1. Frequency Response
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2. Resonance
A network is in resonance (or resonant) when the voltage and current at the network input terminals are in phase.
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3. Resonance: Parallel RLC
The resonant frequency is
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4. Pole-Zero Constellation for Y(s)
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5. Pole-Zero Constellation for Z(s)
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6. The Quality Factor Q
For the parallel RLC circuit, the quality factor at resonance is
Q0 = 2π f0RC = ω0RC
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7. The Damping Factor ζ The damping factor defined as ζ=1/2Qo
Now the quadratic factor can be written in several equivalent ways:
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8. Bandwidth
ω1: the lower half-power frequency ω2: the upper half-power frequency.
The (half-power) bandwidth is defined as the difference of these two half-power frequencies:
B ≡ ω2 − ω1
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9. Bandwidth and Quality Factor
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10. Series RLC Resonance
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11. Scaling
Scaling allows us to design practical circuits at realistic frequencies. The following simple but unrealistic circuit will serve to illustrate the method:
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12. Magnitude Scaling
Km=2000
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13. Frequency Scaling Kf=5×10−6
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14. Bode Diagrams
A Bode diagram or Bode plot is a useful tool for visualizing transfer functions and frequency responses.
A Bode plot shows either magnitude or phase on a logarithmic scale for frequency ω.
Magnitude is shown on a decibel (dB) scale defined as:
HdB = 20 log10 |H( jω)|
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15. Determining Asymptotes for Bode Plots
The two asymptotes intersect at ω = a, the frequency of the zero.
This frequency is also described as the corner, break, 3 dB, or half-power frequency.
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16. Bode Plots: Multiple Terms
H(s)=20+0.2s=20(1+s/100)
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17. Bode Plot: Phase Response
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18. Bode: Complex Conjugate Pairs
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19. Example: Bode Plot
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20. Filters: The Lowpass filter
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21. Filters: The Highpass Filter
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22. Filters: The Bandpass Filter
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23. Filters: The Bandstop Filter
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24. Passive Lowpass Filter
The transfer function is H(s)=1/(1+sRC) and the corner frequency is ω=1/RC.
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25. Passive Highpass Filter
The transfer function is H(s)=sRC/(1+sRC) and the corner frequency is ω=1/RC.
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26. Bandpass Filter with Series RLC
The bandwidth is R/L and the center frequency is ω0 = 1/√LC
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27. Active Filter Design
The following circuit is an active lowpass filter with a corner frequency of 1/R2C and a gain of 1+Rf / R1.
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28. Filter Design: Sallen-Key
Filters can be designed to achieve various attenuation, step-response, and ripple goals
Butterworth and Chebyshev are famous classes of filters with well-designed characteristics, and can be built using the Sallen-Key amplifier shown.
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