Instantaneous Power
The formula for power is: P=V⋅I, Power= Voltage*Current. We call that, instantaneous power. Even when V and I are changing in time, P=V⋅I applies for each instant. It matters not if the V or I changes are sinusoidal, what their frequency is, or even if they are aperiodic. P=V⋅I, is something you can use always. There are no exceptions that I can think of.
You don’t need Ohm’s Law to use V⋅I, because V refers to the voltage at one-point relative to ground, and I refers to the current flowing past that point. But suppose you have a resistor R, and you want to calculate the power dissipated by it. Using Ohm’s Law, the voltage drop across R is V=I⋅R. In this case, V is the voltage drop between two points in a circuit. The power dissipated is (V)⋅I=(I⋅R)⋅I=I2⋅R. If you have a capacitance C instead of R, then you need to use the differential equation I=C dV/dt. If you have an inductance L, then you need to use the differential equation V=L dI/dt.
Sinusoidal Power
Let’s jump directly to sinusoidal waveforms such as we have in AC circuits. Let V and I be sinusoids of the form cos(ωt+ϕ). What happens to the instantaneous power? Well, P=V⋅I still applies, but now P varies with time, and how it varies depends on the phase angle of V relative to the phase angle of I. If the relative phase is zero, we say that voltage and current are “in phase”. That is depicted below.
Note that the frequency of V⋅I is twice the frequency of V or I, and that the value of V⋅I is zero twice in the cycle, and that the average of V⋅I is not zero.
Suppose V and I are not in phase? In this picture, I (dotted line) is shifted
by an angle 90 degrees relative to V (blue). V⋅I is shown as red-green shaded areas; red means power flowing right (+) and green means minus power flowing left (-). Note that the areas of red and green are equal in this case, so they cancel. Power flows just as much to the left as to the right, so the net energy over the entire cycle is zero. What should we call this peculiar state? Referring to the entire cycle, we call it pure imaginary power (also called reactive power, also called VARs). Don’t be fooled, even in this case instantaneous V⋅I remains real.
Below, I is shifted only 45 degrees relative to V. We see that V⋅I is predominantly red, but green for part of the cycle.
We can now generalize to all possible phase shifts. We discuss averages for the entire cycle.
- Phase 0 is pure real power, phase 180 is pure real power flowing in the opposite direction.
- Phase 90 is pure imaginary power, phase 270 is pure imaginary power flowing in the opposite direction.
- Any other phase is a linear combination of nonzero real power and nonzero imaginary power.
Complex Power
In 1893, Charles Proteus Steinmetz published a paper that explained the great advantages of complex AC analysis. The other electrical geniuses of the day (including Nikola Tesla) were all using tedious integral calculus and expressions using definite integrals of cos(ωt+ϕ), or the Euler form e−jωt+ϕ. Steinmetz left them in his dust because he recognized the combination of fortuitous luck with coincidences of mathematics that were relatively obscure at the time. Namely:
- Manufacturers were already making AC generators in 1893 that generated sinusoidal voltages.
- The sinusoid is the only mathematical function that has the property that differentiation and integration return a function of the same form but shifted 90 degrees.
Steinmetz found that by restricting his equations to an integer number of whole cycles, and by replacing real quantities by complex ones, we obtain quasi-static equations that are hugely simplified relative to integral calculus. Simpler how?
- DC P=V⋅I becomes AC S̅=V̅⋅I̅, where S̅ is complex power, usually written as S̅=P+jQ where P is real power and Q is imaginary power. (Actually, it should be S̅=V̅⋅I̅* but, I’m ignoring the sign of Q.)
- DC Ohm’s Law V=I⋅R becomes AC V̅=I̅⋅Z̅ where Z̅ is the complex impedance. Z̅ includes resistance, inductance, and capacitance.
- Differential equation terms in DC (like C dV/dt and L dI/dt ) become algebraic in AC.
- Series, parallel, Kirchhoff’s Laws, mesh analysis, matrix analysis: essentially all the tools and methods of DC circuit analysis become directly applicable to AC if we just use complex and whole cycles.
Next, think once again of the pictures from above with the red-green areas depicting V*I. Instead of time-varying instantaneous V*I, we will focus on just the whole cycle averages, P (as measured by an AC Watt meter) and Q (as measured by an AC VARs meter). P and Q will be constant in time, but they will vary as we change the phase shift. The meter readings versus ϕ are shown in the table.
