ADVANCED ELECTRICAL CIRCUIT: WEEK 4: Frequency Domain Analysis of Simple RLC Circuits

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 $s = σ + i ω$ , 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(s):

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

And rearranging, we have at

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

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 ω_o

$Y(s) = { I(s) \over V(s) } = \frac{s}{ L \left ( s^2 + 2 \alpha s + {\omega_0}^2 \right ) }$

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 $|s| \rightarrow \infty$

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

$s = - \alpha \pm \sqrt{\alpha^2 - {\omega_0}^2}$

Notice that the poles of Y(s) are identical to the roots $λ 1$ and $λ 2$ of the characteristic polynomial.

Sinusoidal steady state

If we now let $s = i ω$ ....

Considering 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

$| I( i \omega ) | = | Y(i \omega) | | V(i \omega) |.\,$

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 $i m a g (ω) = 1$ . This is known as the resonance frequency. Solving for this value, we find:

$\omega_0 = \frac{1}{\sqrt{L C}}.$

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 = V·s/A)

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

The complex admittance of this circuit is given by adding up the admittances of the components:

${1\over Z}={1\over Z_L}+{1\over Z_C}+{1\over Z_R}={1\over{j\omega L}}+{j\omega C}+{1\over R}$

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 $I={V\over Z}$ . 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 resonance frequency $\omega_0={1\over\sqrt{LC}}$ .

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.

ADVANCED ELECTRICAL CIRCUIT

Blog Archive

Sunday, February 28, 2010

WEEK 4: Frequency Domain Analysis of Simple RLC Circuits

Complex admittance

Poles and zeros

Sinusoidal steady state

Parallel RLC circuit

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