CONCEPTUAL
TOOLS
RLC Filter
Neil E. Cotter
Associate Professor (Lecturer)
ECE Department
University of Utah
CONCEPTUAL
TOOLS
Kirchhoff’s Laws
• Same current, i(t), flows through L, C, and R
CONCEPTUAL
TOOLS
Kirchhoff’s Laws
• Same current, i(t), flows through L, C, and R
• Sum of voltages around loop = 0V
CONCEPTUAL
TOOLS
Kirchhoff’s Laws
• Same current, i(t), flows through L, C, and R
• Sum of voltages around loop = 0V
CONCEPTUAL
TOOLS
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
CONCEPTUAL
TOOLS
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)
CONCEPTUAL
TOOLS
Phasors
• Sum of sinusoids becomes sum of complex numbers
• Differentiation becomes multiplication
CONCEPTUAL
TOOLS
Kirchhoff’s Laws
• Same phasor current, I, flows through L, C, and R
CONCEPTUAL
TOOLS
Kirchhoff’s Laws
• Same phasor current, I, flows through L, C, and R
• Sum of phasor voltages around loop = 0V
CONCEPTUAL
TOOLS
Kirchhoff’s Laws
• Same phasor current, I, flows through L, C, and R
• Sum of phasor voltages around loop = 0V
CONCEPTUAL
TOOLS
Ohm’s Law
• Vo = IR = voltage across R
Gain
CONCEPTUAL
TOOLS
• Gain is size of output relative to input
• Gain = |Vo|/|Vi| where |a + jb| = √a2+b2 = A for polar form
or
or
CONCEPTUAL
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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
CONCEPTUAL
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Center Frequency
• Center frequency, ωo, where gain is max
• Occurs where gain = 1
• Solve for ωo using following equation:
CONCEPTUAL
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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:
• Bandwidth = β = ωC2 – ωC1
• Bandwidth is roughly frequency range that gets through filter
CONCEPTUAL
TOOLS
Filter Design
• 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