RLC circuit

It has been requested that this article's formulas be translated to the English language.

An RLC circuit (sometimes known as resonant or tuned circuit) is an electrical circuit comprising a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. A RLC circuit is called a second-order circuit as any voltage or current in the circuit can be described by a second-order differential equation.

Resonant frequency

The resonant or center frequency of an RLC circuit (in hertz) is:

$f_{c}={1 \over 2\pi {\sqrt {LC}}}$

Q factor

The RLC circuit may be used as a bandpass or band-stop filter, and the Q factor is

$Q={f_{c} \over BW}={2\pi f_{c}L \over R}={1 \over {\sqrt {R^{2}C/L}}}$

Alternatively, we can calculate the Q factor in terms of the resonant frequency $\omega$ and the damping factor $\alpha$:

$Q={\omega _{0} \over 2\alpha }$

Configurations

There are two main configurations of RLC circuits: series and parallel.

Series RLC circuit

In this circuit, the three components are all in series with the voltage source.

RLC series circuit

Series RLC Circuit notations: V - the voltage of the power source (measured in volts V) I - the current in the circuit (measured in amperes A) R - the resistance of the resistor (measured in ohms = V/A); L - the inductance of the inductor (measured in henries = H = V·s/A) C - the capacitance of the capacitor (measured in farads = F = C/V = A·s/V)

Given the parameters V, R, L, and C, the solution for the current (I) using Kirchoff's voltage law is:

${V_{R}+V_{L}+V_{C}=V}\,$

For a time-changing voltage V(t), this becomes

$RI(t)+L{{dI} \over {dt}}+{1 \over C}\int _{-\infty }^{t}I(\tau )\,d\tau =V(t)$

Rearranging the equation gives the following second order differential equation:

${{d^{2}I} \over {dt^{2}}}+{R \over L}{{dI} \over {dt}}+{1 \over {LC}}I(t)={1 \over L}{{dV} \over {dt}}$

We now define two key parameters:

$\alpha ={R \over 2L}$

and

$\omega _{0}={1 \over {\sqrt {LC}}}$

Substituting these parameters into the differential equation, we obtain:

${{d^{2}I} \over {dt^{2}}}+2\alpha {{dI} \over {dt}}+\omega _{0}I(t)={1 \over L}{{dV} \over {dt}}$

The ZIR (Zero Input Response) solution

Setting the input (voltage sources) to zero, we have:

${{d^{2}I} \over {dt^{2}}}+{R \over L}{{dI} \over {dt}}+{1 \over {LC}}I(t)=0$

with the initial conditions for the inductor current, I_L(0), and the capacitor voltage V_C(0). In order to solve the equation properly, the initial conditions needed are I(0) and I'(0).

The first one we already have since the current in the main branch is also the current in the inductor, therefore

$I(0)=I_{L}(0)\,$

The second one is obtained employing KVL again:

$V_{R}(0)+V_{L}(0)+V_{C}(0)=0\,$

$\Rightarrow I(0)R+I'(0)L+V_{C}(0)=0\,$

$\Rightarrow I'(0)={1 \over L}\left[-V_{C}(0)-I(0)R\right]$

We have now a homogeneous second order differential equation with two initial conditions. Usually second order differential equations are written as:

$I''+2\xi \omega _{k}I'+\omega _{k}^{2}I=0$

In case of an electrical circuit ω_k > 0 and therefore, there are three possible cases:

Over-damping

RLC serial Over Damping

$\alpha >1\Rightarrow R^{2}C>4L\,$

In this case, the characteristic polynomial's solutions are both negative real numbers. This is called "over damping".

Two negative real roots, the solutions are:

$I(t)=ae^{\lambda _{1}t}+be^{\lambda _{2}t}$

Critical damping

RLC serial Critical Damping

$\alpha =1\Rightarrow R^{2}C=4L\,$

In this case, the characteristic polynomial's solutions are identical negative real numbers. This is called "critical damping".

The two roots are identical ($\lambda _{1}=\lambda _{2}=\lambda$), the solutions are:

$I(t)=(a+bt)e^{\lambda t}$

Under-damping

File:RLC-serial-Under Damping.PNG

RLC serial Under Damping

$\alpha <1\Rightarrow R^{2}C<4L\,$

In this case, the characteristic polynomial's solutions are complex conjugate and have negative real part. This is called "under damping".

Two conjugate roots ($\lambda _{1}={\bar {\lambda }}_{2}=\alpha +i\beta$), the solutions are:

$I(t)=e^{\alpha t}\left[a\sin(\beta t)+b\cos(\beta t)\right]$

The ZSR (Zero State Response) solution

This time we set the initial conditions to zero and use the following equation:

$\left\{{\begin{matrix}{{d^{2}I} \over {dt^{2}}}+{R \over L}{{dI} \over {dt}}+{1 \over {LC}}I(t)={1 \over L}{{dV} \over {dt}}\\I(0^{-})=I'(0^{-})=0\end{matrix}}\right.$

A separate solution for every possible function for V(t) is impossible. However, there is a way to find a formula for I(t) using convolution. In order to do that, we need a solution for a basic input - the Dirac delta function.

In order to find the solution more easily we will start solving for the Heaviside step function and then using the fact that our circuit is a linear system, its derivative will be the solution for the delta function.

The equation will be therefore, for t>0:

$\left\{{\begin{matrix}{{d^{2}I_{u}} \over {dt^{2}}}+{R \over L}{{dI_{u}} \over {dt}}+{1 \over {LC}}I_{u}(t)=0\\I(0^{+})=0\qquad I'(0^{+})={1 \over L}\end{matrix}}\right.$

Assuming λ₁ and λ₂ are the roots of

$P(R)=R^{2}+2\xi \omega _{k}R+\omega _{k}^{2}$

then as in the ZIR solution, we have 3 cases here:

Over-damping

Two negative real roots, the solution is:

$I_{u}(t)={1 \over {L(\lambda _{1}-\lambda _{2})}}\left[e^{\lambda _{1}t}-e^{\lambda _{2}t}\right]$ $\Rightarrow I_{\delta }(t)={1 \over {L(\lambda _{1}-\lambda _{2})}}\left[\lambda _{1}e^{\lambda _{1}t}-\lambda _{2}e^{\lambda _{2}t}\right]$

Critical damping

The two roots are identical ($\lambda _{1}=\lambda _{2}=\lambda$), the solution is:

$I_{u}(t)={1 \over L}te^{\lambda t}$ $\Rightarrow I_{\delta }(t)={1 \over L}(\lambda t+1)e^{\lambda t}$

Under-damping

Two conjugate roots ($\lambda _{1}={\bar {\lambda }}_{2}=\alpha +i\beta$), the solution is:

$I_{u}(t)={1 \over {\beta L}}e^{\alpha t}\sin(\beta t)$ $\Rightarrow I_{\delta }(t)={1 \over {\beta L}}e^{\alpha t}\left[\alpha \sin(\beta t)+\beta \cos(\beta t)\right]$

(to be continued...)

Sinusoidal steady-state analysis

The series RLC can be analyzed in the frequency domain using complex impedance relations. If the voltage source above produces a pure sine wave with amplitude V and angular frequency ω, KVL can be applied:

$V=I\left(R+j\omega L+{\frac {1}{j\omega C}}\right)$

Where I is the complex current through all components. Solving for I:

$I={\frac {V}{R+j\omega L+{\frac {1}{j\omega C}}}}$

Taking the magnitude of the above equation:

$I_{mag}={\frac {V}{\sqrt {R^{2}+\left(\omega L-{\frac {1}{\omega C}}\right)^{2}}}}$

If we choose trivial values where R = 1, C = 1, L = 1, and V = 1, then the graph of magnitude of current as a function of ω is:

Sinusoidal steady-state analysis

Note that there is a peak at $I_{mag}(\omega )=1$. This is known as the resonant frequency. Solving for this value, we find:

$\omega _{o}={\frac {1}{\sqrt {LC}}}$

Parallel RLC circuit

A much more elegant way of recovering the circuit properties of an RLC circuit is through the use of nondimensionalization.

Parallel RLC Circuit notations: V - the voltage of the power source (measured in volts V) I - the current in the circuit (measured in amperes A) R - the resistance of the resistor (measured in ohms = V/A); L - the inductance of the inductor (measured in henries = H = V·s/A) C - the capacitance of the capacitor (measured in farads = F = C/V = A·s/V)

For a parallel configuration of the same components, where Φ is the magnetic flux in the system

$C{\frac {d^{2}\Phi }{dt^{2}}}+{\frac {1}{R}}{\frac {d\Phi }{dt}}+{\frac {1}{L}}\Phi =I_{0}\cos(\omega t)\Rightarrow {\frac {d^{2}\chi }{d\tau ^{2}}}+2\zeta {\frac {d\chi }{d\tau }}+\chi =\cos(\Omega \tau )$

with substitutions

$\Phi =\chi x_{c},\ t=\tau t_{c},\ x_{c}=LI_{0},\ t_{c}={\sqrt {LC}},\ 2\zeta ={\frac {1}{R}}{\sqrt {\frac {L}{C}}},\ \Omega =\omega t_{c}.$

The first variable corresponds to the maximum magnetic flux stored in the circuit. The second corresponds to the period of resonant oscillations in the circuit.

Similarities and differences between series and parallel circuits

The expressions for the bandwidth in the series and parallel configuration are inverses of each other. This is particularily useful for determining whether a series or parallel configuration is to be used for a particular circuit design. However, in circuit analysis, usually the reciprocal of the latter two variables are used to characterize the system instead. They are known as the resonant frequency and the Q factor respectively.

See also

• Electronic oscillator
• Bandwidth
• Resonant frequency
• Quality factor