Given Currents are-
I 1 = 141.42 s i n ( ω t + π 2 ) = 141.42 c o s w t I_1=141.42sin(\omega t+\dfrac{\pi}{2})=141.42coswt I 1 = 141.42 s in ( ω t + 2 π ) = 141.42 cos wt
I 2 = − 70.71 c o s w t I_2=-70.71coswt I 2 = − 70.71 cos wt
I 3 = 50 e j π 3 = 50 ( c o s π 3 + j s i n π 3 ) = 50 ( 1 + 3 j 2 ) = 25 + 25 3 j I_3=50e^{j\frac{\pi}{3}}=50(cos\dfrac{\pi}{3}+jsin\dfrac{\pi}{3})=50(\dfrac{1+\sqrt{3}j}{2})=25+25\sqrt{3}j I 3 = 50 e j 3 π = 50 ( cos 3 π + j s in 3 π ) = 50 ( 2 1 + 3 j ) = 25 + 25 3 j
I 4 = 100 < − 2 π 3 = − 50 − 86.6 j I_4=100<\dfrac{-2\pi}{3}\\=-50-86.6j I 4 = 100 < 3 − 2 π = − 50 − 86.6 j ( convert the polar to rectangular form)
I 5 = 50 ( c o s π 3 + j s i n π 3 ) = 50 ( 1 + 3 j 2 ) = 25 + 25 3 j I_5=50(cos\dfrac{\pi}{3}+jsin\dfrac{\pi}{3})=50(\dfrac{1+\sqrt{3}j}{2})=25+25\sqrt{3}j I 5 = 50 ( cos 3 π + j s in 3 π ) = 50 ( 2 1 + 3 j ) = 25 + 25 3 j
Adding All the currents-
I = I 1 + I 2 + I 3 + I 4 + I 5 I=I_1+I_2+I_3+I_4+I_5 I = I 1 + I 2 + I 3 + I 4 + I 5
= 141.42 c o s ω t − 70.71 c o s ω t + 25 + 25 3 j − 50 − 86.6 j + 25 + 25 3 j =141.42cos\omega t-70.71cos\omega t+25+25\sqrt{3}j-50-86.6j+25+25\sqrt{3}j = 141.42 cos ω t − 70.71 cos ω t + 25 + 25 3 j − 50 − 86.6 j + 25 + 25 3 j
= 141.42 c o s ω t − 70.71 c o s ω t + 50 + 50 × 1.732 j − 50 − 86.6 j =141.42cos\omega t-70.71cos\omega t+50+50\times 1.732j-50-86.6j = 141.42 cos ω t − 70.71 cos ω t + 50 + 50 × 1.732 j − 50 − 86.6 j
= 70.71 c o s ω t + 86.6 j − 86.6 j =70.71cos\omega t+86.6j-86.6j = 70.71 cos ω t + 86.6 j − 86.6 j
= 70.71 c o s ω t + 0 = 70.71 c o s w t = 70.71 s i n ( ω t + π 2 ) =70.71cos\omega t+0=70.71coswt=70.71sin(\omega t+\dfrac{\pi}{2}) = 70.71 cos ω t + 0 = 70.71 cos wt = 70.71 s in ( ω t + 2 π )
Hence The sum of currents is 70.71s i n ( ω t + π 2 ) sin(\omega t+\dfrac{\pi}{2}) s in ( ω t + 2 π ) .
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