James Long, Ph.D., P.E. Analog and RF Consulting Engineer
The Complex Conjugate Match False Fetish-Idol
To a child that has received a hammer for Christmas, everything needs hammering.
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The concept of conjugate matching is probably the most misunderstood and misused concept in electrical engineering, especially among people who memorize instead of reason.
Complex conjugate matching produces the maximum small signal transfer of power from a source that is not a transmission line to a load at the sacrifice of:
Stability (lack of self oscillation)
This is such a great sacrifice of performance parameters that the only time complex conjugate matching is used is when monetary costs are more important than anything else. This is usually the case when the active devices are very expensive and have low gains.
Large signal power
Large signal power conversion efficiency
Amplitude frequency response
Phase frequency response
Differential group delay distortion
Sensitivity to transistor production lot parameter variations
It also involves the situation where the source impedance cannot be changed. Quite frequently, people who memorize instead of reason make the source impedance complex conjugately match the load impedance which degrades the gain by 6 dB and reduces the large signal power conversion efficiency a further compounded 50%.
Making the phase angle of the load voltage and current equal is desirable. A complex load impedance produces an elliptical load line which produces distortion even in a totally "linear" circuit.
Batteries store energy. Because of cost, weight, and volume, designs size batteries for efficient extraction of energy. The only case where maximum power extraction is necessary is for short-term, one use items. Naval torpedoes are one example. To maximize energy extraction efficiency, the power loss in the internal impedance is minimized. This is done by having the equivalent load resistance significantly higher than the battery impedance.
Power utilities, the plug in the wall, supply power to a large number of users. It is not intended to source power to one load. Therefore, as is in batteries, the utility Thevenin impedance is very low and the load impedance is sized to extract the desired power. The reactive part of the load impedance is never reduced to zero by "matching" techniques. This would make a resonant circuit with large circulating currents that would dissipate more power than the poor power factor. Therefore, power factors are rarely corrected to be better than 0.8.
Modular amplifiers are designed with 50 ohm input and output impedance because the manufacturer does not know how the user is going to string the modules together. The gain is reduced by 6 dB (making the output impedance 50 ohms) in order to allow the user to drive filters and other modules whose operation depends upon a specific source and load impedances. Another advantage is the elimination of self oscillation problems because of the known source and load impedances. As in all things physical, you do not get something for nothing. The convenience comes at a drastic economic and system cost.
Power amplifiers' duty is to produce a specified output power with the maximum efficiency. This requires that the voltage and current swings go over the full device output range without exceeding the device limits on voltage, current and power dissipation. It is almost never the case that the load line that fits this slope is related to the slope of the output curves. The triode vacuum tube, in the region of positive grid voltage and high plate current, is the only case where this is true.
Transmission line power transfer to a termination is maximum (100%, no reflections) when the load impedance is the same as the line impedance. This is most important at low frequencies where the line impedance is complex. This caused great problems in the early long distance telephone lines with the production of noticeable echos. In North America the lines were terminated with simple networks and the echos were suppressed with time dependent switches. In Europe, the lines were terminated with complex networks which more exactly matched the line impedance over the whole voice band. Telephone transmissions lines is the application which originated the term "match." The term "matching network" is frequently misused. Impedance transforming networks can transform to a wide range of impedances only one of which is the optimum for a given situation. Impedance transforming networks are rarely used to produce a match or a complex conjugate match. They are used to produce a load or source impedance which produces the desirable performance parameters listed at the top of this page.
Matching network is a misnomer most of the time. Two port networks composed of inductors, capacitors, transmission lines, and transformers form two port networks that have an input impedance different in a controlled way from the load impedance placed on them. Most of the time, the input impedance desired is not equal to the source impedance of the network driving it and the term "match" is totally wrong.