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Tutorial:
Introduction to Transmission Line
Impedance Matching


The impedance matching, or Smith* chart was originally created to enable graphical solutions of transmission line problems. It maps complex load impedances into the G plane of complex reflection coefficients. The grid corresponds to lines of constant real and imaginary impedance, but the native coordinate system is actually that of G.

The transformation from impedance to G is simply that of the reflection coefficient for a transmission line (mis)match:

G = ZL-Z0

ZL+Z0

A G of zero is achieved when ZL = Z0. Let's assume that the line is driven with a generator of the same impedance, Z0 so that under these conditions maximum power is transferred to the load and minimum power reflected back towards the generator. For termination applications the goal is usually to modify or transform the load so that it moves as close as possible to the center of the chart. (green area in Gamma-centric chart: choose from pulldown at bottom-left).

A length of ideal (lossless) transmission line transforms the impedance of a terminating load, ZL according to the equation:
Z = Z0 ZL cos(l 2pi/lmb) + j Z0 sin(l 2pi/lmb)

Z0 cos(l 2pi/lmb) + j ZL sin(l 2pi/lmb)

Where l is the length of the line and l 2pi/lmb is the length of the line in radians. A couple of things can be seen by inspecting this equation. First, if ZL = Z0 then Z will always equal Z0 no matter what the line length. Second, Z is periodic in lmb/2. What may not be obvious is that the transformed load will actually trace a circle clockwise around the chart origin (the center of the circle where G=0 & ZL=Z0) as l is increased from 0 to lmb/2.

This feature is what makes the chart useful for obtaining graphical matches for complex loads. For example, a purely resistive narrowband match can be found by noting that every load circle crosses the horizontal diametric chord of j=0 twice (hide example). Finding the angle on the chart between the no-line load and one of the angles at which j=0 gives the length of line needed to make a load appear purely resistive. (Once that's been achieved, we can of course add an appropriate series or parallel resistor to match generator to line+load.)

However, adding resistors means power is lost and not transferred to the load, an undesirable state of affairs. We'd prefer to use only nominally lossless capacitors or inductors. We note that load circles must also intersect the circle of constant real impedance where Z = Z0 (hide example). As before, we can read from the chart the line length necessary to match the real part of the load to the line. All that is then required is to add a complementary inductor or capacitor between generator and line+load to cancel the reactance (remember, this is a narrowband match). Often, the inductive intersection with Z0 is chosen so that a capacitor can be used to complete the match.

It's not necessary to use any length of transmission line to make a match - matches can be made using only discretes - however, transmission lines and stubs are often preferred for cost and performance reasons as frequencies go higher and geometries shrink. Stubs are pieces of transmission line that go nowhere and are added purely for their reactive effect. The two most basic flavors are open (ZL=inf.) and shorted (ZL=0). Looking at the equation for Z above, we can see that the open stub will have an impedance of -Z0 cot(l 2pi/lmb)j, and the shorted stub Z0 tan(l 2pi/lmb)j, so a short open stub is capacitive and a short shorted stub is inductive.

*   "Smith" is a registered trademark of Analog Instruments, New Providence, NJ. Analog Instruments is not affiliated in any way with Analog Devices nor any of the material presented on this page.
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