RLC circuit – 940nm Diode Laser K94S02F-10.00W – Red Diode Laser K66S06F-0.08W-S
Configurations
Every RLC circuit consists of two components: a power source and resonator. There are two types of power sources Thvenin 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 Thvenin power source
Series LC with Norton power source
Parallel LC with Thvenin power source
Parallel LC with Norton power source.
It is relatively easy to show that each of the two series configurations can be transformed into the other using elementary network transformations specifically, by transforming the Thvenin power source to the equivalent Norton power source, or vice versa. Likewise, each of the two parallel configurations can be transformed into the other using the same network transformations. Finally, the Series/Thvenin and the Parallel/Norton configurations are dual circuits of one another. Likewise, the Series/Norton and the Parallel/Thvenin configurations are also dual circuits.
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 is used to characterize the system instead. They are known as the resonance frequency and the Q factor respectively.
Fundamental parameters
There are two fundamental parameters that describe the behavior of RLC circuits: the resonance frequency and the attenuation (or, alternatively, the damping factor). In addition, other parameters derived from these first two are discussed below.
Resonance frequency
The undamped resonance frequency of an RLC circuit (in radians per second) is given by
In the more familiar unit hertz (or cycles per second), the resonance 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 = 0 and using j2 = 1:
Attenuation
The attenuation is defined as
for the series RLC circuit, and
for the parallel RLC circuit.
Damping factor
The damping factor is the ratio of the attenuation to the resonance frequency 0 :
for a series RLC circuit, and:
for a parallel RLC circuit.
It is sometimes more convenient to use the damping factor, which is dimensionless, instead of the attenuation factor, which has dimensions of radians per second, to analyze the properties of a resonant circuit.
Minimizing the attenuation for oscillator circuits
For applications in oscillator circuits, it is generally desirable to make the attenuation (or equivalently, the damping factor) as small as possible. In practice, this objective requires making the circuit’s resistance R as small as physically possible for a series circuit, or alternatively increasing R to as much as possible for a parallel circuit. In either case, the RLC circuit becomes a good approximation to an ideal LC circuit. However, for very low attenuation circuits (high Q-factor) circuits, issues such as dielectric losses of coils and capacitors can become important.
For applications in band-pass 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 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.
Q factor
The Q factor can be expressed in terms of the three devices in the circuit, from the basis of the definition:
Damped resonance
Main articles: Damping and Damping ratio
The damped resonance frequency can be expressed in terms of the undamped resonance frequency and the damping factor. If the circuit is underdamped, meaning
or equivalently
then we can define the damped resonance as
In an oscillator circuit
.
or equivalently
.
As a result
.
Circuit analysis
Series RLC with Thvenin 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 henrys = H = Vs/A)
C – the capacitance of the capacitor (measured in farads = F = C/V = As/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 domain
The series RLC can be analyzed in the frequency domain using complex impedance relations. If the voltage source above produces a complex exponential waveform with complex amplitude V(s) and angular frequency s = + i , KVL can be applied:
where I(s) is the complex current through all components. Solving for I(s):
And rearranging, we have at
Complex admittance
Next, we solve for the complex admittance Y(s):
Finally, we simplify using parameters and o
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:
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 1 and 2 of the characteristic polynomial.
Sinusoidal steady state
If we now let s = i….
Considering the magnitude of the above equation:
Next, we find the magnitude of current as a function of
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 (in radians per second) is:
Sinusoidal steady-state analysis
Note that there is a peak at imag() = 1. This is known as the resonance frequency. Solving for this value, we find:
Parallel RLC 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 henrys = H = Vs/A)
C – the capacitance of the capacitor (measured in farads = F = C/V = As/V)
The complex admittance of this circuit is given by adding up the admittances of the components:
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 in radians per second:
Sinusoidal steady-state analysis
There is a minimum in the frequency response at the resonance frequency .
A parallel RLC circuit is an 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.
See also
LC circuit
RC circuit
RL circuit
Electrical resonance
Electronic oscillator
Bandwidth (signal processing)
Bandpass filter
Oliver Heaviside
Linear circuit
External links
http://www.falstad.com/circuit/e-lrc.html
a treatment that starts with the mechanical analogy
An interactive simulation on series RCL circuit
Interactive Visual Representation of the LRC Circuit
Pulse Response Examiner freeware (Windows)
Categories: Analog circuitsHidden categories: Articles lacking sources from March 2009 | All articles lacking sources | Articles needing cleanup from January 2010 | All pages needing cleanup | Wikipedia articles needing style editing from January 2010 | All articles needing style editing
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