[PDF] [PDF] Fourier series coefficients for a rectified sine wave

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8 sept 2012 · This is the mathematical theory of Fourier series which uses the periodic waveforms, such as square waves, triangle waves, rectified sinusoids, and so on where T0 is a period of the complex exponential whose frequency is k D 2 The mathematical formula for the full-wave rectified sine signal is just 

[PDF] Fourier series coefficients for a rectified sine wave

The period of the sinusoid (inside the absolute value symbols) is T1 = 2π/ω1 The period of the rectified sinusoid is one half of this, or T = T1/2 = π/ω1 Therefore,

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with period 2π: 1, sin x, cos x, sin2x, cos2x, 2 1 1 0 sin cos 1 2sin 2cos sin cos 1 n n n nx bnx a a x bx ax bx a a Fourier Series: Half-wave Rectifier

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It is also useful to know the values of the cosine, sine, and exponential functions for Determine the Fourier series for the half-wave rectified cosine function

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4 jui 2010 · What is the conditions of Fourier series expansion for a signal? Answer: Exponential Fourier series Half-wave Symmetry; If a signal has an half wave symmetry, only odd (odd cosine and odd sine) Half-wave rectifier

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Table 15 4-1 The Fourier Series of Selected Waveforms Function + ∑ t Half wave rectified sine wave: 0 2 T π ω = ( ) ( )0 0 2 1 cos 2 2 sin 2 4 n n t A A

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This is why the Ck coefficients are 0 (the sine function is odd) Q3 Fourier Series of the output voltage of an ideal half-wave diode rectifier The following circuit is 

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Vm / , so the rms value of a half-wave rectified sine wave is Vrms If periodic voltage and current waveforms represented by the Fourier series case), the exponential term in equation 3 1 becomes very small compared to unity and the diode

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Periodic Function 16 8 The Exponential Form of the Fourier Series sinusoidal rectifier (a) Full-wave rectification (b) Half- wave rectification The triangular waveform Function s(x) (in red) is a sum of sine functions of different amplitudes

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EE341Fall2004

EE341

FourierSeriesforRectifiedSineWave

Considerthesignal

x(t)=Ajsin(!1t)j -2 T-T0T2 T -A 0 A |sin (w 1 t)|

Rectified Sine and Sine

-T10T1 -A 0 A sin (w 1 t) Theperiodofthesinusoid(insidetheabsolutevaluesymbols)isT1=2=!1.Theperiodoftherectified sinusoidisonehalfofthis,orT=T1=2==!1.Therefore, 1=2

T1=T=!o2

Thuswecanwrite

x(t)=A sin!ot 2

TheFourierseriescoefficientsare:

c x k=1 TZ T

0x(t)ejk!otdt

1 TZ T 0A sin!ot 2 ejk!otdt From0toT,jsin(!ot=2)j=sin(!ot=2).Also,replace!oby2=T: c x k=1 TZ T

0AsintT

e j2kt=Tdt 1

EE341Fall2004

cx k=ATZ T 0 ejt=Tejt=Tj2! e j2kt=Tdt A j2TZ T 0 ejt(12k)=Tejt(1+2k)=T dt A j2T" ejt(12k)=Tj(12k)=Tejt(1+2k)=Tj(1+2k)=T# T 0 =A 2T" ej(12k)(12k)=T+ej(1+2k)(1+2k)=T1(12k)=T1(1+2k)=T# Butej(12k)=ejej2k.ej=1andej2k=1,soej(12k)=1.Similarly,ej(1+2k)=1, so: c x k=A 2T

1(12k)=T+1(1+2k)=T1(12k)=T1(1+2k)=T

=A 2

2(12k)2(1+2k)

A

112k+11+2k

2A 114k2

Fork=0

c x 0=1 TZ T

0AsintT

dt A T cos(t=T)=T T 0 2A

Note:Becausethesignaliseven,youcouldalsofindcx

kby: c x k=2 TZ T=2

0x(t)cos(k!ot)dt

Foranoddsignal,

c x k=j2 TZ T=2

0x(t)sin(k!ot)dt

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