Fourier Transform and Transfer Function in Engineering: Understanding Signals and Filters, Slides of Electrical Circuit Analysis

Download Fourier Transform and Transfer Function in Engineering: Understanding Signals and Filters and more Slides Electrical Circuit Analysis in PDF only on Docsity! Engineering 43 Fourier Transfer Fcn Docsity.com Fourier Transform • A Fourier Transform is an integral transform that re-expresses a function in terms of different Sine/Cosine waves of varying amplitudes, wavelengths, and phases. • A Conceptual Example – This Irregular Signal – Is the SAME as the Sum of these Sinusoids Docsity.com Transfer Fuction, H(f) • Consider a “Black Box” that takes Input Power, vin & iin Transforms this Power into an Output, vout & iout – A typical transformation would be to “Filter- Out” certain electrical frequencies. • For Phasor Voltages Vin & Vout Define the voltage Transfer Function as inv outv ini outi   in outfH V V  Docsity.com Transfer Function • Note that the Transfer Function – Is a Function of FREQENCY ONLY – Can Change (and usually does change) the Magnitude and Phase-Angle of many of the incoming, frequency-dependent, electrical signals • Measuring an Unknown “Black Box”   in outfH V V  Apply Sinusoidal Vin (Vin0°), Measure Vout (Voutφ°) and Plot: Vout / Vin and φ Docsity.com Example Transfer Function  Hz f  Hz f   f H    f H   Docsity.com MultiFrequency Example 6.2  Note the THREE Frequencies • 0 Hz • 1000 Hz – 1000∙2π rad/sec • 2000 Hz – 2000∙2π rad/sec Docsity.com Ex6.2 Transfer Function • Apply to vin the Transfer Function • From the Transfer Function find – Apply To components of vin        60220003031000040 HHH Docsity.com Example 6.2 • Using This H(f) Set find • Note that the above Phasors CanNOT be added as they have DIFFERENT Frequencies.        60220003031000040 HHH   1201203040 11  inout H VV    306023031000 22 inout H VV    1027016022000 33 inout H VV Docsity.com 1st Order Lo-Pass Filter • Thus the Behavior of a Cap-Based Impedance – At LO-Frequencies a Cap acts as an OPEN Circuit – At HI-Frequencies a Cap Acts as a SHORT Circuit • Now use Phasor V- Divider on RC ckt • Multiplying Top&Bot by j2πfC inV outV  CfjC 2 1 Z RR Z       inintot C out CfjR Cfj VV Z Z V   21 21     inout RCfj VV 21 1   Docsity.com 1st Order Lo-Pass Filter • Then the Transfer Function • ReWriting • Where • fB is the “Break point” Frequency at which H(f) falls to 70.7% of its Original Magnitude Value. • Note The Mag & Ph of H(f) in terms of fB :    RCfj fH in out 21 1   V V            21 1 11 1 21 1 B B B ff ffj fH fjfRCjf fH         RC fB 2 1         B B fffH ff fH arctan 1 1 2    Docsity.com Lo-Pass Filter • The LoPass Filter Transfer Function • fB : is also call the Half-Power-Frequency – Recall Full Power to a Resistor: – Then HALF Power: RVorRI 22     RVorRI 22 22 inV outV Docsity.com The deciBel (dB) • dB In Terms of Voltage Ratios • Or dB for Currents • Now we Defined • Since |H(f)| is a Voltage Ratio, define                           1 2 2 1 2 2 1 2 2 2 1 2 2 1 2 log20log10log10 log10log10 V V V V V V RV RV P P LdB       inout inout VVfH VVfH   2                           1 2 2 1 2 2 1 2 2 2 2 2 1 1 2 log20log10log10 log10log10 I I I I I I RI RI P P LdB     fHfH dB log20 Docsity.com dB Plots (SemiLog) Plot • Plotting H(f) on the logarithmic dB Scale makes it easier to distinguish Very Large (104 vs 105) or Very Small (10−4 vs 10−5) Points on the Plots     0000562.01010log2085 25.42085  fHfHdb Docsity.com Cascaded NetWork Gain • Consider the Transfer Function of the “BlackBox” at Right • Looking inside the BlackBox find • Note that with Vout1 = Vin2 • Or in dB form   in outfH V V  1 1 1 2 1 2 1 2 1 out out in out in out in out in out V V V V V V V V V V         fHfHfH fH in out in out in out in out out out in out in out 21 2 2 1 1 2 1 1 2 1 1 1 2 :so or   V V V V V V V V V V V V V V                         dBdBdB fHfHfH fHfHfH fHfHfH fHfHfH 21 21 21 21 log20log20log20 log20log20     Docsity.com Octave • An octave is the interval between two points where the frequency at the second point is twice the frequency of the first. • Given Frequencies f1 & f2    2log log log 21 2 1 2 ff N OR f f N oct oct         Octave 1 2 3 4 5 6 7 8 Frequency (Hz) 63 125 250 500 1k 2k 4k 8k Wavelength in air (70oF, 21oC) (ft) 17.92 9.03 4.52 2.26 1.129 0.56 0.28 0.14 Wavelength in air (70oF, 21oC) (m) 5.46 2.75 1.38 0.69 0.34 0.17 0.085 0.043 MUSICAL Octaves Docsity.com WhiteBoard Work  Let’s This Nice Problem   Find the OutPut Voltage for For this Input  tvin     tvout      tVtVVtvin  212000cos3121000cos2317  Docsity.com