PROCEEDINGS

of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Volume 116
1962

PHASE RELATIONS IN THE INPUT CIRCUITS OF TRANSISTOR AMPLIFIERS

A. A. KUZMIN

(Submitted by the scientific seminar of the Radio Engineering Faculty)

Radio-engineering circuits that use phase information must possess good stability of phase relations when the amplitudes of voltages and currents vary. The dependence of phase on amplitude is explained by the presence of nonlinear active and reactive elements.

a

b

Fig. 1.

For the analysis of a large class of circuits, application of the “small parameter” method proves possible. In the present work, as an example, an analysis is made of the stability of phase relations of the input

the circuit of an amplifier using a P-16 transistor in the common-emitter configuration.

The equivalent circuit under investigation is shown in Fig. 1, where
$r_b$ is the base resistance,
$\beta$ is the current gain coefficient,
$r_e, C_e$ are the resistance and capacitance of the emitter junction,
$R_i$ is the internal resistance of the sinusoidal-signal source,
$R_1 C$ are the equivalent input nonlinear resistance and capacitance of the transistor.

The equivalent circuit (Fig. 1b) is obtained from the equivalent circuit shown in Fig. 1a.

As the experiment showed, the input resistance and input capacitance of a P-16 transistor in a common-emitter circuit can, with sufficient accuracy, be approximated by the functions:

\[ R=\frac{R_0}{\left(1+\frac{U_{\sim}}{U_0}\right)^3}; \qquad C=C_0\left(1+\frac{U_{\sim}}{U_0}\right)^3; \tag{1} \]

where $U_{\sim}$ is the variable voltage at the input;
$U_0$ is the constant voltage between base and emitter;
$R_0$ and $C_0$ are the resistance and capacitance at the operating point when $U_{\sim}=0$.

Let us write Kirchhoff’s equations for the input circuit:

\[ i=\frac{U_{\sim}}{R}+C\frac{dU_{\sim}}{dt} +U_{\sim}\frac{dC}{dU_{\sim}}\cdot\frac{dU_{\sim}}{dt}; \tag{2} \]

\[ E\sin\omega t=iR_i+U_{\sim}. \tag{3} \]

Substituting the expressions (2) and (1) into (3) and introducing the new function $y=\dfrac{U_{\sim}}{U_0}$ and the new variable $\varphi=\omega t$, we obtain the equation

\[ \frac{dy}{d\varphi}(1+y)^2(1+4y)+y\{\gamma+\gamma(1+y)^3\} =\alpha\beta\sin\varphi, \]

where

\[ \beta=\frac{1}{\omega C_0R_i}, \qquad \gamma=\frac{1}{\omega C_0R_0}, \qquad \alpha=\frac{E}{U_0}, \]

or

\[ \frac{dy}{d\varphi}+(\gamma+\beta)y =\alpha\beta\sin\varphi -\frac{dy}{d\varphi}y(6+9y+4y^2) -\gamma y^2(3+3y+y^2). \]

This equation is a nonlinear first-order differential equation of the type

\[ \dot y+(\gamma+\beta)y=\alpha\beta\sin\gamma+F(y,\dot y). \tag{4} \]

Since in an amplifier operating in class “A,” $y=\dfrac{U_{\sim}}{U_0}$ is always less than 1, introducing a new function $y=\mu x$ and taking $\alpha=\mu\alpha^{*}$, where $\mu$ is a “small parameter,” we obtain

\[ \dot x+(\gamma+\beta)x=\alpha^{*}\beta\sin\gamma+\mu F(x,\dot x). \tag{5} \]

Equation (5) can be solved by the “small-parameter” method [1, 2]. The solution is represented in the form of a convergent series

\[ x=x_0+\mu x_1+\mu^2 x_2+\ldots \tag{6} \]

It is not difficult to show that, substituting in expression (6) \(x=\dfrac{y}{\mu}\) and \(\alpha^*=\dfrac{\alpha}{\mu}\), we obtain the solution of equation (4) likewise in the form of a convergent series

\[ y=y_0+y_1+y_2+\ldots, \]

where \(y_0\) is the solution of the generating equation

\[ \ddot y_0+(\gamma+\beta)y_0=\alpha\beta\sin\varphi, \tag{7} \]

and \(y_1\) is the solution of the equation

\[ \ddot y_1+(\gamma+\beta)y_1= -\dot y_0 y\bigl(6+9y_0-4y_0^{\delta}(-\gamma y_0^{\delta})3+3y_0+y_0^{\delta}\bigr). \tag{8} \]

We shall restrict ourselves to the first two approximations, i.e. \(y=y_0+y_1\). The zero approximation \(y_0\) describes the input circuit for constant \(R\) and \(C\).

Solving equation (7) and taking into account only the steady-state process, we obtain

\[ y_0=d(\eta\sin\varphi-\cos\varphi), \]

where

\[ d=\frac{\alpha\beta}{1+\eta^2},\qquad \eta=\gamma+\beta. \]

The phase shift in this case is equal to

\[ \varphi=-\operatorname{arctg}\frac{1}{\eta}. \]

Substituting the expression found for \(y_0\) into the right-hand side of equation (8), we obtain a linear equation with respect to \(y_1\), whose solution is found in the form of a sum of harmonics:

\[ y_1=a_0+a_1\sin\varphi+a_2\sin2\varphi+a_3\sin3\varphi+a_4\sin4\varphi+ \]

\[ +b_1\cos\varphi+b_2\cos2\varphi+b_3\cos3\varphi+b_4\cos4\varphi, \]

where

\[ a_1=-\frac{9}{4}d^3\bigl[\beta+\eta(1+\gamma\eta)\bigr], \]

\[ b_1=-\frac{9}{4}d^3\bigl[\eta(\beta-\gamma)-1\bigr]. \]

Adding \(y_0\) and \(y_1\), we find the first harmonic of the complete solution

\[ y_1=d\left\{\eta-\frac{9}{4}d^2\bigl[\beta+\eta(1+\gamma\eta)\bigr]\right\}\sin\varphi- \]

\[ -d\left\{1+\frac{9}{4}d^2\bigl[\eta(\beta-\gamma)-1\bigr]\right\}\cos\varphi. \]

The phase of the first harmonic of the voltage at the transistor input is equal to

\[ \varphi_1=-\operatorname{arctg}\left\{ \frac{1}{\eta} \frac{\left(1+\frac{9}{4}d^2\right)\left[\eta(\beta-\gamma)-1\right]} {\left(1-\frac{9}{4}d^2\right)\left[1+\gamma\eta+\frac{\xi}{\eta}\right]} \right\}. \tag{9} \]

For a sufficiently small amplitude of the input signal, expression (9) may be written approximately as

\[ \varphi_1=-\operatorname{arctg}\frac{1}{\eta} \left[ 1+\frac{9}{4}\frac{z^2\xi}{\eta(1+\eta^2)} \right]; \]

for \(z\to 0\), \(\varphi_1\to\varphi=-\operatorname{arctg}\frac{1}{\eta}\).

The change in the phase shift of the first harmonic as \(z\) varies from zero to \(z=z_1\) is approximately equal to

\[ \Delta\varphi_1=\varphi_1(z_1)-\varphi_1(z=0)= \]

\[ =-\operatorname{arctg}\frac{9}{4}z_1^2 \frac{\omega C_0 R_i R_0^4} {\left[(\omega C_0 R_0 R_i)^2+(R_0+R_i)^2\right]^2}. \]

This expression makes it possible to judge the dependence of \(\Delta\varphi_1\) on \(R_i\), \(R_0\), and \(\omega C_0\). The functions \(\Delta\varphi_1=f(R_i)\), \(\Delta\varphi_1=f(\omega C_0)\) have a maximum, the value of which is not difficult to find.

Physically, the presence of a maximum is explained by the fact that, as \(R_i\) increases, the phase shift increases, while the voltage across \(R\) and \(C\) decreases.

In Fig. 2 the experimental (solid line) and calculated (dashed line) dependences \(\Delta\varphi_1=f(R_i)\) are presented for identical values of \(R_0\), \(\omega C_0\), and \(z_1\). The dependences agree qualitatively.

Fig. 2.

Similarly, one can find the dependence of the phase of the first harmonic of the input current on the circuit parameters and on the amplitude of the input signal.

The present analysis indicates the possibility of applying the “small-parameter” method to the study of the stability of phase relations in nonlinear reactive circuits.

REFERENCES

1. I. G. Malkin. Some Problems in the Theory of Nonlinear Oscillations. State Publishing House of Technical-Theoretical Literature, 1956.

2. A. A. Andronov, A. A. Vitt, and S. E. Khaikin. Theory of Oscillations. Fizmatgiz, 1959.