Chapter 35: Maximum Power Transfer Theorems and Impedance Matching
Date: 11/08/19
Why it is important to understand
This theorem states that maximum power transfer occurs when the impedance of the source equals the impedance of the load, and vice versa. It is also called Jacobi’s Law. A related concept is reflection-less impedance matching. This is where the source impedance, such as on a transmitter, matches the load impedance, such as an antenna, to avoid reflections in the transmission line.
In electronics, impedance matching is the practice of matching the input impedance of an electrical load (or the output impedance of its corresponding signal source) to maximize power transfer or minimize signal reflections from the load. Matching a load to a source for maximum power transfer is extremely important in microwaves, as well as all manner of low frequency applications such as stereo sound systems, electrical generating plants, solar cells and hybrid electrical cars. Impedance matching is very important where small signals are involved.
35.1 Maximum Power Transfer Theorems
A network that contains linear impedances and one or more voltage or current sources can be reduced to a Thevenin equivalent circuit. When a load is connected to the terminals of the equivalent circuit, power is transferred from the source to the load. A Thevenin equivalent circuit with source E, internal impedance z = (r + jx), and a complex load impedance Z = (R + jX) is often analyzed.
The idea is the internal impedance is fixed, and you have to choose a matching load. The conditions for maximum power transfer depend on the following four conditions:
Condition 1
If the load consists of a purely variable resistance R, for maximum power to be transferred:
Thus, the purely resistive load will have to be equal to the magnitude of the source impedance.
Condition 2
If both the load and the source are purely resistive, then R = r, which is the DC condition.
Condition 3
If the load has both variable resistance R and variable reactance X, then for maximum power transfer:
Condition 4
If the load has a variable resistance R and a fixed reactance X, then the resistance R for maximum power transfer is given by:
35.2 Impedance Matching
The mains supply is considered an emf source of infinitely large capacity, such that it is unnecessary to consider the conditions for maximum power transfer. With transmission lines, the lines are usually matched to their characteristic impedance to prevent signal reflection.
With DC generators or secondary cells, the internal impedance is usually very small. In such cases, if an attempt is made to match the load impedance as small as the internal impedance, overheating of the source results.
One method of achieving maximum power transfer between a source and a load is to adjust the value of the load impedance to match the source impedance. This can be achieved using a matching transformer.
For an ideal matching transformer supplying a load impedance, the relationships are:
The primary input impedance (Z1) is given by:
Since the magnitude of the load impedance is |ZL| = V2 / I2, we get:
If the input and load impedance are purely resistive, the equation is:
The goal is to match the impedances by choosing the correct turns ratio.