## Matched Load: Theoretical Foundations of Practical Benefits

In the educational literature, load matching is most often considered in relation to the wave properties of a long line during the transmission of a high-frequency signal. But here the energy relations are considered during the transfer of energy from an active two-terminal network to a passive one.

### What is a matched load

An active two-terminal network is any source of electrical energy, and a passive one is its consumer, which is most often referred to as a load. Let us take the source of EMF as an active two-terminal network with a known output impedance (see the schematic diagram in Fig.1) and calculate at what load resistance the power transmitted to the load will be maximum.

According to Ohm’s law for a complete electrical circuit, the current in the load is:

(1)

where is:

US – open-circuit voltage of an active two-terminal network;
RS – output resistance of an active two-terminal network;

The open circuit voltage of the active two-terminal network is measured with a high-resistance voltmeter at the terminals of the active two-terminal network with the load disconnected.

(2)

(3)

Since both values of RS and RL are measured in ohms, to simplify further mathematical calculations, we express the value of RL through the proportionality coefficient k, showing how many times the load resistance RL differs from the output resistance of the active two-terminal RS:

(4)

Then, after substitution, the expression for the power in the load (3) will take the form:

(5)

Since the left multiplier in the last expression is a constant value, the maximum power in the load will coincide with the maximum of the right multiplier, that is, the function:

(6)

The function f(k) will take the maximum value at such k, at which its derivative with respect to k will be equal to zero. The derivative of the quotient of two functions is defined as:

Consequently:

(7)

Obviously, the derivative takes on a zero value only when k=1, that is, when the equality RL=RS is fulfilled. The maximum power in the load at k=1 is clearly visible on the graph shown in Fig.2:

Thus, the maximum power is transferred to the load when the output resistance of the active two-terminal network and the load resistance are equal. In this case, the load resistance is said to be matched with the output impedance of the electrical energy source or with the output impedance of the signal source.

### Matching transformer

In sound engineering, telephony, radio engineering and telecommunications, a matching transformer is used to match the load with AC signal sources and often also used for galvanic isolation of the signal source and load. The principle of operation of the transformer as such is described in detail in the educational literature, we will only show how to choose the ratio of the number of turns of the primary and secondary windings of the matching transformer to obtain the maximum power in the load.

As an active two-terminal network, we take AC EMF, and connect the load to it through a matching transformer (see the schematic diagram in Fig.3). Let the output resistance of the signal source and the load resistance be known values. Losses and other design limitations in the matching transformer are neglected. Then the power received by the load is equal to the power supplied by the active two-terminal network.

The currents in the windings of an ideal transformer is determined by the ratio of the number of turns of the primary and secondary windings:

(8)

where is:

III – current in the secondary winding of the transformer;
II – current in the primary winding of the transformer;
wI – the number of turns of the primary winding of the transformer;
wII – the number of turns in the secondary winding of the transformer.

The ratio of the voltage values at the terminals of the windings of an ideal transformer is determined by the reciprocal of:

(9)

where is:

UII – voltage at the terminals of the secondary winding;
UI – voltage at the terminals of the primary winding.

If the load is disconnected in the circuit shown in Fig.3, that is, in the absence of current in the secondary winding of an ideal transformer, the voltage at the terminals of the secondary winding will be determined by the open-circuit voltage of the active two-terminal network and the ratio of the number of turns of the transformer windings:

(10)

With a short-circuited secondary winding, when the voltage at the terminals of the primary winding will also be equal to zero, the current in the secondary winding is determined by the value of the open-circuit voltage of the active two-terminal network, its output resistance and the inverse ratio of the number of turns of the transformer windings:

(11)

In accordance with Thévenin’s theorem, the part of the circuit shown in Fig.3, consisting of an active two-terminal network A and an ideal matching transformer, can be replaced by an equivalent active two-terminal network with an open-circuit voltage determined by expression (10), and the output resistance:

(12)

Fig.4 shows the equivalent circuit of such a replacement: