Consider the following illustration in Figure 1 of what we shall see is an oscillator, specifically, a Colpitts oscillator.
Figure 1 The Colpitts oscillator where the passive components are arranged on the right-hand side for easier viewing.
There is an R-L-C network of passive components and an active gain block. For purposes of this model, the input impedance of the gain block is high enough for its loading effect to be ignored, the output impedance of the gain block is zero and the value of its gain, A, is nominally unity or perhaps a little less than unity. The resistance R1 models the output impedance that a real-world gain block might present.
To analyze this circuit, we take the passive components, redraw them as on the right and begin using node analysis (Figure 2). The term G1 = 1 / R1 and the term S = j / (2*π*F).
Figure 2 Node analysis used for circuit presented on the right-hand side of Figure 1.
Having gotten the relationship between Eo and E1 to the exclusion of E2, we do several algebraic steps to get that relationship into a useful form as shown in Figure 3.
Figure 3 An algebraic rearrangement where the transfer function is brought into a more useful form.
Note that the denominator of this last equation is cubic. It is a third order polynomial because there are three independent reactive elements in the circuit, L1, C1 and C2.
Please also note that the order of the polynomial MUST match the number of independent reactive elements in the circuit. If we had come up with an algebraic expression of some other order, we would know we’d made a mistake somewhere.
Graphing the ratio of E1/Eo versus frequency, we see the following in Figure 4.
Figure 4 A plot of E1/Eo versus frequency from algebraic analysis in Figure 3.
The transfer function of the passive R-L-C network has a pronounced peak at a frequency of 1.59 MHz. When we run a SPICE simulation of that transfer function, we find the same result (Figure 5).
Figure 5 A SPICE analysis of the passive R-L-C network showing E1/Eo versus frequency with a defined peak of almost 40 dB at 1.59 MHz.
When we let our gain block be a voltage follower, a JFET source follower as shown in Figure 6, we see oscillation at very nearly the frequency of that transfer function peak.
Figure 6 Colpitts oscillator built from passive network shown in Figure 5 by introducing a JFET source follower, that reflects the frequency of the transfer function peak.
John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).
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