s, we set them equal,

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 R_s > R_L, and a step-up (n > 1) is required when R_s < R_L.

Read also : instrumentation amplifier circuit

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.

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

Figure.(3) shows the result. For maximum power transfer,

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

With the transformer in place, the primary and secondary currents are

Hence,

confirming what was said earlier.