AC, phasors and power

Choose AC and give an ω. Elements keep their own units — henries and farads, not reactances — and the solver does the conversion.

A 10 V source driving a 50 Ω resistor and a 50 mH inductor in series, at 1000 rad/s. We name the three e, r and l, and write the inductor as its inductance:

e,1,0,10
r,1,2,50
l,2,0,0.05

Set Analysis to AC — alternating current, put 1000 in the ω — angular frequency box, and run. At that frequency the inductor is 50 Ω of reactance, so the loop's impedance is 50 + 50j, and the answers say so:

You may also give reactances directly, as ohms, if that is how the problem is stated — a resistor-valued l is not a thing, so put the reactance in as an impedance and leave ω out of it.

Rectangular or polar is a display choice: tick Show AC answers as polar phasors in Settings and 0.1 - 0.1j reads 0.1414∠-45°. Nothing about the answer changes. The aa mini-tool does the same conversion for one value you type.

The four power answers

In AC an element reports more than in DC:

The three are P, Q and S as every book writes them, whichever way the RMS setting is: RMS changes their values, not their names. A source's card shows the delivered forms, -pe, -qe and -se, and the power factor of the power it delivers; the answers pe, qe and se are the consumed values all the same.

The same shape with a 30 Ω resistor and a 40 mH inductor, at 1000 rad/s again, so that the impedance is 30 + 40j, a 3-4-5 triangle:

e,1,0,10
r,1,2,30
l,2,0,0.04

Run in AC at ω = 1000, the source's card reads

the negative of se, since a source's card shows what it delivers. So P = 0.6 W, Q = 0.8 var, |S| = 1.0 VA, and the card's last row gives the power factor of that delivered power, 0.6 lagging. The card's next two rows are those two parts, -pe = 0.6 W and -qe = 0.8 var, and abs(se) in Evaluate is the apparent power.

An inductor or a capacitor reports only its complex power. That is not an omission: a pure reactance consumes no average power, and a zero printed every time would be noise; its reactive power is im(sl) in Evaluate.

The pf mini-tool takes one value, and which of two things it is decides what comes back. Given a complex power such as se — or any complex value, an impedance, an expression with symbols in it — it returns the power factor of that value as given, |P|/|S|, with the word the value carries when it is a number and symbolic when it is not. Given the name of an element of the AC solve, e or r1, it returns the power factor with its lead/lag sense, and it works only when that element's voltage and current came out as numbers.

The two forms read different powers, and that is what makes the word right. A variable such as se, sj or sr1 is read as it stands: the complex power consumed, which is what every element stores, source or load — and its word is that power's own, so se at a source says the opposite word to e, the consumed power being the opposite of the delivered one. A name is read by the element's kind, as the calculator read it: an impedance (r, l, c) on the power it consumes, so an inductive load reads lagging; a source (e, j) on the power it delivers, the current negated first, so a source reads the circuit it sees, and a source feeding an inductive load says lagging like the load. The value is the same either way; only the word depends on it, and a source read on se says the opposite word. Under a name's reading the tool says which power in words — the power delivered by source e, the power consumed by impedance r1; under a value's it says nothing, the value being what you gave it.

Three-phase

There is no three-phase mode, and none is needed. A three-phase circuit is an AC circuit with three sources whose values carry the phase — (120∠0°), (120∠-120°), (120∠120°) — and Y or Δ is just how you wire the nodes. Balanced and unbalanced are the same description with different values.

Line and phase quantities are then read off the elements directly: there is no separate answer for them, because each is some element's own current or voltage.