# Practical Electronics/Parallel RC

## Parallel RC

### Circuit Impedance

${\frac {1}{Z}}={\frac {1}{Z_{R}}}+{\frac {1}{Z_{C}}}$ ${\frac {1}{Z}}={\frac {1}{R}}+j\omega C={\frac {j\omega CR+1}{R}}$ $Z=R{\frac {1}{j\omega CR+1}}$ ### Circuit Response

$I=I_{R}+I_{C}$ $I={\frac {V}{R}}+C{\frac {dV}{dt}}$ $V=(IR-RC{\frac {dV}{dt}})$ ## Parallel RL

### Circuit Impedance

${\frac {1}{Z}}={\frac {1}{Z_{R}}}+{\frac {1}{Z_{L}}}$ ${\frac {1}{Z}}={\frac {1}{R}}+{\frac {1}{j\omega L}}={\frac {R+j\omega L}{j\omega RL}}$ $Z={\frac {j\omega RL}{R+j\omega L}}=j\omega L{\frac {1}{1+j\omega {\frac {L}{R}}}}$ ### Circuit Response

$I=I_{R}+I_{L}$ $I={\frac {V}{R}}+{\frac {1}{L}}\int Vdt$ $V=IR-{\frac {R}{L}}\int Vdt$ ## Parallel LC

### Circuit Impedance

${\frac {1}{Z}}={\frac {1}{Z_{L}}}+{\frac {1}{Z_{C}}}$ ${\frac {1}{Z}}={\frac {1}{j\omega L}}+j\omega C={\frac {(j\omega )^{2}LC+1}{j\omega L}}$ $Z={\frac {j\omega L}{(j\omega )^{2}LC+1}}$ ### Circuit response

$I=I_{L}+I_{C}$ $I={\frac {1}{L}}\int Vdt+C{\frac {dV}{dt}}$ ## Parallel RLC

### Circuit Impedance

${\frac {1}{Z}}={\frac {1}{Z_{R}}}+{\frac {1}{Z_{L}}}+{\frac {1}{Z_{C}}}$ ${\frac {1}{Z}}={\frac {1}{R}}+{\frac {1}{j\omega L}}+j\omega C$ ${\frac {1}{Z}}={\frac {(j\omega )^{2}RLC+j\omega L+R}{j\omega RL}}$ ${\frac {1}{Z}}={\frac {(j\omega )^{2}LC+j\omega {\frac {L}{R}}+1}{j\omega L}}$ ### Circuit response

$I=I_{R}+I_{L}+I_{C}$ $I={\frac {V}{R}}+{\frac {1}{L}}\int Vdt+C{\frac {dV}{dt}}$ $I={\frac {V}{R}}+{\frac {1}{L}}\int Vdt+C{\frac {dV}{dt}}$ $V=IR-{\frac {R}{L}}\int Vdt-CR{\frac {dV}{dt}}$ #### Natural Respond

$0={\frac {V}{R}}+{\frac {1}{L}}\int Vdt+C{\frac {dV}{dt}}$ #### Forced Respond

$I_{t}=IR+L{\frac {dI}{dt}}+{\frac {1}{C}}\int Idt$ 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

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

Circuit Symbol Series Parallel
RC
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 $\omega _{o}=2f_{o}$ $Z_{R}=Z_{C}$ $R={\frac {1}{\omega C}}$ $\omega ={\frac {1}{CR}}$ ${\frac {1}{R}}={\frac {1}{\omega C}}$ ${\frac {1}{R}}=\omega C$ $\omega ={\frac {1}{CR}}$ Voltage V $V=IR+{\frac {1}{C}}\int Idt$ $I={\frac {V}{R}}+C{\frac {dV}{dt}}$ Current I $\int Idt=C(V-IR)$ ${\frac {dV}{dt}}={\frac {1}{C}}(I-{\frac {V}{R}})$ 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