Measurement of Nonlinear Distortions in an Amplifier

M. S. ROITMAN

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

The measurement of nonlinear distortions on the order of 0.05%, and especially smaller ones, presents considerable difficulties. The nonlinear-distortion meter S6-1, produced by domestic industry, is unsuitable for checking the indicated quantities. At the same time, it appears possible, using equipment available in practically any laboratory, to determine the nonlinear distortions of an amplifier comparatively simply, even if they do not exceed 0.01%.

This possibility is based on the fact that the phase shift \(\varphi_\Sigma\) and the frequency distortions

\[ \frac{\Delta K}{K}(\omega) \]

of a measuring amplifier in the operating frequency band usually do not exceed \(n\pi \pm 10^\circ\) and \(\pm 3\%\). Let us first consider an amplifier with a phase shift \(|\varphi| \le 10^\circ\), i.e., an amplifier with \(n\) even. Such amplifiers include all multistage amplifiers with successive voltage feedback, i.e., the most numerous group of measuring amplifiers.

Let us connect to the input of amplifier 2 a four-terminal network 1 with transmission coefficient

\[ \frac{1}{K e^{j\varphi}} \]

(Fig. 1). If a purely sinusoidal voltage from generator \(\Gamma\) is applied to the input of four-terminal network 1 and measuring instrument 3 (a millivoltmeter or oscilloscope) is connected between points \(AB\), then its readings will be equal to

Fig. 1.

\[ U-\left(U\frac{1}{K e^{j\varphi}}\cdot K e^{j\varphi} +\sum_{n=2}^{m} U_n e^{j\varphi_n}+U_{\mathrm{sh}}+U_{\phi}^{r}\right) = \sum_{n=2}^{m} U_n e^{j\varphi_n}+U_{\mathrm{sh}}+U_{\phi}, \]

where

\[ \sum_{n=}^m U_n e^{j\varphi_n} \]

is the sum of the voltages of the harmonics determined by the nonline-

...by the nonlinear distortions of the amplifier; \(U_{\text{n}}\) is the noise voltage; \(U_{\text{b}}\) is the background voltage. Usually

\[ \sum_{n=2}^{m} U_{n\text{y}} e^{j\varphi_n} \gg U_{\text{n}} + U_{\text{b}} \]

and the nonlinear distortions are easy to determine by taking the ratio of the voltage indicated by instrument 3 to the voltage at the output of generator \(G\). If, however, the indicated inequality is not satisfied, then the approximate separation of the voltages is most simply obtained from the voltage curve on the oscilloscope screen.

Thus, in order to measure the nonlinear distortions of amplifier 2, it is necessary to switch in auxiliary voltage divider 1 and, by adjusting its transfer coefficient, obtain a minimum reading of instrument 3. From these readings one can judge the nonlinear distortions of the amplifier.

In fact, the circuit for measuring nonlinear distortions is a compensation circuit at the fundamental harmonic. Since \(|\varphi| < 10^\circ\), balancing of the circuit close to separate balancing is possible. When checking the amplifier in the upper-frequency range, the variable capacitance \(C_1\) should be connected to points \(AD\) (Fig. 1). Adjustment of \(C_1\) makes it possible to change the phase shift of the divider according to the law \(\varphi \approx \dfrac{R_1}{R_1 + R_2} \times \omega (R_1 C_1 - R_2 C_0)\), while the variable resistance \(R_{\text{p}}\) changes the modulus of the transfer coefficient.

Adjustment of the phase of the divider leads to a change in the modulus of the transfer coefficient by an amount \(\delta \approx \dfrac{1}{2}\varphi^2\). When checking the amplifier in the lower-frequency range, capacitance \(C_1\) should be connected to points \(DO\), and the magnitude of the phase shift will be \(\varphi \approx \dfrac{R_2}{R_1 + R_2}\omega R_2(C_1 + C_0)\). It is easy to see that if capacitance \(C_1\) and resistance \(R_{\text{p}}\) have calibrated scales, then, by balancing the circuit at each frequency, one can determine the frequency and phase distortions of the amplifier. From the change in the unbalance signal, one can also determine the instability of the amplifier due to changes in supply voltages, temperature, or other factors.

The described method of checking can be implemented if the phase shift of the amplifier is \(\varphi = 180^\circ \pm 10^\circ\). In this case, to check the amplifier

Fig. 2.

it is required that the signal source have a balanced output with a midpoint, and that instrument 3 be connected not to point \(A\), but to the output of generator \(A\) (Fig. 2).

All of the above is valid in the case where the generator voltage is purely sinusoidal. The presence of harmonic components can substantially increase the error in calibrating the measuring amplifier.

It is advisable to consider the influence of harmonic components in the generator voltage on the error in determining the nonlinear distortions of the amplifier.

Let

\[ U_{\mathrm{in}}(t)=\sum_{n=1}^{m}\dot U_n \sin(n\omega t+\varphi_n) \]

and let the overall transfer coefficients of four-terminal network 1 and of amplifier 2 at the given frequency be equal to

\[ \dot D(\omega)\dot K(\omega)=1. \]

Then the voltage between points \(A-B\) is

\[ U_{\mathrm{in}}(t)= \sum_{n=2}^{m} U_n \sin(n\omega t+\varphi_n) - \sum_{n=2}^{m} U_n D_n K_n \sin(n\omega t+\varphi_n+\varphi_{\mathrm{amp}}-\varphi_{\mathrm{div}}) + \]

\[ + \sum_{n=2}^{m} U_{\mathrm{namp}} \sin(n\omega t+\varphi_{\mathrm{namp}}) + U_{\mathrm{n}}+U_{\mathrm{f}}, \tag{1} \]

where \(\varphi_{\mathrm{amp}}\) is the phase shift of the amplifier for the \(n\)-th harmonic;
\(\varphi_{\mathrm{div}}\) is the phase shift of the divider for the \(n\)-th harmonic.

Since the phase distortions of minimum-phase circuits are more pronounced than the frequency distortions, (1) can be simplified:

\[ U_{\mathrm{out}}(t)\approx \sum_{n=2}^{m} U_{\mathrm{namp}} \sin(n\omega t+\varphi_{\mathrm{namp}}) + \sum_{n=2}^{m} U_n(\varphi_{\mathrm{div}}-\varphi_{\mathrm{amp}})\cdot \cos(n\omega t+\varphi) + \]

\[ +U_{\mathrm{n}}+U_{\mathrm{f}}. \tag{2} \]

The first term of (2) is the useful “product”; the second term is due to the presence of harmonics in the generator voltage. The coefficient of the corresponding harmonic is smaller the higher its ordinal number. Therefore, in practice, only the 2nd and 3rd harmonics should be taken into account. In the case when the phase distortions of the amplifier in the operating-frequency band are equal to \(\pm 10^\circ\), the maximum errors are:

\[ \delta_{\mathrm{n}(2f)} \approx \frac{U_2}{U_1}\cdot \frac{1}{4} \tag{3} \]

and

\[ \delta_{\mathrm{n}(3f)} \approx \frac{U_3}{U_1}\cdot \frac{1}{2}. \tag{4} \]

It should be borne in mind that these errors occur only at the edges of the operating range, where the phase shifts are maximal. For phase shifts less than \(2^\circ\),

\[ \delta_{\mathrm{n}(2f)} < \frac{U_2}{U_1}\cdot \frac{1}{20} \quad \text{and} \quad \delta_{\mathrm{n}(3f)} < \frac{U_3}{U_1}\cdot \frac{1}{10}. \]

We note especially that the higher-quality the amplifier is, the smaller its phase shifts in the operating-frequency band, and the smaller the error with which its nonlinear distortions can be checked.

In conclusion, let us consider one more variant of measuring nonlinear distortions of amplifiers with \(\varphi \approx 180^\circ\). The advantage of this variant is that it eliminates the need to use a generator with a symmetrical output. The voltage from the output of generator \(G\), through divider 1, is applied to amplifier 2 (Fig. 3). Between the input of the divider (point \(a\)) and the output of the amplifier (point \(b\)), two approximately identical active resistances \(Z_{ac}\) and \(Z_{bc}\) are connected in series. Between point \(C\) and

a high-sensitivity measuring instrument 3 is connected to ground (point \(O\)). The fundamental harmonic of the voltage \(U_{co}(t)\) will be equal to zero if the condition

\[ K e^{j\varphi_y} = \frac{Z_{bc}+Z_{ov}} {D e^{-j\varphi_d}(Z_{ac}+Z_{ek})} \simeq \frac{1}{D e^{-j\varphi_d}}\cdot \frac{Z_{bc}}{Z_{oc}} \cdot \left( 1+\frac{Z_{ov}}{Z_{bc}}-\frac{Z_{ek}}{Z_{ac}} \right), \]

is satisfied, where

\[ Z_{ek}=\frac{Z_i Z_{oa}}{Z_i+Z_{oa}}. \]

Balancing of the circuit is carried out by adjusting divider 1 or \(Z_{bc}\) or \(Z_{ac}\). Knowing the divider parameters and the relative values of the resistances, one can easily determine the gain of the amplifier, its amplitude- and phase-frequency characteristics. When the circuit is balanced with respect to the first harmonic, then from the voltage \(U_{co}(t)\) one can find the nonlinear distortions of the amplifier.

Fig. 3.

\[ K_f \simeq \frac{U_{co}(t)}{U_{oa}} \cdot \frac{Z_{ek}+Z_{ac}+Z_{bc}+Z_{ov}} {Z_{ac}+Z_{ek}} \simeq 2\frac{U_{co}(t)}{U_{oa}}. \]

In conclusion, let us note once again that by the methods described above it is comparatively easy to measure, with high accuracy, along with nonlinear distortions, almost all the remaining characteristics of an amplifier—frequency and phase distortions, and instabilities of the gain.