1. RLC Filter Neil E. Cotter Associate Professor (Lecturer) ECE Department University of Utah CONCEPT U AL TOOLS

2. Kirchhoff’s Laws CONCEPT U AL TOOLS Same current, i(t), flows through L, C, and R

3. Kirchhoff’s Laws CONCEPT U AL TOOLS Same current, i(t), flows through L, C, and R Sum of voltages around loop = 0V

4. Kirchhoff’s Laws CONCEPT U AL TOOLS Same current, i(t), flows through L, C, and R Sum of voltages around loop = 0V

5. 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 CONCEPT U AL TOOLS

6. 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) CONCEPT U AL TOOLS

7. Phasors Sum of sinusoids becomes sum of complex numbers Differentiation becomes multiplication CONCEPT U AL TOOLS

8. Kirchhoff’s Laws Same phasor current, I, flows through L, C, and R CONCEPT U AL TOOLS

9. Kirchhoff’s Laws Same phasor current, I, flows through L, C, and R Sum of phasor voltages around loop = 0V CONCEPT U AL TOOLS

10. Kirchhoff’s Laws Same phasor current, I, flows through L, C, and R Sum of phasor voltages around loop = 0V CONCEPT U AL TOOLS

11. V o = IR = voltage across R Ohm’s Law CONCEPT U AL TOOLS

12. Gain Gain is size of output relative to input Gain = |V o | / |V i | where |a + jb| = √a 2 +b 2 = A for polar form CONCEPT U AL TOOLS or

13. Gain versus Frequency Gain is max at “center frequency” denoted by ω o Gain is max/√2 at “cutoff frequencies” denoted by ω C1 and ω C2 CONCEPT U AL TOOLS

14. Center Frequency Center frequency, ω o, where gain is max Occurs where gain = 1 Solve for ω o using following equation: CONCEPT U AL TOOLS

15. Cutoff Frequencies Cutoff frequencies, ω C1 and ω C2, where gain is max/√2 Occurs where gain = 1/√2 Solve for cutoff frequencies using following equation: CONCEPT U AL TOOLS Bandwidth = β = ω C2 – ω C1 Bandwidth is roughly frequency range that gets through filter

16. Filter Design CONCEPT U AL TOOLS Find R and C value for assigned filter: Low-pass filter: ω o = 2π·280 Hz β = 2π·1600 Hz High-pass filter: ω o = 2π·7000 Hz β = 2π·1600 Hz