I used an Android app called Smith Chart Calc, which you can find at the Google Play store, to solve the Figure 5 matching-network problem. Indeed, RF/microwave design-automation suites have Smith-chart functions, and you can find standalone tools as well. It seems there should be ways to automate this process. An app solves the impedance-matching problem. We reverse the normalization to learn that the inductor and capacitor have impedances of 95 Ω and 105 Ω, respectively. We then descend to point C, canceling the reactance added from A to B. Because that’s an admittance, we invert it to find the impedance of -1.05, with the minus sign indicating capacitance. We are just to the right of the blue b=-1 constant-susceptance circle, so let’s call it -0.95. Before we depart, we also log the admittance. We are just to the left of the x=1 red reactance circle, so let’s call it 0.95. At point B, we make our first travelogue entry, noting the reactance. At point B, we see a path straight to the load, a right turn along the blue g=0.25 constant-conductance circle. We’ll head north from A along the red r=0.25 constant-resistance circle, adding positive reactance (inductance) as we go. A Smith-chart map helps design a matching network.īecause we don’t want to add resistance, we must find paths of constant resistance or conductance from point A to point C on the 0.25 red circle. They are both real, so they fall on the horizontal axis, the source impedance where the r=0.25 circle crosses the horizontal axis (point A in Figure 5) and the load impedance where the r=4 circle crosses the horizontal axis (point C). Next, we locate these points on the Smith chart. Our source impedance becomes 0.25 and our load impedance becomes 4. I’m choosing 100 Ω as our normalization impedance because, with the values in Figure 4, the map will spread out on the chart and be easy to follow. We’ll map a journey on the Smith chart and compile a travelogue along the way. You can find L and C using a Smith chart. I propose a simple LC circuit ( Figure 4). Because we’re not building an electric heater, we will use purely reactive components. You can do a lot of algebra to find these circles’ centers and radii, or you can just flip the impedance circles horizontally.Ĭonsider a 1-GHz generator with a 25-Ω source impedance driving a 400-Ω load, for a horrendous Γ=0.882. But I can add some admittance circles to form an immittance chart ( Figure 3), where admittance Y= g+j b=1/ Z. Your chart isn’t as elaborate as some I’ve seen. An immittance chart includes admittance (blue) and impedance (red) circles. Similarly, an impedance of 2+j1 is located where the red r=2 circle crosses the blue x=1 circle (point B), where Γ= 4+j2. A resistance of 2 appears where the red r=2 resistance circle crosses the horizontal axis (point A), where you can see that Γ=0.333. Figure 2 shows some of these circles plotted on a grid representing Γ. Similarly, a locus of points representing constant reactance is a circle of radius 1/ x centered at 1, 1/ x. Noting that Z L is a complex number in the form of r+j y, you can manipulate the equation for Z L to determine that a locus of points representing constant resistance is a circle of radius 1/( r+1) centered at r/( r+1). In a Smith chart, resistance (red) and reactance (blue) are plotted on a grid representing Γ. You’ll also see it represented as the scattering parameter S 11. It’s the reflection coefficient (sometimes denoted as ρ), which we discussed in part 1. Normalization lets one Smith chart work with any characteristic impedance. This part elaborates on the Smith chart’s construction and provides an impedance-matching example. Part 1 of this FAQ looked at why you might use a Smith chart. The Smith chart remains valuable in helping to visualize how such circuits perform. A typical RF/microwave circuit includes a source, transmission line, and load. That circuit includes a source with impedance Z s, transmission line with characteristic impedance Z 0, and load with impedance Z L. Take a journey around a Smith chart to find capacitance and inductance values in a matching network.īefore computers became ubiquitous, the Smith chart simplified calculations involving the complex impedances found in RF/microwave circuits such as the one shown in Figure 1.
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