Electronics/RCL

From Wikibooks, open books for an open world
Jump to: navigation, search

Electronics | Foreword | Basic Electronics | Complex Electronics | Electricity | Machines | History of Electronics | Appendix | edit


RLC Series[edit]

An RLC series circuit consists of a resistor, inductor, and capacitor connected in series:

RLC series circuit v1.svg


By Kirchhoff's voltage law the differential equation for the circuit is:

L \frac{dI}{dt} + IR + \frac{1}{C} \int I dt = V(t)

or

L \frac{d^2I}{dt^2} + R \frac{dI}{dt}  + \frac{I}{C} =\frac{dV}{dt}

Leading to:

s^2 + \frac{R}{L} s + \frac{1}{LC} = 0
s = -\alpha ±  \sqrt{\alpha^2 - \beta^2}

with

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

There are three cases to consider, each giving different circuit behavior,  \alpha^2 = \beta^2, \alpha^2 > \beta^2, or \alpha^2 < \beta^2 .

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

Equation above has only one real root

s = -α = \frac{R}{2L}
I = A e^{(-\frac{R}{2L}) t}


\alpha^2 > \beta^2 ,
\frac{R}{2L} > \sqrt{\frac{1}{LC}}
R > 2\sqrt{\frac{L}{C}}

Equation above has only two real roots

s = -\alpha ± \sqrt{\alpha^2 - \beta^2}
I = e^(-\alpha + \sqrt{\alpha^2 - \beta^2}) t + e^-(\alpha + \sqrt{\alpha^2 - \beta^2}) t
I = e^(-\alpha)  e(\sqrt{\alpha^2 - \beta^2}) t - e^-(\sqrt{\alpha^2 - \beta^2}) t


\alpha^2 < \beta^2 .
R < 2\sqrt{\frac{L}{C}}

Equation above has only two complex roots

s = -\alpha + j\sqrt{\beta^2 - \alpha^2}
s = -\alpha - j\sqrt{\beta^2 - \alpha^2}
I = e^j(-\alpha + \sqrt{\beta^2 - \alpha^2}) t + e^j(-\alpha + \sqrt{\beta^2 -\alpha^2}) t

Circuit Analysis[edit]

R = 0[edit]

If R = 0 then the RLC circuit will reduce to LC series circuit . LC circuit will generate a standing wave when it operates in resonance; At Resonance the conditions rapidly convey in a steady functional method.

Z_L = Z_C
\omega L = \frac{1}{\omega C}
\omega = \sqrt{\frac{1}{LC}}

R = 0 ZL = ZC[edit]

If R = 0 and circuit above operates in resonance then the total impedance of the circuit is Z = R and the current is V / R

At Resonance

Z_L + Z_C = 0 Or Z_L = Z_C
\omega L = \frac{1}{\omega C}
\omega = \sqrt{\frac{1}{LC}}
Z = Z_R + Z_L + Z_C = R + 0 = R
I = \frac{V}{R}

At Frequency

I = 0 . Capacitor opens circuit . I = 0
I = 0 Inductor opens circuit . I = 0

Plot the three value of I at three I above we have a graph I - 0 At Resonance frequency \omega = \sqrt{\frac{1}{LC}} the value of current is at its maximum I = \frac{V}{R} . If the value of current is half then circuit has a stable current I = \frac{V}{2R}does not change with frequency over a Bandwidth of frequencies É1 - É2 . When increase current above I = \frac{V}{2R} circuit has stable current over a Narrow Bandwidth . When decrease current below I = \frac{V}{2R} circuit has stable current over a Wide Bandwidth

Thus the circuit has the capability to select bandwidth that the circuit has a stable current when circuit operates in resonance therefore the circuit can be used as a Resonance Tuned Selected Bandwidth Filter

Further Reading[edit]

  1. RCL circuit analysed in the time domain
  2. RCL circuit analysed in the frequency domain