A periodic signal is repetitive, meaning that it repeats itself cycle after cycle. An ideal periodic signal is everlasting, but in practice if the signal exists for a sufficiently long period, it qualifies to be called a periodic signal. The most commonly used periodic signal is the sinusoidal signal. In this page, we take up for study both the sinusoidal signal and the non-sinusoidal periodic signals.
The topic of sinusoidal signal has already been introduced. Hence the description is kept brief. The waveform of a sinusoidal signal is shown in Fig. 33. This signal can be defined by equation (6.1) shown below:
For the signal in Fig. 33, the period is T, E is its amplitude and the fundamental frequency, f = 1/T Hz . The Laplace transform expression has a complex pair of poles on the imaginary axis of s-plane. A pair of poles on the imaginary system corresponds to an undamped system such as an oscillator. Let equation (6.2) represent a sinusoidal signal.
The signal defined by equation (6.2) repeats every T seconds or 2p radians. In equation (6.2), w is known as the angular frequency, expressed in radians per second. It can be seen from equation (6.1) that (w t) should have the unit of radians. Since t has the unit of second, w is expressed in radians per second. At any instant, the value of the signal can be related to an angle, q. It is shown in Fig. 33 how this angle is measured for a cosine wave. It is the angle corresponding to the time-period that has elapsed after the last positive peak. In equation (6.2), the parameter f is known as the phase angle and it is expressed in radians. Since the cosine function is an even function,the value of phase angle can be obtained as shown by equations (6.3) and (6.4). The sine function is an odd function, and equation (6.4) yields both the sign and the magnitude of phase angle correctly.
In general, a sinusoidal signal can be defined by equation (6.5).
We can express a sinusoidal function in another way, as shown by equation (6.6). Equation (6.6) can be expressed as shown by equation (6.7), which corresponds to the form expressed by equation (6.2).
Equation (6.6) illustrates the additive property of of sinusoids. When two sinusoidal signals at the same frequency are added, the resulting signal has the same frequency, as shown by equation (6.7).
Another property of a sinusoidal signal is that both its derivative and its integral have the same frequency. A sinusoidal signal supplies an alternating current since its value keeps alternating from a positive to a negative value and vice versa and is hence called an ac signal. For an ac signal, various parameters such as the effective or the root-mean square value, absolute average value, form factor, crest factor are defined. Since these parameters can be calculated for any periodic waveform, they are applied to non-sinusoidal periodic signals in the next section.
Equation (6.8) presents the root mean square value of a sinusoidal signal. The average of a signal is defined, as expressed by equation (6.9). Since the average value of a sinusoidal signal is zero, the absolute average of a sinusoidal signal is obtained and it is expressed by equation (6.10).
It is necessary to know the importance of the absolute average value. Some ammeters, voltmeters and multimeters actually read only the absolute average value, but they are calibrated to show the rms value by using the form factor, which is explained next.
The ratio of the effective or rms value of any periodic waveform to its absolute average value is called its form factor. The form factor for a sinusoidal waveform is:
A meter that reads the absolute average value displays the rms value after internally multiplying the read value by 1.11. Such a meter reads accurately the rms value of a sinusoidal signal only. If it is used to read the rms value of a periodic signal that has a form factor different from 1.11. its reading is erroneous.
Form factor gives an indication of the peakiness of the waveform. Normally a peaky periodic signal would have a form factor higher than 1.11. A measure of the peaky nature of a signal is also obtained from its crest factor or peak factor. The ratio of the maximum or the peak value of a periodic waveform to its effective value is defined as the crest factor or the peak factor. The crest factor of a sinusoidal signal is square-root of 2. Usually the crest factor is calculated for a periodic waveform that has zero dc value. The dc value of a periodic signal is its average value over a full-cycle. A sinusoidal signal has zero dc value.
It can be seen that a sinusoid is characterized by three parameters; its amplitude A, its frequency f , and its phase angle f.
A periodical signal, f(t) with a period of T, repeats itself every T seconds. A signal is periodic if
A square wave-signal, shown in Fig. 8.34, is a periodic signal. Its average value, effective value, form factor and crest factor can be found out as follows.
The average value of the waveform in Fig. 34 calculated over its cycle period T is zero. Hence its absolute average value is computed over its positive half-cycle, as shown by equation (6.13).
The rms value of this signal is computed as shown in equation (6.14). The form factor of this waveform is unity and the crest factor of this waveform is also unity. The low values of both form factor and crest factor indicate that the waveform has a flat-top.
A term called duty cycle is often used in connection with periodic signals. The duty cycle D is defined as the ratio of the period for which the signal is positive to its cycle period. For the signal in Fig. 35, the duty cycle D is defined to be:
The average value of the signal in Fig. 35 is obtained, as expressed by equation (6.16).
The rms value of the signal in Fig. 35 is obtained, as expressed by equation (6.17).
It is seen that
This relationship is significant. For example, let the values in equation (6.18) reflect currents through a semiconductor device. For a semiconductor device such as a diode, its rms rating is fixed due to its limited power dissipation capability. Then the average current it can carry falls as the duty cycle of the current waveform reduces, given that its rms value remains fixed.
Using Fourier series, a non-sinusoidal periodic signal can be expressed as the sum of a dc signal and sinusoidal signals at the fundamental frequency and the harmonic frequencies. A harmonic frequency is a multiple of the fundamental frequency. The ratio of harmonic frequency to its fundamental frequency is an integer. We take up the study of Fourier series in one of the pages to follow.
This page has presented what periodic signals are. The next page presents some worked examples to illustrate the concepts associated with this topic. The next topic for study is network functions.
Most of the text books on circuit analysis do not describe the basic aspects of signals. This topic is studied in detail when the subject of Signals & Systems is presented for study. However, it is felt that those studying circuit analysis also need to know the basic aspects of signals used to excite circuits.
The topic of Network Functions is taken up for study next.