Phasor Sum of Two Series Impedances

Two impedances in series do not add. Their magnitudes do not add, at any rate, and the moment a circuit contains anything reactive that stops being a pedantic distinction and becomes the difference between a right answer and a wrong one. An impedance is a vector in the complex plane — a length and a direction — and adding two of them is adding two vectors, which is the law of cosines with the angle between them.

The cleanest way to see it is to work an example both ways. Take $Z_1 = 3 + j4$, a magnitude of 5 at an angle of 53.13°, in series with $Z_2 = 8 + j6$, a magnitude of 10 at 36.87°. Add them in rectangular form and it is trivial: $11 + j10$, whose magnitude is $\sqrt{221} = 14.87$ Ω. Add the magnitudes and you get 15, which is wrong by less than one per cent here only because the two angles are close. Now the phasor form: the angle between them is 53.13 − 36.87 = 16.26°, whose cosine is exactly 0.96, and $25 + 100 + 2(5)(10)(0.96) = 221$. The same 221, from the same triangle, without ever writing a complex number down.

The two limits are what to remember. At Δθ = 0 the cosine is one and the phasor sum collapses to plain addition — which is why series resistances simply add, and why two identical motor feeders in series add arithmetically. At Δθ = 180° the cosine is −1 and it becomes $|Z_1 - Z_2|$: an inductive reactance and an equal capacitive reactance in series leave nothing at all, which is series resonance and is why a resonant circuit can draw a frightening current from a small source. Everything in between is partial cancellation, and the further apart the angles, the more of each phasor is spent pulling against the other rather than adding to it.

Two cautions. First, only the DIFFERENCE of the angles enters, so this page cannot tell you the angle of the result — rotate both phasors together and the sum's length is unchanged while its direction is not. When you need the resultant angle, work in rectangular form: $(R_1 + R_2) + j(X_1 + X_2)$, and take the arctangent at the end. Second, cosine is even, so the sign of Δθ is invisible here: a capacitive element 40° behind its neighbour gives the same magnitude as an inductive one 40° ahead. Which it is has to come from knowing what the elements are, and it matters enormously to everything downstream — to the power factor, to whether correction capacitors help or hurt, and to whether the combination is above or below resonance.