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

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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 e^{ω1 t} + B e^{ω2 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 = ω_o - Δω, ω₂ = ω_o + 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 = ω_o - Δω

ω₂ = ω_o + 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			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	$\omega _{o}=2f_{o}$	$Z_{R}=Z_{C}$ $R={\frac {1}{\omega C}}$ $\omega ={\frac {1}{CR}}$	${\frac {1}{R}}={\frac {1}{\omega C}}$ ${\frac {1}{R}}=\omega C$ $\omega ={\frac {1}{CR}}$
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