rlc filter neil e. cotter associate professor (lecturer) ece department university of utah concept u...
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![Page 1: RLC Filter Neil E. Cotter Associate Professor (Lecturer) ECE Department University of Utah CONCEPT U AL TOOLS](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ec65503460f94bd11c6/html5/thumbnails/1.jpg)
RLC Filter
Neil E. CotterAssociate Professor (Lecturer)
ECE DepartmentUniversity of Utah
CONCEPTUALTOOLS
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Kirchhoff’s Laws CONCEPTUALTOOLS
• Same current, i(t), flows through L, C, and R
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Kirchhoff’s Laws CONCEPTUALTOOLS
• Same current, i(t), flows through L, C, and R• Sum of voltages around loop = 0V
![Page 4: RLC Filter Neil E. Cotter Associate Professor (Lecturer) ECE Department University of Utah CONCEPT U AL TOOLS](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ec65503460f94bd11c6/html5/thumbnails/4.jpg)
Kirchhoff’s Laws CONCEPTUALTOOLS
• Same current, i(t), flows through L, C, and R• Sum of voltages around loop = 0V
![Page 5: RLC Filter Neil E. Cotter Associate Professor (Lecturer) ECE Department University of Utah CONCEPT U AL TOOLS](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ec65503460f94bd11c6/html5/thumbnails/5.jpg)
Phasors
• All signals in circuit are sinusoids of same frequency as input• Use complex numbers to represent sinusoids Capture magnitude Capture phase shift Use j for √-1 (because i was used for current)
• Use phasor transform: P[Acos(2πft +Φ)] = Ae jø
CONCEPTUALTOOLS
![Page 6: RLC Filter Neil E. Cotter Associate Professor (Lecturer) ECE Department University of Utah CONCEPT U AL TOOLS](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ec65503460f94bd11c6/html5/thumbnails/6.jpg)
Phasors
• Treat complex numbers as vectors Sum like vectors Product defined as (a+jb)(c+jd) = ac-bd + j(ad+bc)• Use polar or rectangular form Rectangular form: a+jb
Polar form: Ae jø
• Use right triangle trigonometry to covert forms: Rectangular from polar: a = AcosΦ and b = AsinΦ
Polar from rectangular: A = √a2 + b2 and Φtan-1(b/a)
CONCEPTUALTOOLS
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Phasors
• Sum of sinusoids becomes sum of complex numbers
• Differentiation becomes multiplication
CONCEPTUALTOOLS
![Page 8: RLC Filter Neil E. Cotter Associate Professor (Lecturer) ECE Department University of Utah CONCEPT U AL TOOLS](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ec65503460f94bd11c6/html5/thumbnails/8.jpg)
Kirchhoff’s Laws
• Same phasor current, I, flows through L, C, and R
CONCEPTUALTOOLS
![Page 9: RLC Filter Neil E. Cotter Associate Professor (Lecturer) ECE Department University of Utah CONCEPT U AL TOOLS](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ec65503460f94bd11c6/html5/thumbnails/9.jpg)
Kirchhoff’s Laws
• Same phasor current, I, flows through L, C, and R• Sum of phasor voltages around loop = 0V
CONCEPTUALTOOLS
![Page 10: RLC Filter Neil E. Cotter Associate Professor (Lecturer) ECE Department University of Utah CONCEPT U AL TOOLS](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ec65503460f94bd11c6/html5/thumbnails/10.jpg)
Kirchhoff’s Laws
• Same phasor current, I, flows through L, C, and R• Sum of phasor voltages around loop = 0V
CONCEPTUALTOOLS
![Page 11: RLC Filter Neil E. Cotter Associate Professor (Lecturer) ECE Department University of Utah CONCEPT U AL TOOLS](https://reader030.vdocuments.mx/reader030/viewer/2022032607/56649ec65503460f94bd11c6/html5/thumbnails/11.jpg)
• Vo = IR = voltage across R
Ohm’s Law CONCEPTUALTOOLS
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Gain
• Gain is size of output relative to input• Gain = |Vo|/|Vi| where |a + jb| = √a2+b2 = A for polar form
CONCEPTUALTOOLS
or
or
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Gain versus Frequency
• Gain is max at “center frequency” denoted by ωo
• Gain is max/√2 at “cutoff frequencies” denoted by ωC1 and ωC2
CONCEPTUALTOOLS
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Center Frequency
• Center frequency, ωo, where gain is max• Occurs where gain = 1• Solve for ωo using following equation:
CONCEPTUALTOOLS
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Cutoff Frequencies
• Cutoff frequencies, ωC1 and ωC2, where gain is max/√2• Occurs where gain = 1/√2• Solve for cutoff frequencies using following equation:
CONCEPTUALTOOLS
• Bandwidth = β = ωC2 – ωC1
• Bandwidth is roughly frequency range that gets through filter
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Filter Design CONCEPTUALTOOLS
• Find R and C value for assigned filter:
• Low-pass filter:
ωo = 2π·280 Hz
β = 2π·1600 Hz
•High-pass filter:
ωo = 2π·7000 Hz
β = 2π·1600 Hz
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