#07 Distinguish pointwise power, cycle-average power and energy transfer under the passive sign convention

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Multiply voltage and current at the same instant

   

   Start with single-phase sinusoidal steady state and the passive sign convention.

   Instantaneous absorbed power is the product at the same instant:

   $\displaystyle p\left(t\right)=v\left(t\right)i\left(t\right)$

   For nonzero sinusoidal voltage and current, their product varies at twice the waveform frequency.

   Inspect the instantaneous waveform first; then average it over a full period.

   Narration transcript

   Power in AC circuits is not a fixed number at every instant. At each moment, instantaneous power is simply voltage times current. So if the voltage and current waveforms move, power moves with them. That is why we first study the time-domain power waveform before jumping to average power.

2. 2. Interpret power in an ideal resistor

   

   For an ideal positive resistor, voltage and current are in phase under passive references.

   A positive resistor never has negative instantaneous power:

   $\displaystyle p_{R}\left(t\right)=R i\left(t\right)^{2}\ge 0$

   It dissipates electrical energy as heat; it does not return stored field energy.

   With nonzero excitation, the resistor's cycle-average power is positive:

   $\displaystyle P_{R}=R I_{\mathrm{rms}}^{2}$

   Narration transcript

   For a pure resistor, voltage and current are in phase. Their product never stays negative for long, and the power waveform remains above zero on average. The source keeps delivering energy, and the resistor keeps converting it into heat. This is the cleanest case of continuous power absorption.

3. 3. Follow reversible energy storage

   

   For ideal lossless L or C in sinusoidal steady state, voltage and current differ by 90°.

   The instantaneous power alternates in sign; its full-cycle average is:

   $\displaystyle P_{\mathrm{LC}}=0$

   Zero average does not mean zero instantaneous power or zero stored energy.

   Positive instantaneous power increases stored energy; negative power returns it to the circuit.

   Narration transcript

   For a pure inductor or capacitor, voltage and current are ninety degrees apart. Now the instantaneous power swings positive and negative around zero. That means energy is not only delivered to the element. Part of the time, the element stores energy, and part of the time it gives that energy back to the source.

4. 4. Average over a complete cycle

   

   Average power is net cycle energy divided by the period T:

   $\displaystyle P=\left(\frac{1}{T}\right)\int _{0}^{T} p\left(t\right) \mathrm{d}t$

   Equal positive and negative signed areas give zero net cycle energy:

   $\displaystyle \Delta W=0$

   For an absorbing passive load, a nonzero in-phase current component gives positive average power.

   At fixed RMS amplitudes, the sinusoidal average is:

   $\displaystyle P=V_{\mathrm{rms}} I_{\mathrm{rms}} \cos \left(\varphi \right)$

   Narration transcript

   Average power asks for the net energy transfer over one full cycle. If the positive and negative parts cancel completely, the average is zero. If a real in-phase component survives, the average stays positive. So phase difference decides how much of the instantaneous power becomes useful average power.

5. 5. Choose the passive sign convention

   

   Fix the passive references — reference current enters the voltage reference's positive terminal.

   With those references, positive instantaneous power means absorption.

   Negative instantaneous power means delivery to the rest of the circuit.

   Reversing only the current reference changes the absorbed-power expression to:

   $\displaystyle p\left(t\right)=-v\left(t\right)i_{\mathrm{rev}}\left(t\right)$

   Narration transcript

   The passive sign convention tells us how to read the sign. If current enters the terminal marked positive for voltage, positive power means the element absorbs power. Negative power then means the element is delivering power back. Without that sign rule, the waveform is easy to misread.

6. 6. Compare instantaneous and average power

   

   The same instantaneous product rule always applies:

   $\displaystyle p\left(t\right)=v\left(t\right)i\left(t\right)$

   Use peak amplitudes and a lag angle φ — current is:

   $\displaystyle i\left(t\right)=I_{m} \cos \left(\omega t-\varphi \right)$

   For a positive resistor, the equivalent voltage expression is:

   $\displaystyle P_{R}=\frac{V_{\mathrm{rms}}^{2}}{R}$

   An ideal reactive element in periodic steady state returns the energy it absorbs over a cycle.

   Narration transcript

   The core formula is still simple: p of t equals v of t times i of t. What changes from case to case is the waveform shape and its average value. Resistive parts create positive average power. Purely reactive parts create zero average power over a full cycle.

7. 7. Connect to RMS, complex power and power factor

   

   Instantaneous power is energy-transfer rate at the present instant, measured in watts.

   Average power is also measured in watts; net cycle energy in joules is:

   $\displaystyle \Delta W=P T$

   Next: root-mean-square (RMS) values, complex power and power factor.

   Narration transcript

   Instantaneous power tells us what is happening right now. Average power tells us what remains after one full cycle. That distinction is the doorway to RMS values, complex power, and power factor in the next lesson.

Source video: Circuit Theory-2 #07 Instantaneous and Average Power (2:32)