Circuit Theory/RLC Circuits

Series RLC Circuit

Second Order Differential Equation

$L \frac{dI}{dt} + I R + \frac{1}{C} \int I dt = V$
$\frac{d^2I}{dt^2} + \frac{R}{L} \frac{dI}{dt} + \frac{I}{LC} = 0$

The characteristic equation is

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

Where

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

When $\sqrt{\alpha^2 - \beta^2} = 0$

$\alpha^2 = \beta^2 ; R = 2 \sqrt{\frac{L}{C}}$
The equation only has one real root . $s = -\alpha = - \frac{R}{2L}$
The solution for $I(t) = A e^(-\frac{R}{2L} t)$
The I - t curve would look like

When $\sqrt{\alpha^2 - \beta^2} > 0$

$\alpha^2 > \beta^2$ . R > $\frac{L}{C}$
The equation only has two real root . $s = -\alpha$ ± $\sqrt{\alpha^2 - \beta^2}$
The solution for $I(t) = e^{- \alpha + \sqrt{\alpha^2 - \beta^2} t} + e^{- \alpha - \sqrt{\alpha^2 - \beta^2} t} = e^{-\alpha} e^{j(\sqrt{\alpha^2 - \beta^2})} + e^{-j(\sqrt{\alpha^2 - \beta^2})}$
The I - t curve would look like

When $\sqrt{\alpha^2 - \beta^2} < 0$

$\alpha^2 < \beta^2$ . R < $\frac{L}{C}$
The equation has two complex root . $s = -\alpha$ ± j$\sqrt{\beta^2 - \alpha^2}$
The solution for $I(t) = e^{(- \alpha + \sqrt{\beta^2 - \alpha^2} t)} + e^{(- \alpha - \sqrt{\beta^2 - \alpha^2} t)} = e^{-\alpha} e^{j(\sqrt{\beta^2 - \alpha^2})} + e^{-j(\sqrt{\beta^2 - \alpha^2})}$
The I - t curve would look like

Damping Factor

The damping factor is the amount by which the oscillations of a circuit gradually decrease over time. We define the damping ratio to be:

Circuit Type Series RLC Parallel RLC
Damping Factor $\zeta = {R \over 2L}$ $\zeta = {1 \over 2RC}$
Resonance Frequency $\omega_o = {1 \over \sqrt{L C}}$ $\omega_o = {1 \over \sqrt{L C}}$

Compare The Damping factor with The Resonance Frequency give rise to different types of circuits: Overdamped, Underdamped, and Critically Damped.

Bandwidth

[Bandwidth]

$\Delta \omega = 2 \zeta$

For series RLC circuit:

$\Delta \omega = 2 \zeta = { R \over L}$

For Parallel RLC circuit:

$\Delta \omega = 2 \zeta = { 1 \over RC}$

Quality Factor

[Quality Factor]

$Q = {\omega_o \over \Delta \omega } = {\omega_o \over 2\zeta }$

For Series RLC circuit:

$Q = {\omega_o \over \Delta \omega } = {\omega_o \over 2\zeta } = {L \over R \sqrt{LC}} = {1 \over R} \sqrt{L \over C}$

For Parallel RLC circuit:

$Q = {\omega_o \over \Delta \omega } = {\omega_o \over 2\zeta } = {RC \over \sqrt{LC}} = {R} \sqrt{C \over L}$

Stability

Because inductors and capacitors act differently to different inputs, there is some potential for the circuit response to approach infinity when subjected to certain types and amplitudes of inputs. When the output of a circuit approaches infinity, the circuit is said to be unstable. Unstable circuits can actually be dangerous, as unstable elements overheat, and potentially rupture.

A circuit is considered to be stable when a "well-behaved" input produces a "well-behaved" output response. We use the term "Well-Behaved" differently for each application, but generally, we mean "Well-Behaved" to mean a finite and controllable quantity.

Resonance

With R = 0

When R = 0 , the circuit reduces to a series LC circuit. When the circuit is in resonance, the circuit will vibrate at the resonant frequency.

$Z_L = Z_C$
$\omega L = \frac{1}{\omega C}$
$\omega = \frac{1}{\sqrt{LC}}$
$f = \frac{1}{2\pi} \frac{1}{\sqrt{LC}}$

The circuit vibrates and has the capability of producing a Standing Wave when R = 0 , L = C

With R ≠ 0

When R ≠ 0 and the circuit operates in resonance .

The frequency dependent components L , C cancel out ie ZL - ZC = 0 so that the total impedance of the circuit is $Z_R + Z_L + Z_C = R + [ Z_L - Z_C ] = R + 0 = R$
The current of the circuit is $I = \frac{V}{R}$
The Operating Frequency is $\omega = \frac{1}{\sqrt{LC}}$

If the current is halved by doubling the value of resistance then

$I = \frac{V}{2R}$
Circuit will be stable over the range of frquencies from $\omega_1 - \omega_2$

The circuit has the capability to select bandwidth where the circuit is stable . Therefore, it is best suited for Tuned Resonance Select Bandwidth Filter

Once using L or C to tune circuit into resonance at resonance frequency $f = \frac{1}{2\pi} \frac{1}{\sqrt{LC}}$ The current is at its maximum value $I = \frac{V}{R}$ . Reduce current above $I = \frac{V}{2R}$ circuit will respond to narrower bandwidth than $\omega_1 - \omega_2$. Reduce current below $I = \frac{V}{2R}$ circuit will respond to wider bandwidth than $\omega_1 - \omega_2$.

Conclusion

Circuit General Series RLC Parallel RLC
Circuit
Impedance Z $Z = (j\omega)^2 + (j\omega)\frac{R}{L} + \frac{1}{LC}$ $Z = \frac{1}{RLC} \frac{1}{(j\omega)^2 + j\omega\frac{1}{RC} + \frac{1}{LC}}$
Roots λ λ = $- \zeta \pm \sqrt{\zeta^2 - \omega_o^2}$ λ = $- \zeta \pm \sqrt{\zeta^2 - \omega_o^2}$
I(t) Aeλ1t + Beλ2t Aeλ1t + Beλ2t Aeλ1t + Beλ2t
Damping Factor $\zeta$ $\zeta = {R \over 2L}$ $\zeta = {1 \over 2RC}$
Resonant Frequency $\omega_o$ $\omega_o = {1 \over \sqrt{L C}}$ $\omega_o = {1 \over \sqrt{L C}}$
Band Width $\Delta \omega = 2 \zeta$ ${ R \over L}$ ${ 1 \over CR}$
Quality factor $Q = {\omega_o \over \Delta \omega } = {\omega_o \over 2\zeta }$ $Q = {L \over R \sqrt{LC}} = {1 \over R} \sqrt{L \over C}$ $Q = {CR \over \sqrt{LC}} = {R} \sqrt{C \over L}$