CONCEPTUAL TOOLS RLC Filter Neil E Cotter Associate

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 = 0 V

CONCEPTUAL TOOLS Kirchhoff’s Laws • Same current, i(t), flows through L, C, and R • Sum of voltages around loop = 0 V

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 = √a 2 + b 2 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 = 0 V

CONCEPTUAL TOOLS Kirchhoff’s Laws • Same phasor current, I, flows through L, C, and R • Sum of phasor voltages around loop = 0 V

CONCEPTUAL TOOLS Ohm’s Law • Vo = IR = voltage across R

CONCEPTUAL TOOLS Gain • Gain is size of output relative to input • Gain = |Vo|/|Vi| where |a + jb| = √a 2+b 2 = A for polar form or or

CONCEPTUAL TOOLS Gain versus Frequency • Gain is max at “center frequency” denoted by ωo • Gain is max/√ 2 at “cutoff frequencies” denoted by ωC 1 and ωC 2

CONCEPTUAL TOOLS Center Frequency • Center frequency, ωo, where gain is max • Occurs where gain = 1 • Solve for ωo using following equation:

CONCEPTUAL TOOLS Cutoff Frequencies • Cutoff frequencies, ωC 1 and ωC 2, where gain is max/√ 2 • Occurs where gain = 1/√ 2 • Solve for cutoff frequencies using following equation: • Bandwidth = β = ωC 2 – ωC 1 • 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