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