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

1. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved
RC Computations
by
Anil Kumar Pugalia

2. © 2016 Anil Kumar Pugalia <anil@sysplay.in>. All Rights Reserved
Introduction
~V RI
V = I * R
~V ZI
V = I * Z

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. © 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. © 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. © 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. © 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. © 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. © 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