1	Chapter 25 Nonsinusoidal Waveforms
2	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
3	Waveforms Each sinusoidal components has a unique amplitude and frequency
4	Waveforms These components have many different frequencies Output may be greatly distorted after passing through a filter circuit
5	Composite Waveforms Waveform made up of two or more separate waveforms Most signals appearing in electronic circuits Comprised of complicated combinations of dc and sinusoidal waves
6	Composite Waveforms Once a periodic waveform is reduced to the summation of sinusoidal waveforms Overall response of the circuit can be found
7	Composite Waveforms 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
8	Composite Waveforms RMS voltage of composite waveform is determined as Referred to as true RMS voltage
9	Composite Waveforms Waveform containing both dc and ac components Power is determined by considering effects of both signals
10	Composite Waveforms Power delivered to load will be determined by
11	Fourier Series Any periodic waveform Expressed as an infinite series of sinusoidal waveforms Expression simplifies the analysis of many circuits that respond differently
12	Fourier Series A periodic waveform can be written as: f(t) = a₀ + a₁cos wt + a₂cos 2wt + ∙∙∙ + a_n cos nwt + ∙∙∙ + b₁sin wt + b₂ sin 2wt + ∙∙∙ + b_n sin nwt + ∙∙∙
13	Fourier Series Coefficients of terms of Fourier series Found by integrating original function over one complete period
14	Fourier Series Individual components combined to give a single sinusoidal expression as:
15	Fourier Series 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 c_n coefficients are amplitudes of sinusoidal terms
16	Fourier Series Sinusoidal term with n = 1 Same frequency as original waveform First term Called fundamental frequency
17	Fourier Series All other frequencies are integer multiples of fundamental frequency These frequencies are harmonic frequencies or simply harmonics
18	Fourier Series Pulse wave which goes from 0 to 1, then back to 0 for half a cycle, will have a series given by
19	Fourier Series Average value a₀= 0.5 It has only odd harmonics Amplitudes become smaller
20	Even Symmetry Symmetrical waveforms Around vertical axis have even symmetry Cosine waveforms Symmetrical about this axis Also called cosine symmetry
21	Even 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
22	Odd Symmetry Odd symmetry Waveforms that overlap terms on opposite sides of vertical axis if rotated 180° Sine symmetry Sine waves that have this symmetry
23	Odd 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
24	Half-Wave Symmetry Portion of waveform below horizontal axis is mirror image of portion above axis
25	Half-Wave Symmetry These waveforms will always be of the form Series will have only odd harmonics and possibly a constant term
26	Shifted Waveforms 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
27	Shifted Waveforms Select which of the known waveforms best describes the given wave
28	Shifted Waveforms 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
29	Shifted Waveforms If given waveform leads the known waveform Add phase angle If it lags, subtract phase angle
30	Frequency Spectrum Waveforms may be shown as a function of frequency Amplitude of each harmonic is indicated at that frequency
31	Frequency Spectrum True RMS voltage of composite waveform is determined by considering RMS value at each frequency
32	Frequency Spectrum 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
33	Frequency Spectrum To calculate power Convert all voltages to RMS Frequency spectrum may then be represented in terms of power
34	Frequency Spectrum 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
35	Frequency Spectrum When displaying power levels 50-W reference load is used Horizontal axis is in hertz Vertical axis is in dB
36	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
37	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