RLC circuit

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An RLC circuit (sometimes known as resonant or tuned circuit) is an electrical circuit consisting of 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.

Contents

Fundamental Parameters

There are two fundamental parameters that describe the behavior of RLC circuits: the resonant frequency and the damping factor. In addition, there are several other parameters that can be derived from these first two (see next section).


Resonant frequency

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

\omega_o = {1 \over \sqrt{L C}}


In the more familiar unit hertz, the natural frequency becomes

f_o = {\omega_o \over 2 \pi} = {1 \over 2 \pi \sqrt{L C}}


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

ZLC = ZL + ZC = 0

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

Z_C = { 1 \over Cs }
ZL = Ls

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

 s = \pm j \omega_o = \pm j {1 \over \sqrt{L C}}

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


Damping factor

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

 \zeta = {R \over 2L}


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. In this case, the RLC circuit becomes a good approximation to an ideal LC circuit, which is not realizable in practice (even if the resistor is removed from the circuit, there is always a small but non-zero amount of resistance in the wiring and interconnects between the other circuit elements that can never be eliminated entirely).

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.

Derived Parameters

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

Bandwidth

The RLC circuit may be used as a bandpass or band-stop filter, and the bandwidth (in radians per second) is

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


Alternatively, the bandwidth in hertz is

\Delta f =   {  \Delta \omega \over 2 \pi  } =  { \zeta \over \pi }= { R \over 2 \pi L}


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  { 1 \over \sqrt{2} } at the half-power frequencies.

Quality or Q factor

The Quality of the tuned circuit, or Q factor, is calculated as the ratio of the resonance frequency ωo to the bandwidth Δω (in radians per second):

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


Or in hertz:

Q = {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}

Q is a dimensionless quantity.

Damped Resonance

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

 \zeta \ < \ \omega_o

then we can define the damped resonance as

 \omega_d = \sqrt{ \omega_o^2 - \zeta^2 }


In an oscillator circuit

 \zeta \ \ << \ \ \omega_o .

As a result

 \omega_d \ \ = \ \ \omega_o \ \ (approx).


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

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.

Circuit Analysis

Series RLC with Thévenin power source

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

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:

 \zeta = {R \over 2L}
and
\omega_0 = { 1 \over \sqrt{LC}}

both of which are measured as radians per second.

Substituting these parameters into the differential equation, we obtain:


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



The Zero Input Response (ZIR) solution

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



{{d^2 i} \over {dt^2}} + 2 \zeta {{di} \over {dt}} + \omega_o^2  i(t) = 0


with the initial conditions for the inductor current, IL(0), and the capacitor voltage VC(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. Substituting the two parameters ζ and ω0, we have



i''+2\zeta i' + \omega_0^2 i = 0


We now convert the form of this equation to its characteristic polynomial

\lambda^2 + 2 \zeta \lambda + \omega_0^2 = 0

Using the quadratic formula, we find the roots as

 \lambda = -\zeta \pm \sqrt{\zeta^2 - \omega_0^2}

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

Over-damping
File:RLC-serial-Over Damping.PNG
RLC series Over Damped Response

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

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)=A e^{\lambda_1 t} + B e^{\lambda_2 t}


Critical damping
File:RLC-serial-Critical Damping.PNG
RLC series Critically Damped

\zeta=\omega_0 \Rightarrow RC=4 { L \over R } \,

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

The two roots are identical (λ1 = λ2 = λ), the solutions are:

I(t) = (A + Bt)eλt
for arbitrary constants A and B



Under-damping
File:RLC-serial-Under Damping.PNG
RLC series Under Damped

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

In this case, the characteristic polynomial's solutions are complex conjugate and have negative real part. This is called "under damping" and results in oscillations or ringing in the circuit. The solution consists of two conjugate roots

λ1 = − ζ + iωc

and

λ2 = − ζ − iωc

where

\omega_c = \sqrt{\omega_o^2 - \zeta^2}


The solutions are:

i(t) = Ae^{-\zeta + i \omega_c} + Be^{-\zeta - i \omega_c}
for arbitrary constants A and B.


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

i(t)=e^{-\zeta t} \left[ C \sin(\omega_c t) + D \cos(\omega_c t) \right]
for arbitrary constants C and D.


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.



The Zero State Response (ZSR) 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.


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


There are two approaches we can take to finding the ZSR: (1) the Laplace Transform, and (2) the Convolution Integral.

Laplace Transform

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

 (s^2 + 2\zeta s + \omega_o^2) I(s) = {s \over L } V(s)


where V(s) is the Laplace Transform of the input signal:
V(s) = \mathcal{L} \left\{ v(t) \right\}


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

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


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):

  I(s) = Y(s) \times V(s)


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

i(t) = \mathcal{L}^{-1} \left\{ I(s) \right\}


Example:

Suppose v(t) = Au(t)

where u(t) is the Heaviside step function.

Then

 V(s) = { A \over s }


 I(s) = { A \over L (s^2 + 2\zeta s + \omega_o^2)  }
Convolution Integral

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(\lambda)= \lambda^2+2 \zeta \lambda + \omega_o^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 (λ1 = λ2 = λ), the solution is:


I_u(t)={1 \over L} t e^{\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 = \zeta + i\omega_c), the solution is:



I_u(t)={1 \over {\omega_c L}} e^{\zeta t} \sin(\omega_c t)

\Rightarrow I_{\delta}(t)={1 \over {\omega_c L}} e^{\zeta t} \left[ \zeta \sin(\omega_c t) + \omega_c \cos(\omega_c t) \right]

(to be continued...)

Frequency Domain

The series RLC can be analyzed in the frequency domain using complex impedance relations. If the voltage source above produces a complex exponential wave form with amplitude V(s) and angular frequency s = σ + iω , KVL can be applied:

V(s) = I(s) \left ( R + Ls + \frac{1}{Cs} \right )

where I(s) is the complex current through all components. Solving for I:

I(s) = \frac{1}{ R + Ls + \frac{1}{Cs} } V(s)


And rearranging, we have

I(s) = \frac{s}{ L \left ( s^2 + {R \over L}s + \frac{1}{LC} \right ) } V(s)


Complex Admittance

Next, we solve for the complex admittance Y(s):

 Y(s) = { I(s) \over V(s) } = \frac{s}{ L \left ( s^2 + {R \over L}s + \frac{1}{LC} \right ) }


Finally, we simplify using parameters α and ωo

 Y(s) = { I(s) \over V(s) } = \frac{s}{ L \left ( s^2 + 2 \alpha s + \omega_o^2 \right ) }


Notice that this expression for Y(s) is the same as the one we found for the Zero State Response.

Poles and Zeros

The zeros of Y(s) are those values of s such that Y(s) = 0:

s = 0 and  s = \infty


The poles of Y(s) are those values of s such that  Y(s) = \infty:

 s = - \zeta \pm \sqrt{\zeta^2 - \omega_o^2}


Notice that the poles of Y(s) are identical to the roots λ1 and λ2 of the characteristic polynomial.

Sinusoidal Steady State

If we now let s = iω....

Taking the magnitude of the above equation:

 | Y(s=i \omega) | = \frac{1}{\sqrt{ R^2 + \left ( \omega L - \frac{1}{\omega C} \right )^2 }}


Next, we find the magnitude of current as a function of ω

 | I( i \omega  ) |  =  | Y(i \omega) | \times | V(i \omega) |


If we choose values where R = 1 ohm, C = 1 farad, L = 1 henry, and V = 1 volt, then the graph of magnitude of the current I (in amperes) as a function of ω (in radians per second) is:

File:RLC series imag.png
Sinusoidal steady-state analysis

Note that there is a peak at Imag(ω) = 1. This is known as the resonance frequency. Solving for this value, we find:

\omega_o = \frac{1}{\sqrt{L C}}

Parallel RLC circuit

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

RLC Parallel circuit

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 = L I_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 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.

Applications of tuned circuits

There are many applications for tuned circuits especially in radio and communication systems. They can be used to select a certain narrow range of frequencies from the total spectrum of radio waves.

See also

de:Schwingkreis fr:Circuit RLC it:Circuito RLC zh:RLC电路

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