RLC circuit

来源:百度文库 编辑:神马文学网 时间:2024/04/29 12:12:34
From Wikipedia, the free encyclopedia
Jump to:navigation,search
An RLC circuit (also known as aresonant circuit or atuned circuit) is anelectrical circuit consisting of aresistor (R), aninductor (L), and acapacitor (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 totalspectrum 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-orderdifferential equation for circuit analysis.
Contents
[hide]
1 Configurations2 Similarities and differences between series and parallel circuits3 Fundamental Parameters3.1 Resonant frequency3.2 Damping factor
4 Derived Parameters4.1 Bandwidth4.2 Resonance Damping
5 Circuit Analysis5.1 Series RLC with Thévenin power source5.1.1 Frequency Domain5.1.1.1 Complex Admittance5.1.1.2 Poles and Zeros5.1.1.3 Sinusoidal Steady State
5.2 Parallel RLC circuit
6 See also7 External links
[edit] Configurations
Every RLC circuit consists of two components: a power source and resonator. There are two types of power sources –Thévenin andNorton. Likewise, there are two types of resonators – seriesLC 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 is used to characterize the system instead. They are known as theresonant frequency and theQ factor respectively.
[edit] Fundamental Parameters
There are two fundamentalparameters 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
Theundampedresonance or natural frequency of an RLC circuit (inradians per second) is given by

In the more familiar unithertz (or inverse seconds), the natural frequency becomes

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

Both of these impedances are functions of complexangular frequency s:

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

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

[edit] Damping factor
Thedamping factor of the circuit (inradians 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 idealLC 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 abandpass orband-stop filter by replacing R with a receiving device with the same input resistance, and thebandwidth (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 electricalpower is proportional to the square of the circuit voltage (or current), the frequency response will drop toat the half-power frequencies.
[edit] Resonance Damping
Thedamped 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
(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 thevoltage source.

Series RLC Circuit notations:
v - the voltage of the power source (measured involts V) i - the current in the circuit (measured inamperes A) R - theresistance of the resistor (measured inohms = V/A); L - theinductance of the inductor (measured inhenrys = H = V·s/A) C - thecapacitance of the capacitor (measured infarads = F =C/V = A·s/V) q - the charge across the capacitor (measured incoulombs C)
Given the parameters v, R, L, and C, the solution for the current q usingKirchhoff‘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
both of which are measured asradians per second.
Substituting these parameters into the differential equation, we obtain:

or

[edit] Frequency Domain
The series RLC can be analyzed in thefrequency domain usingcompleximpedance relations. If the voltage source above produces a complex exponential wave form with amplitude v(s) andangular frequency s = σ + iω ,KVL can be applied:

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

And rearranging, we have

[edit] Complex Admittance
Next, we solve for the complexadmittance 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.
[edit] Poles and Zeros
Thezeros of Y(s) are those values of s such that Y(s) = 0:
s = 0 and
Thepoles of Y(s) are those values of s such that. By thequadratic formula, we find

Notice that the poles of Y(s) are identical to the roots λ1 and λ2 of the characteristic polynomial.
[edit] Sinusoidal Steady State
If we now let s = iω....
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 imag(ω) = 1. This is known as theresonant frequency. Solving for this value, we find:

[edit] Parallel RLC circuit
A much more elegant way of recovering the circuit properties of an RLC circuit is through the use ofnondimensionalization.
The remainder of this article may requirecleanup to meet Wikipedia‘squality standards.
Please discuss this issue on thetalk page, or replace this tag with amore specific message. This article has been tagged since August 2006.

Parallel RLC Circuit notations:
V - the voltage of the power source (measured involts V) I - the current in the circuit (measured inamperes A) R - theresistance of the resistor (measured inohms = V/A); L - theinductance of the inductor (measured inhenrys = H = V·s/A) C - thecapacitance of the capacitor (measured infarads = F =C/V = A·s/V)
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.
[edit] See also
Resonant frequencyElectronic oscillatorLC circuitBandwidthBandpass filterQuality factorOliver HeavisideRC circuit
[edit] External links
a treatment that starts with the mechanical analogyAn interactive simulation on series RCL circuit
Retrieved from "http://en.wikipedia.org/wiki/RLC_circuit"