Question #291938

A 50 Hz sinusoidal voltage has rms value of 200 V. At t=0, the instantaneous value is positive and half of its maximum value. Write down expression for voltage and sketch the waveform


1
Expert's answer
2022-01-30T13:51:29-0500

Using the provided data, we proceed to find the elements for V(t)=Vmaxsin(ωt+ϕ)V_{(t)}=V_{max}\sin{(\omega t+\phi)}:


f=50Hzω=2πf=314.16rad/s\\ f = 50\, Hz \\ \therefore \omega=2\pi f= 314.16\,{ rad/s}


We find the maximum value with Vrms and the phase angle as:


Vrms=200VVmax=2Vrms=(1.4142)(200V)=282.84Vϕ=arcsin(V(0)Vmax)ϕ=arcsin(½VmaxVmax)=arcsin(½)=30°=π6V_{rms}=200\,V \\V_{max}=\sqrt{2}V_{rms}= (1.4142)(200\,V)=282.84\,V \\ \phi= \arcsin{\Big( \dfrac {V_{(0)}}{V_{max}}\Big)} \\ \text{} \\ \phi= \arcsin{\Big( \dfrac {½V_{max}}{V_{max}}\Big)} =\arcsin{(½)}= 30°=\frac{\pi}{6}


In conclusion, the expression for the voltage will be V(t)=(282.84V)sin(100πt+π6)V_{(t)}=(282.84\,V)\sin{(100\pi t+\frac{\pi}{6})}.


The graphic for the voltage will be:


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