# Combining Phasors (Wave Forms)

#### mrdannydandan

The instantaneous values of two alternating voltages are given by:

v1= 150sin(ωt+ π/3) Volt
v2= 90sin(ωt- π/6) Volts

Derive a sinusoidal expression-resultant v1+ v2 in form Rsin(ωt+ α)

Combine the phasors (Vectors) of each wave form?

#### skeeter

MHF Helper
$150\sin \left(\omega t + \dfrac{\pi}{3}\right) = 150\bigg[\sin(\omega t)\cos\left(\dfrac{\pi}{3}\right) + \cos(\omega t)\sin\left(\dfrac{\pi}{3}\right)\bigg] = 75\sin(\omega t)+75\sqrt{3}\cos(\omega t)$

$90\sin \left(\omega t - \dfrac{\pi}{6}\right) = 90\bigg[\sin(\omega t)\cos\left(\dfrac{\pi}{6}\right) - \cos(\omega t)\sin\left(\dfrac{\pi}{6}\right)\bigg] = 45\sqrt{3}\sin(\omega t)-45\cos(\omega t)$

$V_1+V_2 = (75+45\sqrt{3})\sin(\omega t) + (75\sqrt{3}-45)\cos(\omega t)$

$A\sin(\omega t) + B\cos(\omega t) = R\sin(\omega t + \alpha)$

$R = \sqrt{(75+45\sqrt{3})^2 + (75\sqrt{3}-45)^2} = 30\sqrt{34}$

$\alpha = \arctan\left(\dfrac{5\sqrt{3}-3}{5+3\sqrt{3}}\right) = \arctan(30-17\sqrt{3}) \approx 0.5068$

$V_1+V_2 = 30\sqrt{34} \sin(\omega t + \alpha)$

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#### mrdannydandan

Thank you very much skeeter! I was forgetting to square root the resultant! Kicking myself now - it's all come clear now many thanks