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ENCYCLOPEDIA OF RADIO ELECTRONICS AND ELECTRICAL ENGINEERING Amplitude, average, effective. Encyclopedia of radio electronics and electrical engineering
Encyclopedia of radio electronics and electrical engineering / Beginner radio amateur The article is devoted to the influence of the form of electric current on its action. It also talks about measuring the voltage and current of various electrical waveforms. First of all, it should be recalled that an alternating electric current, regardless of its form, is characterized by amplitude Iampl (synonymous with maximum), average Iav and effective Ieff (rms, effective) values. The effect of current on different loads when its shape changes varies in different ways. For example, the charging current of a battery during the transition from full-wave rectification to single-half-wave rectification is halved. If the load of the rectifier is a heater, with such a change in shape, not the current, but the power is halved. Since, as you know, the power P is proportional to the square of the current (P \u2d IXNUMXR), then for half-wave rectification, the current does not decrease by half, but by The first of the three quantities characterizing alternating current is its peak valuee Iampl. It is equal to the maximum instantaneous value of the current for the period of its change. Oddly enough, from the point of view of the impact of different forms of current on different loads, the current amplitude is the least informative. That is why the value of alternating current is determined by comparing its action with that of direct current. Secondary the value of an alternating current is the value of such a direct current that carries the same charge of electricity in the same period of time as an alternating current. For an alternating current whose shape is symmetrical about the time axis (for example, a sinusoidal signal), the average value of the current is zero. Therefore, the average value is usually understood as average rectified, i.e., the average value of the current after it has been rectified. The average value of the current characterizes its action, for example, when charging a battery. Effective the value of alternating current is the value of direct current, which, passing through an active linear load (say, a resistor), releases the same amount of heat in the same period of time as the alternating current will release in this load. It is the effective value of the current that is important in relation to heating devices. To characterize the shape of periodic signals, two parameters are introduced: the amplitude factor ka=Iampl/Ieff and the shape factor kf=Ieff/Iav.rect. With the most common waveform - sinusoidal - the considered values are:
Graphically, the average value of alternating current is the area under the curve that characterizes the dependence of current on time. The effective value corresponds to the square root of the area under the curve describing the dependence of the square of the current on time. On fig. 1 shows graphs for a conventional sinusoidal signal I(t)/Iampl and its square (I(t)/Iampl)2. From a comparison of the graphs, it can be seen that the square of the current (and the instantaneous power is proportional to it) pulsates at a double frequency compared to the current. In addition, the deviation of the curve of the square of the current relative to the line at the level of 0,5 up and down is the same. When calculating the area under this curve, the deviations are compensated, which means that it is half that of the area under the straight line characterizing direct current. Since the effective value of the current is proportional to the square root of the area, it is obvious that it is in
Unfortunately, the area under the sinusoid I(t) / Iampl cannot be determined without knowledge of the integral calculus, you will have to believe the above ratios. For the voltage of an alternating electrical signal, there are the same characterizing values as for the current - amplitude Uampl, average Uav and effective Ueff. The relationship between them is the same. With an effective mains voltage of 220 V, the amplitude voltage is 311 V, the average rectified voltage is 198 V. In practice, a radio amateur has to meet with electrical signals of various shapes. Let's consider some of them. Sinusoidal voltage (Fig. 2, a) with full-wave rectification (Fig. 2, b) retains its characteristics, and the average voltage becomes strictly equal to the average rectified one.
It was said above that with single-half-wave rectification (Fig. 2, c), the average voltage value decreases by half compared to full-wave rectification, and the effective value decreases by a factor of. It is easy to understand that if in any power controller one out of N half-cycles is passed to the load, the average voltage decreases by N times (the power in the load decreases by the same amount), and the effective voltage - by Meander (Fig. 2d). This is the name of a signal that one half of the period is equal to its maximum value, and the other half is equal to zero (Fig. 2d). For him, the average value is equal to half the amplitude. The power dissipated by the current of this form in the load is half that of the power from direct current, so the effective value of the signal in Rectangular pulse train (Fig. 2, f) with a duration t with a repetition period T. For such a signal, there is the concept of "duty cycle", which is usually denoted by the letter Q and is defined as the ratio of the period to the pulse duration: Q = T/t. Since the signal current of this form acts Q times less time than direct current, the average value of the signal is Q times less than the amplitude, and the effective value is sawtooth signal (Fig. 2, g, h). For him, the average value (average rectified for bipolar) is equal to half the amplitude (the area of the triangle is equal to half the product of the base and the height). To calculate the effective value, it is necessary to determine the area under the parabola describing the dependence of the square of the signal on time. It’s not so easy to calculate this area; in a mathematical calculation, the effective value is obtained in The same relationship is also valid for a triangular signal (Fig. 2i), including a bipolar one (Fig. 2j). Voltage at the output of the phase-pulse controller (Fig. 2, l). Its shape is characterized by the conduction angle a, which can generally vary from 0 to
average - effective - where Uampl.s is the peak voltage of the network at the input of the regulator, and the angle a in the last formula must be substituted in radians. On fig. Figure 3 shows the dependencies described by these formulas.
How do measuring instruments respond to different waveforms? We note, first of all, that almost all pointer and digital multimeters in the mode of measuring direct voltage and current determine the average value of the signal under study. Instruments of the electromagnetic system are suitable for measuring the effective voltage and current - an image of the corresponding sign is plotted on their scale (Fig. 4, a). These devices are commonly used in various panels to control mains voltage. They are relatively simple and cheap, but consume significant power, operate in a narrow frequency range, and have a non-linear scale.
Special devices for accurate measurement of the effective voltage in a wide frequency range are complex and expensive. To determine the amplitude value of the voltage, a diode rectifier loaded on a DC voltmeter and a large capacitor is usually used (Fig. 4b). The accuracy of such a measurement is sufficient for a voltage that is much higher than the drop across the diode (about 0,6 V). Pointer and digital multimeters, when monitoring AC voltage and current, determine the average rectified value and multiply it by the shape factor of the sinusoidal signal. As a result, when measuring a sinusoidal voltage, we see its effective value on the indicator of the device. With any other form of signals, the interpretation of the results of measurements with an AC voltmeter is difficult. For example, when connecting an AC voltmeter that uses a half-wave rectifier and does not have a coupling capacitor at the input, it will show either zero or a value twice as high as the effective value depending on the polarity of the connection to the output of a full-wave rectifier. If it is connected to the output of a half-wave rectifier, then it will show either zero or the effective voltage of the unrectified signal. In both cases, the measurement results are unreliable. In the presence of a coupling capacitor, the interpretation of the readings is even more difficult. Therefore, to measure alternating unipolar voltage in the absence of specialized instruments, a DC voltmeter should be used. Such a voltmeter measures, as already mentioned, the average voltage, and to obtain an effective value, its readings should be multiplied by the shape factor. And to get the amplitude value, it is enough effective to multiply by the amplitude factor. Knowing the amplitude value of the voltage of a sequence of rectangular pulses, it is not difficult to determine the duty cycle of the pulses from the result of measuring the average value, which is sometimes very convenient.
The table shows the ratios of the average and effective values to the amplitude, as well as the shape and amplitude coefficients for the considered signals.
Author: S. Biryukov, Moscow
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