Resource Lesson
LC Circuit
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The LC circuits we will be investigating are those involving a DC power supply. Let's begin with a simple circuit containing a DC power supply (battery), two switches, a resistor, a capacitor, and an inductor.
When only switch A is closed (both B switches are open), only the left circuit containing the resistor, battery, and capacitor is connected and the capacitor becomes charged. Once the capacitor is fully charged and has attained the voltage the battery, switch A is opened and both switch B's are closed. At that time only the capacitor and the inductor are components in the active circuit.
Initially the charges on the positive capacitor plate (we are discussing conventional current) will begin circulating counter-clockwise. As the current from the capacitor dies out, the inductor reverses its emf to keep the charges flowing until the bottom plate of the capacitor becomes positively charged and the top plate holds all of the negative charge. Then the process reverses and the current flows clockwise until the capacitor's top plate is once again positively charged and its bottom plate negatively charged. This process of "filling and emptying" the capacitor's plates, and its subsequent electric field, continues at a frequency which we will define a little later in the lesson.
Note that in the animated gif provided by
Chetvorno (Own work) [CC0], via Wikimedia Commons
, the capacitor and inductor are presented on opposite sides compared to the introductory circuit presented at the top of the page.
Analogies to vibrating spring-mass systems
In a vibrating spring-mass system, the energy is shared between the elastic potential energy in the spring, U
_{s}
=
, and the kinetic energy, KE =
of the vibrating mass.
At any intermediate position during a vibration, some of the energy is kinetic and some is potential elastic, but the total amount of energy remains constant.
In an oscillating LC circuit, the energy is shared between the amount stored in the electric field of the capacitor and the amount storied in the magnetic field of the inductor.
Here are the analogies that equate the behavior of an oscillating spring-mass system and an resonating LC circuit.
mass
become
inductance, L
velocity
becomes
current, i
spring constant
becomes
C
^{-1}
displacement from equilibrium
becomes
charge, q
maximum displacement (amplitude)
becomes
Q
_{o}
(the maximum charge on the capacitor)
Substituting our new variables into our equation for the energy of a vibrating mass-spring system we get,
But what is Q
_{max}
or Q
_{o}
? This value comes from the functional equation for a capacitor: Q = CV where C is the capacitance and V is the voltage of the charging battery. When there is no charge on the capacitor (q = 0) we can calculate the maximum current.
But what about the frequency of the circuit mentioned earlier? What would be its expression? Once again, we will turn to our analogies.
where the units of "LC" are sec
^{2}
. The resonant frequency of the LC circuit is merely the reciprocal of its period,
In this presentation, the resistance in the circuit is considered minimal. That is, there are no energy losses to heat. In real circuits, the oscillations would eventually decay and die out.
Using the resonant frequency
So now that we have an expression for the frequency of an oscillating LC circuit, let's examine the position, velocity, and acceleration functions of our vibrating mass-spring system and the analogies for an LC circuit. In the top diagrams of a vibrating mass-spring, we started at a position of maximum compression. Let's "define" that amplitude to be negative and the amplitude at maximum extension to be positive. This would give us a negative cosine function for the instantaneous position of the vibrating mass.
spring-mass
LC circuit
position/charge
To obtain an equation for the instantaneous velocity of the vibrating mass, we take the derivative with respect to time of the mass' position function.
spring-mass
LC circuit
velocity/current
An one more time, to obtain an equation for the instantaneous acceleration of the vibrating mass, we would take the derivative with respect to time of the mass' velocity function.
spring-mass
LC circuit
acceleration/(di/dt)
Recall that the emf induced in an inductor is calculated according to the equation
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