Why do we say that the load impedance must be equal to the characteristic impedance to have adaptation/maximum power transfer?

Question

If impedance matching requires conjugate complex matching of the transmission line to the generator impedance (for optimal power transfer), why do we say that the load impedance must be equal to the characteristic impedance?

So in order to have optimal power transfer, \$Z_{in} = Z_g^*\$, (\$Z_g\$ is the source impedance and \$Z_{in}\$ the input of the network or load). How does that equal have \$Z_g = Z_{characteristic}\$ of the transmission line?

I got from class notes that in a transmission line with no losses the value of the power transmitted is constant along the line and that value depends on the load reflection coefficient given by:

$$P = \frac{|V_{i2}|^2}{2Z_L}\times(1 - |K_s|^2) $$ where,

\$V_{i2}\$ is the reflection component of the voltage, \$Z_L\$ is the characteristic impedance, and
\$K_s\$ is the load reflection coefficient.

This power can be maximized if \$K_s = 0\$, which means \$Z_L = Z_s\$.

Answer 1

The common network theorem by Thevenin says that in linear circuits every signal source can be modeled as an ideal voltage source and a series impedance. Together they are the Thevenin equivalent of the source. The equivalent is, of course, valid only at the frequency where the equivalent was calculated. As you have already caught, for max. power transmission the load impedance must be the complex conjugate of the series impedance of the source Thevenin equivalent assuming the load is the adjustable thing, not the source.

A transmission line has 2 ports - the input and the output. If you insert one between the signal source and the load it cannot be handled only as an extra series impedance. The common model (by O.Heaviside in 1885) for a practical transmission line (parallel wires, coax) presents the line as a ladder where capacitance, inductance and maybe also resistance (for presenting losses in practical materials) are distributed along and between the wires. The idea and also its basic math is shown here https://en.wikipedia.org/wiki/Transmission_line

Quite a tricky math (but compulsory for successful design calculations) shows how the impedance of the load is affected when its seen through a transmission line. Engineering students study to use the Smith Chart to speed up the calculations.

One fact: A non-lossy tranmission line with characteristic impedance Zc shows the load as a resistor only in certain cases. The commonly used case is to have a resistive load with resistance =Zc. That's also the case when there's no wave reflection from the load. We say the load is matched. The reflection coefficient from the matched load becomes zero, if one calculates it.

The total impedance of a circuit which has a lossless transmission line + a matched load is also Zc. If that load must be connected to a signal source which has a resistive series impedance A which cannot be taken off nor changed, the highest power to the load is got when Zc = A.

If A can be changed, but the line+the matched load are given, the highest power to the load is got when A=0. If A=0 one can get to the matched load even higher power by feeding the line through a network which makes the voltage higher - for ex through a transformer.

If an ideal voltage source with no series resistance(or at least the series resistance is other than Zc) feeds a load through a transmission line the case depends heavily on the frequency and the length and the properties of the line and what's the load. Generally it's false to say that the maximum power transmission happens when the load is a resistor =Zc. The maximum power is outputted from the source when the source sees the complex conjugate of its Thevenin equivalent series impedance. But if the line is lossy quite a big part of that power maybe doesn't reach the load.

As seen from the load side the Thevenin equivalent of the transmission line + signal source define which load takes the biggest power when the line and the source cannot be changed. The load must be the complex conjugate. If the line happens to be lossless and the source has zero series resistance the power to the load can be as high as wanted by decreasing the load resistance and by cancelling the Thevenin equivalent impedance (=reactive) by inserting the opposite series reactance. Of course, there's reflections forth and back the line, but if they are not considered harmful for any practical reasons, the power to the load can be as high as wanted.

In practical circuits we see matching networks for 3 reasons:

1. Reflections are harmful. They distort modulated signals and can generate overvoltages and too high currents in practical signal sources.

2. Power transmission must be effective enough for the purpose

3. With weak signals like in radio receivers the noise performance must be optimized One wants the highest possible signal to noise ratio even if the power transmission becomes less than optimal.

Answer 2

It's much easier to look at the DC case, which is exactly the same (just read Impedance as Resistance). https://www.electronics-tutorials.ws/dccircuits/dcp_9.html The formulas leading up to "Graph of Power against Load Resistance" are very simple.