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Tutorial: Impedance Matching

Last month, instructor Bob Froelich gave a brief introduction to the concepts behind impedance matching for RF circuits. A recorded version of his presentation can be viewed on the Microwave Journal website. Some of the main points of the presentation are summarized as follows.

One can begin to understand the need for impedance matching by reviewing some basic AC circuit analysis principles. The simple mnemonic “ELI the ICE Man” helps us remember the relationship between voltage and current in inductors and capacitors. In inductors, the current lags the voltage by 90 degrees for a sinusoidal signal (E:electric field or voltage – L:inductor – I:current). Conversely for capacitors the current leads the voltage by 90 degrees (I:current – C:capacitor – E:voltage). In basic terms, maximum power transfer occurs when the current and voltage are in phase. Therefore if your circuit has a net capacitance it would be beneficial to cancel it out with some added inductance from an AC analysis perspective. This supports the rational for conjugate matching of complex loads.

Complex impedances can be visualized conveniently on the Smith Chart. In the chart, inductive terminations appear on the upper half and capacitive terminations on the lower half. The complex conjugate of any given termination appears as the mirror image across the horizontal axis of the chart. Any termination on the horizontal axis of the chart represents a termination with no reactance (a simple resistance load, for example). The very center point of the chart represents a load with the same impedance as the characteristic impedance of the system (usually 50 ohms) — a perfect match.

If you start with a certain impedance point on the chart and add an inductance in series, the new impedance seen looking into the added inductor terminated with the existing impedance will be shifted upwards on the chart by a variable amount which is directly proportional to the inductor value. The upward movement will follow a circle of constant resistance — a family of terminations which share the same resistance, but different reactance values. Adding a capacitor will move the termination in the opposite direction. Matching networks can be designed on the chart graphically by starting at the desired termination and adding components to move the impedance of the termination plus the added components to the center of the chart. Since inductor and capacitor reactances are frequency-dependent, this matching network will only work perfectly at one frequency.

Higher bandwidth matching networks can be achieved by keeping the Q factor of the circuit as low as possible. On the Smith Chart this means keeping the values as close to the center horizontal axis as possible. This can be achieved by breaking up your matching network into more segments. For example, instead of one L-C combination to shift the termination to the center of the chart, split the network into two pairs: L-C-L-C. This will have the effect of keeping the matching network traces closer to the horizontal axis on the chart. Note that if the termination that you are trying to match lies far from the horizontal axis, then you are limited in the broadband matching network that you can create with this technique. This limitation is quantified by Fano’s Theorem.

To learn more about impedance  matching as well as using the Smith Chart to design small signal amplifiers, attend our upcoming course, RF Fundamentals, this June 14-18. This course is offered via five 90 minute live web conference sessions. Attend the course with no travel expenses and continue your job with minimal interruption!

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