Impedance Matching: Minimum Reflection vs Maximum Power Transfer

Question

1. The minimum reflection is achieved when the two "circuits" being connected have the same impedance. This makes sense because all the components of the wave can propagate without needing to be modified.

2. The maximum power transfer is achieved via complex conjugate matching. This is because the real parts of the source and load impedance must be equal in order to maximize the power transfer and reactances cancel out when complex conjugate impedances are used.

These two statements seem contradictory...

How is it that the maximum power is transfered into a system when the reflection is not minimized?

Answer 1

Reflections occur or are noticeable when there is a transmission line involved and that transmission line is long enough for significant reflections to occur. This is generally accepted to be a length of about one-tenth of a wavelength. So, at 1 MHz, the wavelength is 300 metres and so unmatched transmission line problems start at about 30 metres. Higher frequencies naturally have unmatched problems on shorter line lengths.

However, the impedance of a transmission line for radio frequencies of about 1 MHz and above can be taken to be purely resistive. In other words it doesn't present a complex impedance hence it should be matched with an equivalent resistance to avoid reflection problems and this also ties in with the maximum power transfer. So no real problems here.

For an antenna, it can have a highly capacitive impedance if it is regarded as "short". An example being a monopole that is less than one-quarter of a wavelength. The radiation resistance it would naturally present when a quarter wave long would fall from 37 ohms to a much smaller figure when the antenna is shortened. The effective series capacitive reactance rises from near-zero at a quarter wave to tens, hundreds or thousands of ohms as the antenna shortens.

So this is an example of where using an inductor (a conjugate component) can cancel the short antenna's capacitive impedance and allow a better transfer of power.

An antenna is used to match a circuit impedance (50 Ohm) to the free space impedance (120 pi). Ideally we get no reflection, thus I would expect that all the power has been transferred to the environment

Of course there is a reflection - that is the mechanism by which we get an impedance transformation to that of free space at a particular frequency. And, adding a conjugate component to cancel out the inherent capacitive reactance of a "short" antenna doesn't alter how the antenna works but it does allow a better transfer of power.

Answer 2

Power is real - the in phase (instantaneous) product of current and voltage.

Currents into a reactance (imaginary) load do not result in actual power transfer. From one point of view any reactive component is reflection.

When you present the conjugate it is cancelled at the junction point.

If you had matched sign reactances, then you'd be just adding them, and doubling the reflection.

Answer 3

From a circuit design point of viee, in a Smith chart, we want the output impedance of a source to be close to either the point of maximum output power or the point of conjugate impedance of zero reflection.

Let's say the source that gives power to the next stage is A, and the next stage is B. Maximum output power means that A puts maximum power into this point.

Conjugate impedance of zero relection means that the energy from A will be minimally reflected back from B to A. When it goes: 'maximum power is transfered', it means not only that A supplies the most, also that B will minimally reflect the power back.