Chapter 25. Waveforms

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1 Chapter 5 Nosiusoidal Waveforms Waveforms Used i electroics except for siusoidal Ay periodic waveform may be expressed as Sum of a series of siusoidal waveforms at differet frequecies ad amplitudes 1

2 Waveforms Each siusoidal compoets has a uique amplitude ad frequecy 3 Waveforms These compoets have may differet frequecies Output may be greatly distorted after passig through a filter circuit 4

3 Composite Waveforms Waveform made up of two or more separate waveforms Most sigals appearig i electroic circuits Comprised of complicated combiatios of dc ad siusoidal waves 5 Composite Waveforms Oce a periodic waveform is reduced to the summatio of siusoidal waveforms Overall respose of the circuit ca be foud 6 3

4 Composite Waveforms Circuit cotaiig both a ac source ad a dc source Voltage across the load is determied by superpositio Result is a sie wave with a dc offset 7 Composite Waveforms RMS voltage of composite waveform is determied as V rms V dc V ac Referred to as true RMS voltage 8 4

5 Composite Waveforms Waveform cotaiig both dc ad ac compoets Power is determied by cosiderig effects of both sigals 9 Composite Waveforms Power delivered to load will be determied by P out V R rms load 10 5

6 Fourier Series Ay periodic waveform Expressed as a ifiite series of siusoidal waveforms Expressio simplifies the aalysis of may circuits that respod differetly 11 Fourier Series A periodic waveform ca be writte as: f(t) = a 0 + a 1 cos t + a cos t + + a cos t + + b 1 si t + b si t + + b si t + 1 6

7 Fourier Series Coefficiets of terms of Fourier series Foud by itegratig origial fuctio over oe complete period a a b 0 1 T T T t1 T t1 t1 T t1 t1 T t1 f ( t) dt f ( t) cos ( f ( t) si ( t) dt t) dt 13 a c Fourier Series Idividual compoets combied to give a sigle siusoidal expressio as: c ad cos where si ta x a ( x 1 a b b b si ) x a si ( x 90 ) b si x 14 7

8 Fourier Series Fourier equivalet of ay periodic waveform may be simplified to f(t) = a 0 + c 1 si( t + 1 ) + c si( t + ) + a 0 term is a costat that correspods to average value c coefficiets are amplitudes of siusoidal terms 15 Fourier Series Siusoidal term with = 1 Same frequecy as origial waveform First term Called fudametal frequecy 16 8

9 Fourier Series All other frequecies are iteger multiples of fudametal frequecy These frequecies are harmoic frequecies or simply harmoics 17 Fourier Series Pulse wave which goes from 0 to 1, the back to 0 for half a cycle, will have a series give by v( t) 0.5 si( t) 1, 3, 5,, 18 9

10 Fourier Series Average value a 0 = 0.5 It has oly odd harmoics Amplitudes become smaller 19 Eve Symmetry Symmetrical waveforms Aroud vertical axis have eve symmetry Cosie waveforms Symmetrical about this axis Also called cosie symmetry 0 10

11 Eve Symmetry Waveforms havig eve symmetry will be of the form f( t) = f(t) A series with eve symmetry will have oly cosie terms ad possibly a costat term 1 Odd symmetry Odd Symmetry Waveforms that overlap terms o opposite sides of vertical axis if rotated 180 Sie symmetry Sie waves that have this symmetry 11

12 Odd Symmetry Waveforms havig odd symmetry will always have the form f( t) = f(t) Series will cotai oly sie terms ad possibly a costat term 3 Half-Wave Symmetry Portio of waveform below horizotal axis is mirror image of portio above axis 4 1

13 Half-Wave Symmetry These waveforms will always be of the form T f t f t Series will have oly odd harmoics ad possibly a costat term 5 Shifted Waveforms If a waveform is shifted alog the time axis Necessary to iclude a phase shift with each of the siusoidal terms To determie the phase shift Determie period of give waveforms 6 13

14 Shifted Waveforms Select which of the kow waveforms best describes the give wave 7 Shifted Waveforms Determie if give waveform leads or lags a kow waveform Calculate amout of phase shift from = (t/t) 360 Write resultig Fourier expressio for give waveform 8 14

15 Shifted Waveforms If give waveform leads the kow waveform Add phase agle If it lags, subtract phase agle 9 Frequecy Spectrum Waveforms may be show as a fuctio of frequecy Amplitude of each harmoic is idicated at that frequecy 30 15

16 Frequecy Spectrum True RMS voltage of composite waveform is determied by cosiderig RMS value at each frequecy V rms V dc V 1 V V 3 31 Frequecy Spectrum If a waveform were applied to a resistive elemet Power would be dissipated as if each frequecy had bee applied idepedetly Total power is determied as sum of idividual powers 3 16

17 Frequecy Spectrum To calculate power Covert all voltages to RMS Frequecy spectrum may the be represeted i terms of power 33 Frequecy Spectrum Power levels ad frequecies of various harmoics of a periodic waveform may be measured with a spectrum aalyzer Some spectrum aalyzers display either voltage levels or power levels 34 17

18 Frequecy Spectrum Whe displayig power levels 50- referece load is used Horizotal axis is i hertz Vertical axis is i db 35 Circuit Respose to a Nosiusoidal Waveform Whe a waveform is applied to iput of a filter Waveform may be greatly modified Various frequecies may be blocked by filter 36 18

19 Circuit Respose to a Nosiusoidal Waveform A composite waveform passed through a badpass filter May appear as a sie wave at desired frequecy Method is used to provide frequecy multiplicatio 37 19

20 This documet was created with WiPDF available at The uregistered versio of WiPDF is for evaluatio or o-commercial use oly.

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