Use a Smith chart for component characterizations

In response to the prior Smith chart essay, several comments were offered via LinkedIn groups. One comment in particular, from Dan Bullard, caught my attention: “One of my interview questions is, what does a quartz crystal look like on a Smith chart? That throws most people.” The question is worth a look-see. The equivalent circuit of a crystal is taken as follows.

Figure 1 Here is the equivalent circuit of a quartz crystal.

Doing a Smith chart plot of this circuit yields the following result.

Figure 2 This is a Smith chart plot for the circuit in Figure 1.

The left side of Figure 2 is simply the Smith chart for limited ranges of the real and imaginary coefficients based on the previously-presented equations. The right side of Figure 2 includes in red going clockwise as frequency is swept upward through series resonance, the crystal's Smith chart track for ESR = 6 ohms, L = 10 mHy, C1 chosen for series resonance at 10 MHz, and for C2 = 30 pF. I don't know for sure that these L1, C1, and C2 values are real-world proper, but lowering L1 to only 1 mHy yields a Smith chart result that is visually indistinguishable from the track shown. This article was originally published on EDN. John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE). Related articles:

• The math behind the Smith chart
• The Smith chart: more vital after all these years
• Making oscillator selection crystal clear
• Crystal Oscillator Fundamentals and Operation
• Crystal series and parallel resonances