Calculation of nonlinear circuits. Encyclopedia of radio electronics and electrical engineering

Encyclopedia of radio electronics and electrical engineering / Beginner radio amateur

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Linear circuits are those whose properties do not depend on the applied voltage or current. A linear element is a resistor (as long as the current is not too high and the resistor does not overheat and burn out), a capacitor (as long as the voltage across it is below the breakdown voltage), and many others. Until now, we have only dealt with such. However, in some cases, the properties of the elements change depending on the voltage on them or the current. Such elements and the circuits in which they are included are called non-linear.

Typical and most common non-linear elements are semiconductor devices (diodes, transistors), gas-discharge devices, and vacuum tubes. There are non-linear resistors (varistors) and non-linear capacitances (varicaps). An inductor with a magnetic core is always non-linear to some extent. Depending on the purpose of the element, they try either to reduce the nonlinearity (for example, in amplifiers), or, conversely, to emphasize it as much as possible (in detectors and rectifiers, in voltage and current stabilizers).

Consider first the behavior of semiconductor non-linear elements at direct current, moving from simple to complex. Even the current-voltage characteristic of a conventional diode can only be approximately described analytically (using a formula). It can be set in the form of a table relating the current through the element to the voltage at its terminals, but it is best done graphically. It is not for nothing that the characteristics of diodes and transistors are given in the reference books in the form of graphs!

On fig. 18 shows the current-voltage characteristic of current i through some abstract diode depending on the voltage at its terminals U. With reverse voltage across the diode (to the left of point 0 on the graph), the current through the diode is very small (reverse current). At a forward voltage below a certain threshold Upop, the current is also small, but the situation changes when U>Upor. Now the current rises sharply and the curve goes steeply upwards. The threshold voltage depends on the substance of the semiconductor. For germanium diodes, it is approximately 0,15 V, for silicon - 0,5 V.

The slope of the current-voltage characteristic at each point determines the differential resistance of the diode. It is easy to determine it by setting some voltage increment D11, and finding the corresponding current increment Δi1; Vdiff = ΔU1/Δi1. On the left side of the graph it is large, and on the right side it is small - there the same voltage increment ΔU2 = ΔU1 corresponds to a much larger current increment Δi2. The strong dependence of Vdiff on voltage or current through a diode is widely used in radio engineering.

Let's calculate, for example, the simplest voltage regulator (Fig. 19), containing a semiconductor diode VD1 and a current-limiting resistor R1. It is quite obvious that the sum of the voltage drops across the resistor and across the diode is equal to the input voltage Uin. Let's call the drop across the diode stabilization voltage Ust. Then Ust = Uin - iR1. But the current in the circuit depends on Ust, therefore it is not possible to solve this equation analytically, but it is easy to do it graphically.

Let's plot Uin on the horizontal axis and draw a load characteristic corresponding to the selected resistor R1 (straight line in Fig. 18). Recall that it is drawn through two points on the axes: Uin and iK3 = Uin/R1. Only at one point, the currents through the diode and the resistor coincide - at the point of intersection of the diode characteristic with the load line - other modes in the circuit are impossible. The intersection point and gives the desired Ust. Graphically, you can see how Ust changes when Uin or the resistance of the resistor R1 changes.

In practice, conventional voltage stabilization diodes are rarely used, only when low voltages are required. Zener diodes are widely used, produced for a wide variety of voltages. These are also diodes, but working on the reverse branch of the characteristic. At a certain voltage, a reversible avalanche breakdown occurs in them and the current increases sharply. The circuit for switching on a zener diode instead of a diode is shown in fig. 19 dashed lines.

Since the characteristic of the zener diode in the Ust region is very steep and Ust is almost independent of the current, the calculation of the circuit is simplified: given the current through the zener diode i, we find R1 = (Uin-Ust) / i. If a load is connected in parallel to the zener diode, consuming some current iH, then i = ist + iH, where ist is the current through the zener diode. It should be noted that stabilization is the better, the greater the current of the zener diode compared to the load current.

As another example, let's calculate the mode of a simple transistor amplifying stage (Fig. 20).

A silicon transistor, for example, the KT315 series, opens at a base voltage of about 0,5 V, however, it is impossible to apply such a bias from a voltage source (a source with low internal resistance), since the slightest change in the bias voltage will lead to a large change in current through transistor. It is advisable to supply a bias current through a resistor with a large resistance R1, but not from a power source (as is sometimes done incorrectly), but to stabilize the mode from the transistor collector.

It is advisable to set the voltage on the collector equal to half the supply voltage: UK = Upit/2. This will ensure good amplifier linearity and symmetrical clipping of strong signals. We set the collector current of the transistor (for reasonable reasons - for low-power cascades from fractions to several milliamps) and find R2 = Upit / 2iK. The output impedance of the cascade will be the same. Now we take the current transfer coefficient of the transistor h21E from the reference book and find the base current ib = iK / h21E- It remains to find the resistance of the bias resistor R1 = Upit / 2ib. It is easy to see that R1 =R2 h21E.

The calculation is completed, however, if the h21E of the transistor is very different from the value taken from the reference data, it may be necessary to select the resistor R1 until UK = Upit / 2 is obtained.

Let us briefly dwell on the behavior of nonlinear circuits when exposed to alternating current, and as an example, consider the operation of a symmetrical limiter made on two silicon diodes connected in anti-parallel (Fig. 21).

If the input voltage Uvx is much greater than Uthr, the current in the circuit is determined only by the input voltage and the resistance of the resistor R1: i = Uvx / R1. The current-voltage characteristic of the diodes will be displayed as a symmetrical curve, shown in fig. 22. Having built a current graph on the left (in the example, a sinusoid), it is easy to plot the voltage graph on the diodes point by point (curve below). We see that the resulting voltage shape is close to rectangular, with an amplitude of about 0,5 V.

Similarly, you can find the form of current or voltage in any other circuits with non-linear characteristics.

We note one important circumstance. If in linear circuits with a sinusoidal action with a certain frequency f no signals with other frequencies arise, then in nonlinear circuits everything is different. In our example, a sinusoidal voltage of one frequency f was applied to the limiter, and the output voltage already contains a whole spectrum of frequencies, in this case f, 3f, 5f, etc. Multiple frequencies are called harmonics. If one of the diodes is turned off, only half-waves of one polarity will be limited, and even harmonics will appear.

The picture is even more complicated, if the sum of oscillations with different frequencies f1 and f2 enters the nonlinear circuit - then combination frequencies f1 + f2, f1 - f2, and others will appear, in the general case mf1 ± nf / 2, where min are integers. Since the amplitude of these non-linear distortion products is directly related to the coefficient of non-linearity, it becomes possible to estimate the latter, for example, in audio frequency amplifiers, by applying a two-tone signal to the input and measuring the amplitude of the side components at the output of the amplifier.

Question for self-test. Plot the current-voltage characteristic of an ordinary incandescent light bulb, given that the resistance of the filament is directly proportional to the absolute temperature (normal room temperature is 300°K, the temperature of the filament at full heat is 3000°K).

Of course, we cannot strictly solve the thermodynamic problem of the dependence of the temperature of the lamp filament on the applied voltage, current or power, since this will require the solution of differential equations. However, we can build an approximate graph of the current-voltage characteristic (CVC) of the lamp based on the following: at zero voltage, there is no current, the temperature of the filament is 300 K, and its resistance is Ro. This is the differential resistance at the zero point of the VAC, which determines the slope of the curve: α0~ΔI/ΔU=1/R0. We denote the coordinates of the end point of the CVC as Unom and Inom.

These are the nominal voltage and current of the lamp. The differential resistance at this point is 10 times greater (since the temperature is 3000 K). Accordingly, α1 will be less: α~ 1/10Ro What remains, having two points of the CVC and two directions of the curve at these points, connect them with a smooth line (Fig. 62).

As you can see, an ordinary incandescent lamp has the properties of a current stabilizer - a barter, since with significant changes in the voltage on the lamp (especially near UHOM), the current through the lamp changes little.

Author: V.Polyakov, Moscow

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