A circuit with R=.002, L=12.5, and C=50 can carry a maximum...

A circuit with $R=.002, L=12.5$, and $C=50$ can carry a maximum current of 250. Ignoring the effect of the transient, what is the maximum allowable amplitude $\alpha$ of an applied periodic current $F(t)=\alpha \cos \gamma t$ at frequency $\gamma=(a) .04$ ? (b) .05? (c) .1?

Key Concepts

RLC Circuit Analysis

This concept involves modeling circuits that contain a resistor (R), inductor (L), and capacitor (C) as systems governed by second?order differential equations. In these circuits, the interplay between resistance, inductance, and capacitance defines the system's behavior, such as natural frequencies, damping, and oscillatory responses. Understanding the RLC circuit is essential for predicting how it responds to external forcing functions and for assessing its limits, such as maximum output currents.

Steady-State Response to Forcing Functions

In many physical systems, after initial transients decay, the system reaches a steady state where the response is periodic and directly related to the periodic forcing function. In the context of an RLC circuit subjected to a harmonic forcing term, the steady-state analysis focuses on the long?term behavior without the influence of the transient response. This enables a clear analysis of how the amplitude of the input current translates into the amplitude of the circuit’s current at different driving frequencies.

Frequency Response and Impedance

The frequency response of an RLC circuit describes how the magnitude and phase of its output change with the frequency of the applied input. This is critically influenced by the circuit’s impedance, which is frequency dependent. By calculating the impedance at a given frequency, one can determine the gain of the system (the ratio of the output amplitude to the input amplitude). This concept is vital for ensuring that, for any driving frequency, the maximum input amplitude is set so that the resulting current does not exceed the circuit’s limitations.

Transcript

00:01 31 .18.

00:02 So we have an oscillating lc circuit and the amplitudes of current and voltage are 7 .5 mliamps and 250 millibolds respectively and the capacitor has a value of 220 nanophares.

00:16 So we want to find some things, one of which is the period of the oscillation, the maximum energies stored in the inductor and the capacitor, the maximum rate of change of current, and the maximum rate of change of current, and the maximum rate of change of the inductor's energy.

00:40 So we know from the sort of general om's law for a capacitor that the amplitude of the voltage equals the amplitude of the current times the reactants, in this case of the capacitor.

01:00 So omega is going to be i over cb.

01:11 Now the period is two pi over omega and so then just evaluating that we find the period is 46 .1 microseconds.

01:50 So now the maximum energy the capacitor stores, and its electric field is one half cv squared.

02:03 And so if this is the amplitude, then that's the maximum value it will take.

02:10 And so this is 6 .88 times 10 to the negative 9th joules.

02:30 So now the maximum energy that the inductor stores, you know, in principle, you know, is l or i squared over over two which is a problem because we don't know what l is we know what c is and we know the frequency so we could figure it out but also we don't have to because of the conservation of energy and we know that it goes back and forth between all being in the capacitor and all being in the inductor that its maximum has to be the same and in fact this is the total energy in the system so let's start a new one.

03:30 So now we want to find the idt, or the maximum bidt.

03:38 So the voltage across the inductor is equal to l, the amplitude of that voltage, going to be didt maximum...

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