In a series oscillating R L C circuit, R=16.0 Ω, C= 31.2 μF, L=9.20 mH, and ℰm=ℰm sinωd t with 𝒢

step 1 31 .60. So we have an RLC circuit that's being driven here. These are the values of all the thin

In a series oscillating R L C circuit, R=16.0 Ω, C= 31.2 μF,...

In a series oscillating $R L C$ circuit, $R=16.0 \Omega, C=$ $31.2 \mu \mathrm{F}, L=9.20 \mathrm{mH},$ and $\mathscr{E}_{m}=\mathscr{E}_{m} \sin \omega_{d} t$ with $\mathscr{G}_{m}=45.0 \mathrm{V}$ and $\omega_{d}=3000$ rad/s. For time $t=0.442 \mathrm{ms}$ find $(\mathrm{a})$ the rate $P_{g}$ at which energy is being supplied by the generator, (b) the rate $P_{C}$ at which the energy in the capacitor is changing, (c) the rate $P_{L}$ at which the energy in the inductor is changing, and (d) the rate $P_{R}$ at which energy is being dissipated in the resistor. (e) Is the sum of $P_{C}, P_{L},$ and $P_{R}$ greater than, less than, or equal to $P_{g} ?$

Key Concepts

Energy Storage in Capacitors

Capacitors store energy in the form of an electric field. The energy stored in a capacitor is given by the expression U = 1/2 C V^2. The rate of change of this stored energy, derived from differentiating U with respect to time, allows analysis of how quickly energy is being accumulated or released in the capacitor over time.

Series RLC Circuit

A series RLC circuit consists of a resistor, inductor, and capacitor connected in a single loop. It exhibits oscillatory behavior when energy is exchanged between the inductor and capacitor, while the resistor dissipates energy as heat. This overall behavior is governed by differential equations describing the voltage and current relationships within the circuit.

Instantaneous Power Calculation in AC Circuits

In AC circuits, instantaneous power is determined using the product of the instantaneous voltage and current (P = V × I). This calculation applies both to the power supplied by a generator and the power consumed or stored by individual circuit elements, making it crucial for analyzing transient or dynamic energy transfer in oscillating circuits.

Energy Storage in Inductors

Inductors store energy in a magnetic field created by the current flowing through them. The stored energy is expressed as U = 1/2 L I^2. Similar to capacitors, the time rate of change of energy in the inductor, found by differentiating U with respect to time, is essential in understanding the energy dynamics within oscillatory circuits.

Energy Dissipation in Resistors

Resistors convert electrical energy into thermal energy through the process of Joule heating. The power dissipated in a resistor is given by P = I^2 R. This concept is important in analyzing energy loss within circuits and is integral to energy conservation considerations in oscillating systems.

Energy Conservation in Oscillating Circuits

Energy conservation in an RLC circuit involves accounting for the power supplied by the generator, the rate of energy storage changes in the capacitor and inductor, and the power lost in the resistor. The instantaneous energy balance shows that the sum of the rates of change of energy in the reactive elements (capacitor and inductor) plus the power dissipated in the resistor equals the power delivered by the generator.

Transcript

0:00 31 .60.

00:02 So we have an rlc circuit that's being driven here.

00:06 These are the values of all the things and at 0 .442 milliseconds we want to determine what the power being delivered by the our power source here is, or the generator or whatever, and then we want to find the power in the capacitor, the inductor and the resistor and then determine whether the sum of these is equal less than and or greater to the power that the generator is delivering.

00:36 So first what we want to do is determine the current in the circuit.

00:52 It's going to be the same for all of the elements because it's a serious circuit.

00:59 So the impedance is going to be the usual formula here.

01:06 R squared plus omega d l minus 1 over omega dc squared squared.

01:24 Close up our square root, try not to run out of space vertically.

01:36 All right.

01:42 So this works out to 1 .93 amps, and the phase is the arc tangent of the inductive reactants minus the capacitive reactants divided by the resistance.

02:14 This would be 46 .5 degrees.

02:22 And so the current is a function of time.

02:27 Will be i sine omega d t minus the phase and we saw we just found the amplitude of the current and the phase so now the power is applied by the generator is just the current at that time times its voltage at that time is going to be i times e sub m times a sign of omega t minus phi times a sign of omega t minus phi times a sign of omega t.

03:34 So we know what all of these are.

03:35 Omega's are 3 ,000 radiance per second, t .442 milliseconds, 5, 46 .5 degrees, etc.

03:46 So we find 41 .4 watts.

04:03 So now for the capacitor, the voltage across it is b, sub c, sine of omega d t minus phi, minus the pi over two radiance.

04:30 We get from a capacitor negative ec cosine omega t minus phi and the amplitude of this ec is just i over omega c so now the rate at which we're adding energy to the electric field in the capacitor is the rate at which the energy is changing this is the energy in a capacitor.

05:27 Now the derivative of the charge is the current...

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