RLC circuit

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An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel.

Tuned circuits have many applications particularly for oscillating circuits and in radio and communication engineering. They can be used to select a certain narrow range of frequencies from the total spectrum of ambient radio waves. For example, AM/FM radios with analog tuners typically use an RLC circuit to tune a radio frequency. Most commonly a variable capacitor is attached to the tuning knob, which allows you to change the value of C in the circuit and tune to stations on different frequencies.

An 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 for circuit analysis.

[edit] Configurations

Every RLC circuit consists of two components: a power source and resonator. There are two types of power sources – Thévenin and Norton. Likewise, there are two types of resonators – series LC and parallel LC. As a result, there are four configurations of RLC circuits:

Series LC with Thévenin power source
Series LC with Norton power source
Parallel LC with Thévenin power source
Parallel LC with Norton power source.

[edit] 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 particularly 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.

[edit] Fundamental Parameters

There are two fundamental parameters that describe the behavior of RLC circuits: the resonant frequency and the damping factor. In addition, other parameters derived from these first two are discussed below.

[edit] Resonant frequency

The undamped resonance or natural frequency of an RLC circuit (in radians per second) is:

<math>\omega_o = {1 \over \sqrt{L C}}</math>

In the more familiar unit hertz, the natural frequency becomes

<math>f_o = {\omega_o \over 2 \pi} = {1 \over 2 \pi \sqrt{L C}}</math>

Resonance occurs when the complex impedance Z_LC of the LC resonator becomes zero:

Both of these impedances are functions of complex angular frequency s:

Setting these expressions equal to one another and solving for s, we find:

<math> s = \pm j \omega_o = \pm j {1 \over \sqrt{L C}}</math>

where the resonance frequency ω_o is given in the expression above.

[edit] Damping factor

The damping factor of the circuit (in radians per second) is:

for a series RLC circuit, and:

for a parallel RLC circuit.

For applications in oscillator circuits, it is generally desirable to make the damping factor as small as possible, or equivalently, to increase the quality factor (Q) as much as possible. In practice, this requires decreasing the resistance R in the circuit to as small as physically possible for a series circuit, and increasing R to as large a value as possible for a parallel circuit. In this case, the RLC circuit becomes a good approximation to an ideal LC circuit.

Alternatively, for applications in bandpass filters, the value of the damping factor is chosen based on the desired bandwidth of the filter. For a wider bandwidth, a larger value of the damping factor is required (and vice versa). In practice, this requires adjusting the relative values of the resistor R and the inductor L in the circuit.

[edit] Derived Parameters

The derived parameters include Bandwidth, Q factor, and damped resonance frequency.

[edit] Bandwidth

The RLC circuit may be used as a bandpass or band-stop filter by replacing R with a receiving device with the same input resistance, and the bandwidth (in radians per second) is

<math> \Delta \omega = 2 \zeta = { R \over L}</math>

Alternatively, the bandwidth in hertz is

<math> \Delta f = { \Delta \omega \over 2 \pi } = { \zeta \over \pi } = { R \over 2 \pi L }</math>

The bandwidth is a measure of the width of the frequency response at the two half-power frequencies. As a result, this measure of bandwidth is sometimes called the full-width at half-power. Since electrical power is proportional to the square of the circuit voltage (or current), the frequency response will drop to <math> { 1 \over \sqrt{2} } </math> at the half-power frequencies.

[edit] Quality or Q factor

The Quality of the series tuned circuit, or Q factor, is calculated as the ratio of the resonance frequency <math>\omega_o</math> to the bandwidth <math>\Delta \omega </math> (in radians per second):

<math>Q_s = {\omega_o \over \Delta \omega } = {\omega_o \over 2\zeta } = {L \over R \sqrt{LC}} = {1 \over R} \sqrt{L \over C}</math>

Or in hertz:

<math>Q_s = {f_o \over \Delta f} = {2 \pi f_o L \over R} = {1 \over \sqrt{R^2 C / L}} = {1 \over R} \sqrt{L \over C}</math>

For the parallel tuned circuit:

Q is a dimensionless quantity.

[edit] Resonance Damping

The damped resonance frequency derives from the natural frequency and the damping factor. If the circuit is underdamped, meaning

then we can define the damped resonance as

<math> \omega_d = \sqrt{ \omega_o^2 - \zeta^2 } </math>

In an oscillator circuit

<math> \zeta \ \ << \ \ \omega_o </math>.

As a result

<math> \omega_d \ \ = \ \ \omega_o \ \ </math> (approx).

See discussion of underdamping, overdamping, and critical damping, below.

[edit] Circuit Analysis

[edit] Series RLC with Thévenin power source

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

Image:RLC series circuit.png

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 Kirchhoff's voltage law (KVL) is:

<math> {v_R+v_L+v_C=v} \,

</math>

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

<math>

Ri(t) + L { {di} \over {dt}} + {1 \over C} \int_{-\infty}^{t} i(\tau)\, d\tau = v(t)

</math>

Rearranging the equation gives the following second order differential equation:

<math>

{{d^2 i} \over {dt^2}} +{R \over L} {{di} \over {dt}} + {1 \over {LC}} i(t) = {1 \over L} {{dv} \over {dt}}

</math>

We now define two key parameters:

and

<math>\omega_0 = { 1 \over \sqrt{LC}} </math>

both of which are measured as radians per second.

Substituting these parameters into the differential equation, we obtain:

<math>

{{d^2 i} \over {dt^2}} + 2 \zeta {{di} \over {dt}} + \omega_0^2 i(t) = {1 \over L} {{dv} \over {dt}} </math>

<math>

i+2\zeta i' + \omega_0^2 i = {1 \over L} {{dv} \over {dt}} </math>

[edit] The Zero Input Response (ZIR) solution

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

<math>

i+2\zeta i' + \omega_0^2 i = 0 </math>

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

<math>

i(0)=i_L(0) \, </math>

The second one is obtained employing KVL again:

<math>

v_R(0)+v_L(0)+v_C(0)=0 \, </math>

<math>

i(0)R+i'(0)L+v_C(0)=0 \, </math>

and

<math>

i'(0)={1 \over L}\left[-v_C(0)-I(0)R \right] </math>

We have now a homogeneous second order differential equation with two initial conditions. Substituting the two parameters ζ and ω₀, we have

<math>

i+2\zeta i' + \omega_0^2 i = 0 </math>

We now form the equation’s characteristic polynomial

<math>\lambda^2 + 2 \zeta \lambda + \omega_0^2 = 0 </math>

Using the quadratic formula, we find the roots as

<math> \lambda = -\zeta \pm \sqrt{\zeta^2 - \omega_0^2} </math>

Depending on the values of ζ and ω₀, there are three possible cases:

[edit] Over-damping

RLC series Over-Damped Response

<math>

\zeta>\omega_0 \Rightarrow RC>4 { L \over R} \, </math>

In this case, the characteristic polynomial's solutions are both negative real numbers.

Two negative real roots, the solutions are:

<math>

I(t)=A e^{\lambda_1 t} + B e^{\lambda_2 t} </math>

This is called "over-damping".

[edit] Critical damping

RLC series Critically Damped

<math>

\zeta=\omega_0 \Rightarrow RC=4 { L \over R } \, </math>

In this case, the characteristic polynomial's solutions are identical negative real numbers.

The two roots are identical (<math> \lambda_1=\lambda_2=\lambda </math>), the solutions are:

<math>I(t)=(A+Bt) e^{\lambda t}</math>

for arbitrary constants A and B

This is called "critical damping".

[edit] Under-damping

RLC series Under-Damped

<math>

\zeta<\omega_0 \Rightarrow RC<4 { L \over R } \, </math>

In this case, the characteristic polynomial's solutions are complex conjugate and have negative real parts. The solution consists of two conjugate roots

<math>\lambda_1 = -\zeta + i\omega_c</math>

and

<math>\lambda_2 = -\zeta - i\omega_c</math>

where

<math>\omega_c = \sqrt{\omega_o^2 - \zeta^2}</math>

The solutions are:

<math>i(t) = Ae^{(-\zeta + i \omega_c)t} + Be^{(-\zeta - i \omega_c)t} </math>

for arbitrary constants A and B.

Using Euler's formula, we can simplify the solution to

<math>i(t)=e^{-\zeta t} \left[ C \sin(\omega_c t) + D \cos(\omega_c t) \right]</math>

for arbitrary constants C and D.

This is called "under-damping" and results in oscillations or ringing in the circuit.

These solutions are characterized by exponentially decaying sinusoidal response. The time required for the oscillations to "die out" depends on the Quality of the circuit, or Q factor. The higher the Quality, the longer it takes for the oscillations to decay. As before, for a series RLC circuit,

which is a dimensionless parameter.

[edit] The Zero State Response (ZSR) solution

This time we set the initial conditions to zero and have:

<math>

i+2\zeta i' + \omega_0^2 i = {1 \over L} v' </math>

with the initial condition

<math>

i(0^{-})=i'(0^{-})=0 </math>

There are two approaches we can take to finding the ZSR: (1) the Laplace transform, and (2) the convolution integral.

[edit] Laplace Transform

We first take the Laplace transform of the second order differential equation:

<math> (s^2 + 2\zeta s + \omega_o^2) i(s) = {s \over L } v(s) </math>

<math> i(s) = { s \over L (s^2 + 2\zeta s + \omega_o^2) } v(s) </math>

where v(s) is the Laplace transform of the input signal:

<math>v(s) = \mathcal{L} \left\{ v(t) \right\} </math>

We then solve for the complex admittance Y(s) (in siemens):

<math> Y(s) = { I(s) \over V(s) } = { s \over L (s^2 + 2\zeta s + \omega_o^2) } </math>

We can then use the admittance Y(s) and the Laplace transform of the input voltage v(s) to find the complex electrical current i(s):

Finally, we can find the electrical current in the time domain by taking the inverse Laplace transform:

<math>i(t) = \mathcal{L}^{-1} \left\{ i(s) \right\} </math>

Example:

Suppose <math>v(t) = Au(t) </math>

where u(t) is the Heaviside step function.

Then

<math> i(s) = { A \over L (s^2 + 2\zeta s + \omega_o^2) } </math>

[edit] Convolution Integral

A single closed 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.

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

The equation will be therefore, for t>0:

<math>

\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. </math>

Assuming λ₁ and λ₂ are the roots of

<math>

P(\lambda)= \lambda^2+2 \zeta \lambda + \omega_o^2 </math>

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