RLC circuit

A RLC circuit (also known as a resonant circuit, tuned circuit, or LCR circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. This configuration forms a harmonic oscillator.

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.

ConfigurationsEdit

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.

Similarities and differences between series and parallel circuitsEdit

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 is used to characterize the system instead. They are known as the resonant frequency and the Q factor respectively.

Fundamental parametersEdit

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.

Resonant frequencyEdit

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

 

In the more familiar unit hertz (or cycles per second), the natural frequency becomes

 

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

 

Both of these impedances are functions of angular frequency  :

 
 

Setting the magnitude of the impedance to be zero at   and using  :

 
 

Damping factorEdit

The normalized damping factor of the circuit is:

 

for a series RLC circuit, and:

 

for a parallel RLC circuit. It's desirable to use the normalized forced damping factor (non-dimensional) instead of the regular one (in radians per second) to analyze the properties of the resonant 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.

Derived parametersEdit

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

BandwidthEdit

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. In the Series case the bandwidth (in radians per second) is

 

Alternatively, the bandwidth in hertz is

 

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   at the half-power frequencies.

Resonance dampingEdit

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

 

In an oscillator circuit

 .

As a result

 .

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

Circuit analysisEdit

Series RLC with Thévenin power sourceEdit

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 henrys = H = V·s/A)
C - the capacitance of the capacitor (measured in farads = F = C/V = A·s/V)
q - the charge across the capacitor (measured in coulombs C)

Given the parameters v, R, L, and C, the solution for the charge, q, can be found using Kirchhoff's voltage law. (KVL) gives

 

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

 

Using the relationship between charge and current:

 

The above expression can be expressed in terms of charge across the capacitor:

 

Dividing by L gives the following second order differential equation:

 

We now define two key parameters:

  and  

Substituting these parameters into the differential equation, we obtain:

 

or

 

Frequency domainEdit

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   , KVL can be applied:

 

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

 

And rearranging, we have at

 
Complex admittanceEdit

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

 

Finally, we simplify using parameters ζ and ωo

 

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

Poles and zerosEdit

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

  and  

The poles of Y(s) are those values of s such that  . By the quadratic formula, we find

 

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

Sinusoidal steady stateEdit
Now let  ....

Taking the magnitude of the above equation:

 

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

 

If we choose values where R = 1 ohm, C = 1 farad, L = 1 henry, and V = 1.0 volt, then the graph of magnitude of the current i (in amperes) as a function of ω (in radians per second) is:

 
Sinusoidal steady-state analysis

Note that there is a peak at  . This is known as the resonant frequency. Solving for this value, we find:

 

Parallel RLC circuitEdit

 

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 henrys = H = V·s/A)
C - the capacitance of the capacitor (measured in farads = F = C/V = A·s/V)

The complex impedance of this circuit is given by adding up the impedances in parallel:

 

The change from a series arrangement to a parallel arrangement has some very real consequences for the behaviour. This can be seen by plotting the magnitude of the current  . For comparison with the earlier graph we choose values where R = 1 ohm, C = 1 farad, L = 1 henry, and V = 1.0 volt and ω in radians per second:


Sinusoidal steady-state analysis

There is a minimum in the frequency response at the resonant frequency  .

A parallel RLC circuit is a example of a band-stop circuit response that can be used as a filter to block frequencies at the resonance frequency but allow others to pass.

Template:Cleanup-remainder

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

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

 

with substitutions

 

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.

See alsoEdit

External linksEdit