Driven LCR Oscillator Animation

Introduction

The Driven LCR oscillator (LCR) is an exact analog of the Forced Harmonic Oscillator shown under the Mechanics chapter.  Its behavior goes a long way toward understanding the electrodynamics of the real world.

Calculations:

Figure 1: Circuit diagram showing the Signal Generator, Vs, Capacitor C, Resistance, R, and Inductor, L.    R is a light bulb which changes brightness as current flows through it.

 

 

The picture above shows the important parameters of the LCR. They include a Capacitor, C with variable capacitance, a damping Resistor (shown here as

Spring

Constant=k
 
a light bulb with a variable resistance R ohms) and an Inductance, L, which has an inductance of 1 Henry, at the bottom.  The function of the capacitor is to trade off energy between itself and the magnetic field of the inductance and the function of the light bulb is to realistically absorb energy from the circuit.  Again, the layout of this electrical circuit is exactly analogous to the layout of the Forced Harmonic Oscillator under the Mechanical chapter.

 

The equation that describes the charge of the capacitor is:

                                                                                        (1)

where Vs is the voltage provided by the signal generator.

To solve equation 1 we make the substitution:

                                                                                                      (2)

where the real part of q will be the charge on the capacitor, i is the square root of -1, Real[] denotes the real part of the resulting value, Q is the peak capacitor charge, t is time.  Using equation 2 in equation 1 we have very easily:

                                                                                                 (3)

For notation convenience we make the definitions:                                                                    

                                                                                                                         (4)

                                                                                                                                 (5)

Using equations 4 and 5 in equation 3 we have

or

                                        (6)

from which using equation 2 we have

                                                 (7)

Separating Q into real and imaginary parts we get:

                                                                                 (8)

Note that, if Vs<0, the imaginary part of Q is always negative and the real part is negative when w>w0  and positive when w<w0.

Computing the real part of equation 7 we obtain:

                                               (9)

We can obtain the phase of q using the following equation:

                                                   (10)

where the function atan2 evaluates the arc tangent on the interval (0 -> -p) without the usual discontinuity of  the arctangent.

Dielectric Slab Animation

 

Text Box:

 

 

 

 

 

 


Shown above is the dielectric slab in the capacitor gap.  The color and depth of the charge in the two plates indicate both the sign and the value of their charge.  The rotation angle of dipoles, which are shown as dumbbells with a red end and a green end, is animated. When the imposed electric field is large, they are aligned anti-parallel to the electric field.  This is the effect of a dielectric slab in a capacitor: Since the slab’s dipoles align opposite to the imposed electric field they cancel a large fraction of that field and therefore a larger charge on the red and green plates is required to achieve the same voltage across the capacitor.  When the imposed electric field is small or zero, the orientation of the dipoles becomes random as can be seen from the animation  

Summary

The meaning of equation 10 is that the capacitor charge, q, will follow the voltage applied by the signal generator and will thus have a cosine periodic behavior with the phase lag given in equation 10.