Practical Electronics/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

 * $$I = I_R + I_L$$
 * $$I = \frac{V}{R} + \frac{1}{L} \int V dt$$


 * $$V = IR - \frac{R}{L} \int V dt$$

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

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

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

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


 * $$V = IR - \frac{R}{L} \int V dt - CR \frac{dV}{dt}$$

Natural Respond

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

Forced Respond

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

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
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