Practical Electronics/Parallel RC

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Parallel RC[edit]

A parallel RC Circuit

Circuit Impedance[edit]

\frac {1}{Z} = \frac{1}{Z_R} + \frac{1}{Z_C}
\frac{1}{Z} = \frac{1}{R} + j\omega C = \frac {j\omega CR + 1}{R}
Z = R \frac {1}{j\omega CR + 1}

Circuit Response[edit]

I = I_R + I_C
I = \frac {V}{R} + C \frac{dV}{dt}
V =  (IR - RC \frac {dV}{dt})

Parallel RL[edit]

An RL parallel circuit\

Circuit Impedance[edit]

\frac{1}{Z} = \frac{1}{Z_R} + \frac{1}{Z_L}
\frac{1}{Z} = \frac{1}{R} + \frac{1}{j\omega L} = \frac{R + j\omega L}{j\omega RL}
Z = \frac{j\omega RL}{R + j\omega L} = j\omega L\frac {1} {1 + j\omega \frac{L}{R}}

Circuit Response[edit]

I = I_R + I_L
I = \frac{V}{R} + \frac{1}{L} \int V dt
V = IR - \frac{R}{L} \int V dt

Parallel LC[edit]

LC circuit diagram

Circuit Impedance[edit]

\frac{1}{Z} = \frac{1}{Z_L} + \frac{1}{Z_C}
\frac{1}{Z} = \frac{1}{j\omega L} + j\omega C = \frac{(j\omega)^2 LC + 1}{j\omega L}
Z = \frac{(j\omega)^2 LC + 1}{j\omega L}

Circuit response[edit]

I = I_L + I_C
I = \frac{1}{L} \int V dt + C \frac{dV}{dt}

Parallel RLC[edit]

RLC parallel circuit.png

Circuit Impedance[edit]

\frac{1}{Z} = \frac{1}{Z_R} + \frac{1}{Z_L} + \frac{1}{Z_C}
\frac{1}{Z} = \frac{1}{R} + \frac{1}{j\omega L} + j\omega C
\frac{1}{Z} = \frac{(j\omega)^2 LC + j\omega L + R }{j\omega RL}
\frac{1}{Z} = \frac{(j\omega)^2 \frac{LC}{R} + j\omega \frac{L}{R} + 1 }{j\omega L}

Circuit response[edit]

I = I_R + I_L + I_C
I = \frac{V}{R} + \frac{1}{L} \int V dt + C \frac{dV}{dt}
I = \frac{V}{R} + \frac{1}{L} \int V dt + C \frac{dV}{dt}
V = IR - \frac{R}{L} \int V dt - CR \frac{dV}{dt}

Natural Respond[edit]

0 = \frac{V}{R} + \frac{1}{L} \int V dt + C \frac{dV}{dt}

Forced Respond[edit]

I_t = IR + L \frac{dI}{dt} + \frac{1}{C} \int I dt

Second ordered equation that has two roots

ω = -α ± \sqrt {\alpha^2 - \beta^2}

Where

\alpha = \frac{R}{2L}
\beta = \frac{1}{\sqrt{LC}}

The current of the network is given by

A eω1 t + B eω2 t

From above

When {\alpha^2 = \beta^2}, there is only one real root
ω = -α
When {\alpha^2 > \beta^2}, there are two real roots
ω = -α ± \sqrt {\alpha^2 - \beta^2}
When {\alpha^2 < \beta^2}, there are two complex roots
ω = -α ± j\sqrt {\beta^2 - \alpha^2 }

Resonance Response[edit]

At resonance, the impedance of the frequency dependent components cancel out . Therefore the net voltage of the circui is zero

Z_L - Z_C = 0 and V_L + V_C = 0

\omega L = \frac{1}{\omega C}
\omega = \sqrt {\frac{1}{LC}}
Z = Z_R + (Z_L - Z_C) = Z_R = R
I = \frac{V}{R}

At Resonance Frequency

\omega = \sqrt {\frac{1}{LC}} .
I = \frac{V}{R} . Current is at its maximum value

Further analyse the circuit

At ω = 0, Capacitor Opened circuit . Therefore, I = 0 .
At ω = 00, Inductor Opened circuit . Therefore, I = 0 .


With the values of Current at three ω = 0 ,  \sqrt {\frac{1}{LC}} , 00 we have the plot of I versus ω . From the plot If current is reduced to halved of the value of peak current I = \frac{V}{2R} , this current value is stable over a Frequency Band ω1 - ω2 where ω1 = ωo - Δω, ω2 = ωo + Δω


  • In RLC series, it is possible to have a band of frequencies where current is stable, ie. current does not change with frequency . For a wide band of frequencies respond, current must be reduced from it's peak value . The more current is reduced, the wider the bandwidth . Therefore, this network can be used as Tuned Selected Band Pass Filter . If tune either L or C to the resonance frequency \omega = \sqrt {\frac{1}{LC}} . Current is at its maximum value I = \frac{V}{R} . Then, adjust the value of R to have a value less than the peak current I = \frac{V}{R} by increasing R to have a desired frequency band .


  • If R is increased from R to 2R then the current now is I = \frac{V}{2R} which is stable over a band of frequency
ω1 - ω2 where
ω1 = ωo - Δω
ω2 = ωo + Δω

For value of I < I = \frac{V}{2R} . The circuit respond to Wide Band of frequencies . For value of I = \frac{V}{R} < I > I = \frac{V}{2R} . The circuit respond to Narrow Band of frequencies

Summary[edit]

Circuit Symbol Series Parallel
RC
RLC series circuit.png
A parallel RC Circuit
Impedance Z Z_t = R + \frac {1}{\omega C} = \frac {\omega CR + 1}{\omega C} \frac{1}{Z_t} = \frac{1}{Z_R} + \frac{1}{Z_C} = \frac{1}{R} + \omega C = \frac {R}{\omega CR + 1}
Frequency \omega_o = 2 f_o Z_R = Z_C
R = \frac{1}{\omega C}
\omega = \frac{1}{CR}
\frac{1}{R} = \frac{1}{\omega C}
\frac{1}{R} = \omega C
\omega = \frac{1}{CR}
Voltage V V = IR + \frac {1}{C} \int I dt I = \frac {V}{R} + C\frac{dV}{dt}
Current I \int I dt = C (V - IR) \frac {dV}{dt} = \frac{1}{C}(I - \frac{V}{R})
Phase Angle Tan θ = 1/2πf RC
f = 1/2π Tan CR
t = 2π Tan CR
Tan θ = 1/2πf RC
f = 1/2π Tan CR
t = 2π Tan CR