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