Answer to Question #167094 in Electric Circuits for Vinit Kumawat

Question #167094

Draw the phasor diagram and find the sum of the currents: i1=141.42Sin(ωt +  π/2) , i2 = -70.71 Cos ωt , i3= 50 ej π/3 , i4=100∠-2 π/3 and i5= 50 (Cosπ/3 + j Sinπ/3) 


1
Expert's answer
2021-03-05T07:19:27-0500

Given Currents are-

I1=141.42sin(ωt+π2)=141.42coswtI_1=141.42sin(\omega t+\dfrac{\pi}{2})=141.42coswt


I2=70.71coswtI_2=-70.71coswt


I3=50ejπ3=50(cosπ3+jsinπ3)=50(1+3j2)=25+253jI_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


I4=100<2π3=5086.6jI_4=100<\dfrac{-2\pi}{3}\\=-50-86.6j ( convert the polar to rectangular form)


I5=50(cosπ3+jsinπ3)=50(1+3j2)=25+253jI_5=50(cos\dfrac{\pi}{3}+jsin\dfrac{\pi}{3})=50(\dfrac{1+\sqrt{3}j}{2})=25+25\sqrt{3}j


Adding All the currents-


I=I1+I2+I3+I4+I5I=I_1+I_2+I_3+I_4+I_5


=141.42cosωt70.71cosωt+25+253j5086.6j+25+253j=141.42cos\omega t-70.71cos\omega t+25+25\sqrt{3}j-50-86.6j+25+25\sqrt{3}j


=141.42cosωt70.71cosωt+50+50×1.732j5086.6j=141.42cos\omega t-70.71cos\omega t+50+50\times 1.732j-50-86.6j


=70.71cosωt+86.6j86.6j=70.71cos\omega t+86.6j-86.6j


=70.71cosωt+0=70.71coswt=70.71sin(ωt+π2)=70.71cos\omega t+0=70.71coswt=70.71sin(\omega t+\dfrac{\pi}{2})


Hence The sum of currents is 70.71sin(ωt+π2)sin(\omega t+\dfrac{\pi}{2}) .


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