Скачать или смотреть Power in AC circuit elements - full calculus derivations with phasors [AC circuit physics]

Learn why only resistors actually dissipate power in an AC circuit (with all the calculus!) as we derive the instantaneous and average power for resistors, inductors and capacitors.

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00:00 Introduction and RLC circuit animation of phasor diagram and voltage waveforms.

00:45 Instantaneous power for a resistor in an AC circuit: we start by viewing the animated phasor diagram for the resistor and the corresponding waveforms for the current and voltage functions. Remember, current and voltage are in phase for the resistor! Next, we compute instantaneous power by taking the product of the current and voltage functions, so p_R(t)=i(t)v_R(t). This yields an instantaneous AC power of I^2*Rcos^2(omega*t).

02:32 Average power for a resistor in an AC circuit: now we take the average value integral of the instantaneous power, and I usually do this over the interval 2pi/omega - note that this is actually two periods of the squared cosine function, but any integer number of periods will do. When we compute the average power integral, we arrive at p_R-bar = 1/2*I^2R, which can also be written 1/2*V^2/R or 1/2*IV, where V is the voltage amplitude across the resistor and I is the current amplitude through the resistor. Note that by introducing the RMS (root mean square) voltage and current, we can rephrase this average power in a beautiful way: p_R-bar = I_RMS^2*R - the analogous formula to what we found in DC circuits! Similarly, we can say p_R-bar=V_RMS^2/R and p_R-bar=I_RMS*V_RMS, so the RMS current and voltage allow us to write down formulas for average power that are all analogous to the DC circuit power formulas for a resistor!

08:27 Instantaneous power for an inductor: to calculate the instantaneous power function for an inductor, we simply multiply the current and voltage functions again, but this time we end up with a product of two cosines with different phase angles: p_L(t)=IX_L^2*cos(omega*t)cos(omega*t+pi/2), where X_L is the inductive reactance. We can simplify by using the fact that cos(omega*t+pi/2) is the same thing as -sin(omega*t), then we use the identity sin(2x)=2sin(x)cos(x) to rewrite the inductor power function as -1/2sin(2*omega*t). Note that this also has a period half as great as the original sinusoidal functions!

11:20 Average power for an inductor: because we're integrating a sinusoidal function over an integer number of periods, the average power for an inductor is zero in an AC circuit. The inductor is periodically storing energy in its magnetic field, but then it gives the energy back - releasing the magnetic field energy back into the circuit periodically!

12:30 Instantaneous power for a capacitor: we calculate instantaneous power for the capacitor by taking the product of current and voltage functions, and this time we end up with p_C(t)=I^2*X_C*cos(omega*t)*cos(omega*t-pi/2), where X_C is the capacitive reactance. We use the fact that our phase shifted cosine is equivalent to a simple sine function, then exploit the double angle identity one more time to arrive at 1/2I^2*X_C*sin(2*omega*t).

15:14 Average power for a capacitor: again we find ourselves integrating a sinusoidal function over an integer number of periods, so the average power integral for the capacitor vanishes. This time, we say the capacitor is storing energy in the electric field between its plates, then periodically releasing that energy back into the circuit.

15:52 Review of all results for power with resistors, inductors and capacitors in the AC circuit.

This video is Part 6 of my AC circuits mini-series:

Part 1: the current phasor and the voltage phasor for a resistor:
👉    • The current phasor and resistor voltage ph...  

Part 2: the voltage phasor for an inductor:
👉    • The inductor voltage phasor and inductive ...  

Part 3: the voltage phasor for a capacitor:
👉    • The capacitor voltage phasor and capacitiv...  

Part 4: phasor analysis of the RLC circuit:
👉    • RLC series circuit phasor analysis, impeda...  

Part 5: RMS values for current and voltage - full calculus derivations.
👉    • RMS values for current and voltage - full ...  

Part 6: Power calculations for inductors, capacitors and resistors.
👉    • Power in AC circuit elements - full calcul...  

Part 7: Power and the power factor in the RLC series circuit.
👉    • Power and the power factor in the series R...  

Part 8: Resonance in the RLC circuit + RLC power calculations. [TBA]

#physics #ACcircuits #ACpower