# Addition and Subtraction of Nonsinusoidal Waveforms

The Fourier series expression for the waveform resulting from the addition or subtraction of two nonsinusoidal waveforms can be found using phasor algebra if the terms having the same frequency are considered separately.
For example, the sum of the following two nonsinusoidal waveforms is found using this method:
$$v_1 = 30 + 20 \sin 20t + 5 sin(60t + 30^\circ)$$
$$v_2 = 60 + 30 \sin 20t + 20 \sin 40t + 10 \cos 60t$$
1. dc terms:
$$V_{T_{0}} = 30 V + 60 V = 90 V$$
2. $\omega = 20$:
$$V_{T_{1}} = 20 \sin 20t + 30 \sin 20t = 50 V \sin 20t$$
3. $\omega = 40$:
$$V_{T_{2}}= 20 \sin 40t$$
4. $\omega = 60$:
$$5 \sin(60t + 30^\circ) = (0.707)(5) V \angle 30^\circ = 3.54 V \angle 30^\circ$$ $$10 \cos 60t = 10 sin(60t + 90^\circ) ⇒ (0.707)(10) V \angle 90^\circ= 7.07 V \angle 90^\circ$$ $$V_{T_{3}} = 3.54 V \angle 30^\circ + 7.07 V \angle 90^\circ$$ $$= 3.07 V + j 1.77 V + j 7.07 V = 3.07 V + j 8.84 V$$ $$V_{T_{3}} = 9.36 V \angle 70.85^\circ$$
and
$$v_{T_{3}} = 13.24 \sin(60t + 70.85^\circ)$$
with
$$v_T = v_1 + v_2 = 90 + 50 \sin 20t + 20 \sin 40t + 13.24 \sin(60t + 70.85^\circ)$$