Conjugate Matching vs. Reflectionless Matching in Electronic Circuits
To transfer maximum power, the source and the load impedance should be the complex conjugate of each other—this is called conjugate matching. To minimize reflections, the two impedances are kept equal. This type of impedance matching is called reflectionless matching.
Conjugate and reflectionless impedance matching become identical when all the components in a circuit are purely resistive.
If the circuit operation is limited to the narrow frequency range to which the matching circuits are designed, the circuit is both conjugate matched and reflectionless matched.
In electronics circuits, impedance matching is employed to maximize power transfer or to minimize reflections from the load. Impedance matching is always performed between two terminations, commonly the source and the load.
To transfer maximum power, the source and the load impedance should be the complex conjugate of each other. This is called conjugate matching. To minimize reflections, the two impedances are kept equal. This type of impedance matching is called reflectionless matching.
Conjugate matching requires load impedance to be conjugate of source impedance, whereas reflectionless matching needs load impedance to be the same as source impedance. When we compare conjugate matching to reflectionless matching, there is some confusion regarding the conditions that must be satisfied by the source and load impedances to achieve maximum power transfer and minimum reflection simultaneously.
Let's first discuss the basics of comparing conjugate matching vs. reflectionless matching.
Conjugate vs. Reflectionless Matching
Matching circuits connected between source and load
Consider a system with a source impedance equal to ZS and a load impedance equal to ZL. When impedance matching is required to achieve maximum power transfer, then conjugate matching is employed. The condition for conjugate matching is:
If the circuit impedances are matched by achieving minimum signal reflection, then it is considered reflectionless matching. The condition for reflectionless matching is:
Conjugate and reflectionless impedance matching become identical when all the components in a circuit are purely resistive. Since there is no imaginary part in purely resistive systems, there are no conjugate impedances possible or available. To summarize, in linear systems, conjugate matching is the same as reflectionless matching.
When matching the impedance in DC circuits, the source resistance is matched with the load resistance. This helps to maximize power transfer as well as minimize signal reflection in DC circuits. The presence of reactive components in the circuit requires complex conjugate impedances in the load end.
Let’s take a closer look at the specifics of conjugate matching versus reflectionless matching.
Conjugate matching is also called complex conjugate matching or maximum power transfer matching. In the presence of reactive components, the maximum power is transferred to load when the load resistance and source resistance are equal and the load reactance is the negative of the source reactance.
It can be mathematically explained as:
If , then the condition for conjugate matching is:
As shown in the figure above, matching circuits should be introduced between the load and source to make the condition given in the equation above true. The matching circuits are realized using resistance, inductance, and capacitance. As these circuits utilize inductance and capacitance, they are designed for a particular frequency or narrow band of frequency.
Here is where confusion often arises—if the circuit is complex conjugate matched, how about the signal reflections in the circuit? We discussed how the frequency of operation of the circuit influences the design of matching circuits. If the circuit operation is limited to the narrow frequency range to which the matching circuits are designed, the circuit is both conjugate matched and reflectionless matched. In the designed narrow frequency range, the signal reflections are reduced along with maximum power transfer.
In transmission lines, for reduced reflections, re-radiation, and voltage nulls, reflectionless matching is used. The transmission line has characteristic impedance and it plays an important role in reflectionless matching. When the load impedance and source impedance are equal to the transmission line characteristic impedance, reflectionless matching functions as maximum power transfer matching as well. The condition for reflectionless matching in a transmission line can be mathematically expressed as the following, where Zo is the transmission line characteristic impedance:
In electronic circuits, sometimes we need to focus on maximum power transfer, and other times the focus needs to be on reflections. Depending upon the requirements, conjugate matching or reflectionless matching can be used. In both cases, the source and load resistances satisfy the same condition. It is the presence of reactances that makes conjugate matching different from reflectionless matching. If we are operating the impedance matched circuits in the designed frequency range of matching circuits, maximum power transfer and minimum reflections can be achieved simultaneously.
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