| Nonsinusoidal Waveforms |
| Used in electronics except for sinusoidal | ||
| Any periodic waveform may be expressed as | ||
| Sum of a series of sinusoidal waveforms at different frequencies and amplitudes | ||
| Each sinusoidal components has a unique amplitude and frequency |
| These components have many different frequencies | ||
| Output may be greatly distorted after passing through a filter circuit | ||
| Waveform made up of two or more separate waveforms | ||
| Most signals appearing in electronic circuits | ||
| Comprised of complicated combinations of dc and sinusoidal waves | ||
| Once a periodic waveform is reduced to the summation of sinusoidal waveforms | ||
| Overall response of the circuit can be found | ||
| Circuit containing both an ac source and a dc source | ||
| Voltage across the load is determined by superposition | ||
| Result is a sine wave with a dc offset | ||
| RMS voltage of composite waveform is determined as | |
| Referred to as true RMS voltage |
| Waveform containing both dc and ac components | ||
| Power is determined by considering effects of both signals | ||
| Power delivered to load will be determined by |
| Any periodic waveform | ||
| Expressed as an infinite series of sinusoidal waveforms | ||
| Expression simplifies the analysis of many circuits that respond differently | ||
| A periodic waveform can be written as: | ||
| f(t) = a0 + a1cos wt + a2cos 2wt + ∙∙∙ + an cos nwt + ∙∙∙ + b1sin wt + b2 sin 2wt + ∙∙∙ + bn sin nwt + ∙∙∙ | ||
| Coefficients of terms of Fourier series | ||
| Found by integrating original function over one complete period | ||
| Individual components combined to give a single sinusoidal expression as: |
| Fourier equivalent of any periodic waveform may be simplified to | ||
| f(t) = a0 + c1sin(wt + q1) + c2sin(2wt + q2) + ∙∙∙ | ||
| a0 term is a constant that corresponds to average value | ||
| cn coefficients are amplitudes of sinusoidal terms | ||
| Sinusoidal term with n = 1 | ||
| Same frequency as original waveform | ||
| First term | ||
| Called fundamental frequency | ||
| All other frequencies are integer multiples of fundamental frequency | |
| These frequencies are harmonic frequencies or simply harmonics |
| Pulse wave which goes from 0 to 1, then back to 0 for half a cycle, will have a series given by |
| Average value | ||
| a0 = 0.5 | ||
| It has only odd harmonics | ||
| Amplitudes become smaller | ||
| Symmetrical waveforms | ||
| Around vertical axis have even symmetry | ||
| Cosine waveforms | ||
| Symmetrical about this axis | ||
| Also called cosine symmetry | ||
| Waveforms having even symmetry will be of the form f(–t) = f(t) | |
| A series with even symmetry will have only cosine terms and possibly a constant term |
| Odd symmetry | ||
| Waveforms that overlap terms on opposite sides of vertical axis if rotated 180° | ||
| Sine symmetry | ||
| Sine waves that have this symmetry | ||
| Waveforms having odd symmetry will always have the form f(–t) = –f(t) | |
| Series will contain only sine terms and possibly a constant term |
| Portion of waveform below horizontal axis is mirror image of portion above axis |
| These waveforms will always be of the form | |
| Series will have only odd harmonics and possibly a constant term |
| If a waveform is shifted along the time axis | ||
| Necessary to include a phase shift with each of the sinusoidal terms | ||
| To determine the phase shift | ||
| Determine period of given waveforms | ||
| Select which of the known waveforms best describes the given wave |
| Determine if given waveform leads or lags a known waveform | |
| Calculate amount of phase shift from f = (t/T)•360° | |
| Write resulting Fourier expression for given waveform |
| If given waveform leads the known waveform | ||
| Add phase angle | ||
| If it lags, subtract phase angle | ||
| Waveforms may be shown as a function of frequency | ||
| Amplitude of each harmonic is indicated at that frequency | ||
| True RMS voltage of composite waveform is determined by considering RMS value at each frequency |
| If a waveform were applied to a resistive element | ||
| Power would be dissipated as if each frequency had been applied independently | ||
| Total power is determined as sum of individual powers | ||
| To calculate power | ||
| Convert all voltages to RMS | ||
| Frequency spectrum may then be represented in terms of power | ||
| Power levels and frequencies of various harmonics of a periodic waveform may be measured with a spectrum analyzer | |
| Some spectrum analyzers display either voltage levels or power levels |
| When displaying power levels | ||
| 50-W reference load is used | ||
| Horizontal axis is in hertz | ||
| Vertical axis is in dB | ||
Circuit Response to a Nonsinusoidal Waveform
| When a waveform is applied to input of a filter | ||
| Waveform may be greatly modified | ||
| Various frequencies may be blocked by filter | ||
Circuit Response to a Nonsinusoidal Waveform
| A composite waveform passed through a bandpass filter | ||
| May appear as a sine wave at desired frequency | ||
| Method is used to provide frequency multiplication | ||