Determination of the gain coefficient of an amplifier consisting of nonlinear frequency-independent elements
Abstract Generated abstract
This paper addresses the calculation of the effective gain and nonlinear distortion coefficients of an amplifier made of several cascaded nonlinear, frequency-independent stages. It argues that replacing the cascade by a single equivalent nonlinearity through direct sequential expansion is algebraically cumbersome, and derives compact formulas for the coefficients of a polynomial approximation when the relative nonlinear gain variations are small. The resulting expressions give the equivalent linear, quadratic, and cubic coefficients for an arbitrary number of stages, including terms associated with combination products, with a two-stage case shown explicitly. These formulas are presented as a simpler and sufficiently accurate tool for analyzing nonlinear distortions in amplifiers and self-oscillators.
Full Text
Determination of the Gain of an Amplifier Consisting of Nonlinear Frequency-Independent Elements
M. S. Roitman
(Presented by the scientific seminar of the Department of Radio Engineering)
Let us consider a system with several nonlinear stages connected in series (see Fig. 1).
The problem reduces to replacing the system of series-connected nonlinearities by a single equivalent one. Such a replacement is associated with very cumbersome computations and with a decrease in the “transparency” of the analysis.
Fig. 1.
In general form we could write:
\[ u_{\Sigma\,\mathrm{out}}(t) = U_m \sin \omega t \prod_{g=1}^{n} K_g \left\{ 1+\frac{\Delta K_g}{K_g}\,[u_{g-1}(t)] \right\}, \tag{1} \]
where
\[ K_g \left\{ 1+\frac{\Delta K_g}{K_g}\,[u_{g-1}(t)] \right\} \]
is the gain of the \(g\)-th stage.
The amplifier gain can be rewritten in the form:
\[ \begin{aligned} K_{\Sigma}[u(t)] &= \prod_{g=1}^{n} \cdot K_g + \sum_{g=1}^{n} \frac{\Delta K_g}{K_g}\,[u_{g-1}(t)] \prod_{g-1}^{n} \cdot K_g \\ &\quad + \sum_{g=2}^{n} \frac{\Delta K_g}{K_g}\,[u_{g-1}(t)] \frac{\Delta K_{g-1}}{K_{g-1}}\,[u_{g-2}(t)] \prod_{g=1}^{n} \cdot K_g + \tag{2} \\ &\quad + \prod_{g=1}^{n} \cdot K_g \prod_{g=1}^{n} \frac{\Delta K_g}{K_g}\,[u_{g-1}(t)] . \end{aligned} \]
But since
\[ \frac{\Delta K_g}{K_g} \ll 1, \]
one may restrict oneself to the first three terms of the series.
The third term reflects the combination products in a multistage nonlinear circuit, and taking it into account in the analysis of generators is fundamentally necessary [1].
In many cases, however, it is legitimate to restrict oneself to only the first two terms. If the approximation of all stages is taken to be polynomial, then, taking (2) into account, one can obtain convenient formulas that sharply reduce the cumbersome calculations and make it possible to determine the coefficients of the polynomial approximating the equivalent nonlinearity of the entire circuit.
In the case of calculation in the usual order, a signal \(U_m \sin \omega t\) is applied to the input of the first stage.
At its output we obtain
\[ u_{n\,\mathrm{out}}(t)=K_1 U_m \sin \omega t+K'_1(U_m \sin \omega t)^2+K''_1(U_m \sin \omega t)^3+\cdots \]
At the output of the second stage,
\[ u_{\mathrm{out}}(t)=K_2 u_{1\mathrm{out}}(t)+K'_2[u_{1\mathrm{out}}(t)]^2+K''_2[u_{1\mathrm{out}}(t)]^3+\cdots \]
and so on. Carrying out all the calculations successively and grouping like terms, we can find the equivalent approximation
\[ u_{n\,\mathrm{out}}(t)\simeq K_{\mathrm{e}} U_m \sin \omega t+K'_{\mathrm{e}}(U_m \sin \omega t)^2+K''_{\mathrm{e}}(U_m \sin \omega t)^3. \]
But this procedure, as already indicated, leads to very cumbersome calculations and complicates the analysis.
The coefficients of the polynomial approximating \(n\) cascade-connected stages can be obtained directly from the formulas*):
\[ K_{\mathrm{e}}=\prod_{q=1}^{n}\cdot K_q-\sum_{q=2}^{n}\frac{K'_q}{K_q}\cdot K_{\mathrm{e}(q-1)}\cdot U_m^2; \tag{3} \]
\[ K'_{\mathrm{e}}=\prod_{q=1}^{n}\cdot K_q \left[ \frac{K'_1}{K_1} +\frac{K'_2}{K_2}\cdot K_1+\cdots+ \frac{K'_q}{K_q}\prod_{q=1}^{n-1}\cdot K_q \right] = \]
\[ \prod_{q=1}^{n}\cdot K_q \left[ \frac{K'_1}{K_1} +\sum_{q=2}^{n}\frac{K'_q}{K_q}\prod_{1}^{q-1}\cdot K_{q-1} \right]; \tag{4} \]
\[ K''_{\mathrm{e}}= \prod_{q=1}^{n}\cdot K_q \left[ \frac{K''_1}{K_1} +\sum_{q=2}^{n}\frac{K''_q}{K_q}\prod_{1}^{q-1}(K_{q-1})^2 \right] + \]
\[ +2\sum_{q=2}^{n}\frac{K'_q}{K_q}\cdot K^1_{\mathrm{e}(q-1)}, \tag{5} \]
where
\[ K^1_{\mathrm{e}(q-1)} = \prod_{1}^{q-1}\cdot K_{(q-1)} \left[ \frac{K'_1}{K_1} + \sum_{q=2}^{q-1}\frac{K'_q}{K_q}\prod_{1}^{q-2}\cdot K_{q-2} \right] \]
is the coefficient of the second term of the equivalent polynomial for the cascade-connected stages from \(1\) to \(q-1\).
For \(q=1\), \(\prod_{1}^{q-1}\cdot K_{q-1}=1\). The last term in (5) determines the combination components.
In the particular case of two stages we have
\[ K_{\mathrm{e}}=K_1\cdot K_2-\frac{K'_1K''_2}{K_2} \]
*) The derivations have been omitted because of their cumbersomeness.
\[ K_{\mathrm{э}}^{1}=K_{2}K_{1}^{1}+K_{2}^{1}K_{1}^{2} =K_{1}K_{2}\left(\frac{K_{1}^{1}}{K_{1}}+\frac{K_{2}^{1}}{K_{2}}\cdot K_{1}\right); \]
\[ K_{\mathrm{э}}^{\prime\prime}=K_{2}K_{1}^{\prime\prime}+K_{1}^{3}K_{2}^{\prime\prime}+2K_{1}K_{1}^{\prime}K_{2}^{\prime} =K_{1}K_{2}\left(\frac{K_{1}^{\prime\prime}}{K_{1}}+\frac{K_{2}^{\prime\prime}}{K_{2}}K_{1}^{2} +\frac{2K_{1}^{1}K_{2}^{1}}{K_{2}}\right). \]
The expressions obtained by us are simple, transparent, and sufficiently accurate for the analysis of nonlinear distortions in amplifiers and in self-oscillators.
REFERENCES
- M. S. Roitman. On one method for finding the stationary solution for self-oscillators. The present collection.