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• Notes

$\begin{array}{l}{\text { A sinusoidal current has a maximum amplititude of } 20 \mathrm{A} \text { . The current passes through one complete cycle in } 1 \mathrm{ms} \text { . }} \\ {\text { the magnitude of the current at zero time is } 10 \mathrm{A} \text { . }} \\ {\text { a) What is the frequency of the current in hertz? }} \\ {\text { b) What is the frequency in radians per second? }} \\ {\text { c) Write the expression for it }(t) \text { using the cosine function. Express } \phi \text { in degrees. }} \\ {\text { d) What is the rms value of the current? }}\end{array}$

$T=1 m S$

$f=\frac{1}{T}=\frac{1}{10^{-3}}=1000 \mathrm{Hz}$

$w=2 \pi f=2 \pi * 1000=2000 \pi \ rod/sec$

$i(t)=I_{max}cos [w t+\phi]$

$i(t)=20 cos [2000 \pi t+\phi]$

$t=0 \Rightarrow i(t)=10A$

$10=20 cos [2000 \pi*0+\phi]$

$10=20 \cos \phi$

$\phi=60^{\circ}$

$i(t)=20 \cos [2000 \pi t+60]$

$I_{rms}= \sqrt{\frac{1}{T} \int_{0}^{T} i^2 (t) d t}$

$I_{rms}=\sqrt{\frac{1}{T} \int_{0}^{T} [20cos (2000 \pi t+60)]^2d t}$

$I_{rms}=\sqrt{\frac{1}{10^{-3}} \int_{0}^{li3} [20 \cos [2000 \pi t+60]]^{2} d t}$

$I_{r M S}=14.14 A=\frac{I_{max}}{\sqrt{2}}= \frac{20}{\sqrt{2}}$

$\begin{array}{l}{\text { A sinusoidal voltage is given by the expression: } v=300 \cos \left(120 \pi \mathrm{t}+30^{\circ}\right) .} \\ {\text { a) What is the period of the voltage in milliseconds? } \quad \text { b) What is the frequency in hertz? }} \\ {\text { c) What is the rms value of } \mathrm{V} ?}\end{array}$

$V=300 \cos \left[120 \pi t+30^{\circ}\right]$

$W=120 \pi=\frac{2 \pi}{T}$

$T=\frac{2 \pi}{120 \pi}=\frac1{60} sec$

$F=\frac{1}{T}= \frac{ 1}{1 /60} =60 Hz$

$V_{rms}=\frac{V_{max}}{\sqrt{2}}=\frac{300}{\sqrt{2}}=212.13 volt$