On One Circuit for Frequency Summation Using Semiconductor Triodes

L. P. Olomutskii, M. S. Roitman

(Presented at the Tomsk City Scientific and Technical Conference dedicated to the 90th anniversary of the birth of V. I. Lenin)

A frequency-summation circuit is easily reduced to circuits of “pure” multiplication of sinusoidal functions. Indeed,

\[ u = A\sin(\omega_1+\omega_2)t = u_1 \cdot u_2 = \]

\[ = A\sin\omega_1 t \cos\omega_2 t + A\cos\omega_1 t \cdot \sin\omega_2 t . \tag{1} \]

The creation of circuits that provide pure multiplication of sinusoidal functions reduces to the problem of creating controlled resistances whose conductance varies according to a sinusoidal law.

Figure 1 gives a skeleton circuit for “pure” multiplication of sinusoidal functions. Here \(R\) is the controlled resistance, whose conductance varies according to a sinusoidal law with frequency \(\omega_2\); \(r\) is the load resistance; \(B(t)\) is the control factor; and \(A\sin\omega_1 t\) is the input voltage, hereinafter called the signal.

Fig. 1.

Let us consider the conditions under which “pure” multiplication of sinusoidal functions can be achieved.

1. The conductance of the control element must depend linearly on the magnitude of the control factor \(Y=cB(t)\), where \(B(t)\) is the control factor (voltage, current, electric-field or magnetic-field strength). In this case, variation of the factor according to any law will lead to variation of the conductance according to the same law.

It should be noted that, in the general case, condition (1) is not obligatory for “pure” multiplication of sinusoidal functions. However, in that case, in order for the conductance to vary according to a sinusoidal law, the control factor must vary according to another, more complex law, the technical realization of which is considerably more complicated.

1. The magnitude of the controlled resistance must depend only on the controlling factor, i.e., it is necessary that the resistance be linear.

2. The control circuit and the signal circuit must not be galvanically connected, or this connection must be minimal. Obviously, direct passage of the control current in the case where there is a galvanic connection will lead to a violation of “pure” multiplication of functions.

3. The conductivity of the circuit for the signal must be symmetrical. The presence of asymmetry will lead to the appearance of an additional frequency spectrum.

4. The ratio of the load resistance to the magnitude of the controlled resistance $\alpha$ must be less than 1.

The voltage across the load when conditions (1)—(4) are satisfied is equal to

\[ u=\frac{\alpha A \cos \omega_2 t \cdot \sin \omega_1 t}{1+\alpha \cos \omega_2 t}, \tag{2} \]

where

\[ \alpha=\frac{r}{R}, \qquad R=\frac{1}{cB(t)}. \]

For $\alpha \ll 1$ we obtain

\[ u=aA \cos \omega_2 t \cdot \sin \omega_1 t, \]

i.e., we have pure multiplication of sinusoidal functions.

As the controlled resistance, semiconductor triodes in inverse connection may be used. A circuit for multiplying sinusoidal functions using semiconductor triodes as the controlled resistance is shown in Fig. 2.

Fig. 2.

Let us consider the fulfillment of the conditions for “pure” multiplication of sinusoidal functions as applied to this controlled resistance.

Figure 3 gives the dependence of the conductivity of the controlled element on the control current. It is evident from the figure that the dependence of conductivity on the control current is close to linear. By the method of five ordinates one can estimate the nonlinear distortions due to the influence of the nonlinearity of the controlled element. From the same Fig. 3 it follows that the conductivity of the controlled element is almost independent of the signal voltage.

To check the third condition, the dependence of the load voltage on the magnitude of the controlling factor was taken at different ambient temperatures (Fig. 4). The graphs show that the coupling between the control and signal circuits is sufficiently small.

If the conductivity of the circuit for the signal voltage is symmetrical, then the circuit must contain no dc component of current and no even harmonics. In this case an integrating instrument (for example, an instrument of a magnetoelectric system) should give no reading. Figure 5 gives the dependence $u_{\mathrm{av}}=f(u_c)$ for frequencies $f_1=30$ cps and $f_2=250$ cps, confirming good linearity of the circuit in the operating region.

Fig. 3.

Axis labels and curve annotations visible in the figure:

• Vertical axis: \(Y\) (mmho)
• Horizontal axis: \(I_{\mathrm{e}}\), \(\mu\text{A}\)
• Curve labels: \(U_{\mathrm{e}} = 0.75\), \(U_{\mathrm{e}} = 1.06\), \(U_{\mathrm{e}} = 1.25\)

Fig. 4.

Axis labels and curve annotations visible in the figure:

• Vertical axis: \(U\), mV
• Horizontal axis: \(U\), V
• Curves: \(t = 60^\circ\text{C}\), \(t = 50^\circ\text{C}\), \(t = 40^\circ\text{C}\), \(t = 30^\circ\text{C}\), \(t = 22^\circ\text{C}\)

Fig. 5.

Text and annotations visible in the figure:

• Vertical axis: \(U^{\mathrm{n}}\) (mV)
• Inside the plot: “operating region”
• Curve annotation: \(U^{\sim} = 1.6\) \((f = 30\ \text{Mc/s})\)
• Horizontal-axis annotation: \([[unclear: symbol]];\ f = 230\ \text{kHz}\)

Fulfillment of the fifth condition is easily ensured by choosing a sufficiently small load resistance.

A block diagram for frequency summation is shown in Fig. 6. According to (2), the voltage at the output of the summing stage is equal to

\[ u=\frac{zA\cos\omega_2 t\cdot \sin\omega_1 t}{1+z\cos\omega_2 t} +\frac{zA\sin\omega_2 t\cos\omega_1 t}{1+z\sin\omega_2 t}. \tag{3} \]

Fig. 6

Expanding the denominators of (3) in a series by Newton’s formula and retaining two terms, we have:

\[ u=zA\cos\omega_2 t\sin\omega_1 t(1-z\cos\omega_2 t) +zA\sin\omega_2 t\cos\omega_1 t \]
\[ \cdot(1-z\sin\omega_2 t) =zA\sin(\omega_1+\omega_2)t-\frac{z^2A}{4} \left[ \cos(2\omega_2-\omega_1)t \right. \]
\[ \left. +\sin(2\omega_2-\omega_1)t +\cos(2\omega_2+\omega_1)t-\sin(2\omega_2+\omega_1)t -2\cos\omega_1 t-2\sin\omega_1 t \right]. \tag{4} \]

It follows from (4) that the frequency of the control circuit must be chosen much higher than the frequency of the signal voltage; in this case the parasitic frequencies at the output will be separated from the useful frequency \(\omega_1+\omega_2\), and they can easily be filtered out. To reduce the amplitudes of the parasitic frequencies, \(z\) must be chosen sufficiently small.

Tests carried out on the summing circuit have shown that even without using a filter, the voltage of the useful frequency exceeds the voltage of the parasitic frequencies by 30 db. With the use of a filter, the results improve sharply.

It should be emphasized that a very great advantage of the described circuit is the very small passage of the control voltage.

Failure to satisfy conditions 1, 2, 4, and 5 leads to the appearance of a frequency spectrum across the load. But even in this case, provided that the frequencies \(\omega_1\) and \(\omega_2\) differ significantly from one another (which is almost always the case), the combination frequencies are considerably separated from the useful signal, and the signal can easily be selected by means of a band-pass filter.

If condition 3 is not fulfilled, then even with ideal fulfillment of conditions 1, 2, 4, and 5, selection of the useful signal is extremely difficult, since the frequency of the useful signal is very close to the frequency of the control current.