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R C Computations

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Computations showing derivations for basic high pass & low pass filters using R C circuit.

Published in: Devices & Hardware
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R C Computations

  1. 1. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved RC Computations by Anil Kumar Pugalia
  2. 2. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved Introduction ~V RI V = I * R ~V ZI V = I * Z
  3. 3. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved Impedance (Z) Fundamentals ω = 2πf C ⇒ 1 jωC = −j ωC L⇒ jω LR⇒ R
  4. 4. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved Basic RC Circuit ~V R I C A=∣Z∣= √ω2 R2 C2 + 1 ωC tan(ϕ)=arg Z= 1 ω RC V =I∗Z ,Z=R− j ωC Z=R− j ωC =A e −j ϕ I= V Z = V A e j ϕ = V A < ϕ>
  5. 5. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved Voltage across the Resistor ~V R VR I C VC A= √ω 2 R 2 C 2 +1 ωC ,tan(ϕ)= 1 ω RC , I= V A <ϕ> V R=I∗R= V∗R A <ϕ> V =V m∗sin(ωt) V R=(Vm∗ R A )∗sin(ωt+ ϕ) R A = ω RC √(ω RC)2 + 1 =α
  6. 6. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved Voltage across the Capacitor ~V R VR I C VC A= √ω 2 R 2 C 2 +1 ωC ,tan(ϕ)= 1 ω RC , I= V A <ϕ> VC=I∗ −j ω∗C = V A∗ω∗C <ϕ− π 2 > V =V m∗sin(ωt) VC=( Vm A∗ω∗C )∗sin(ωt+ϕ− π 2 ) 1 A∗ω∗C = 1 √(ω RC)2 +1 =β
  7. 7. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved RC Conclusion tan(ϕ)= 1 ω RC V =V m∗sin(ωt) V R=(V m∗α)∗sin(ωt+ϕ) α= ω RC √(ω RC)2 +1 ~V I C VC R VR VC=(V m∗β)∗sin(ωt+ϕ−π 2 ) β= 1 √(ω RC)2 +1 =√1−α 2
  8. 8. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved RC Predictions α= ω RC √(ω RC)2 + 1 , tan(ϕ)= 1 ω RC ,R=220Ω f (Hz) 1000 2000 C (μF) α Ф (deg) α Ф (deg) 100 1.00 0.4 1.00 0.2 10 1.00 4.1 1.00 2.1 1 0.81 35.9 0.94 19.9 0.1 0.14 82.1 0.27 74.5 0.01 0.01 89.2 0.03 88.4
  9. 9. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved Cut-Off Frequency of Filters tan(ϕ)= 1 ω RC V =V m∗sin(ωt) V R=(Vm∗α)∗sin(ωt+ ϕ) α= ω RC √(ω RC)2 + 1 =cos(ϕ) ~V I C VC R VR VC=(V m∗√1−α2 )∗sin(ωt+(ϕ− π 2 )) HPF => LPF => Ex: C = 1μF, R = 10Ω => fc = 16kHz α= 1 √(2) ⇒ωc= 1 RC ⇒f c= 1 2π RC

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