91Ó°ÊÓ

We need to find out that,

Peak current is,$I=I_{max}\cos \left(\mathrm{Ï•}\right)$

Let the maximum current in circuit is, $I_{max}$

And the peak current in circuit is,$I=\frac{V_0}{Z}$

And the peak voltage is, $V_0$

And the impedance is, $Z$

And the resistance is, $R$

And power factor is,

localid="1650314468329" $$\begin{gathered} \cos \left(\mathrm{Ï•}\right)=\frac{R}{Z} \\ R=Z\cos \left(\mathrm{Ï•}\right) \end{gathered}$$

We know that,

$$\begin{gathered} I_{max}=\frac{V_0}{R} \\ {I}_{max}=\frac{V_0}{Z\cos \left(\mathrm{Ï•}\right)} \\ {I}_{max}\cos \left(\mathrm{Ï•}\right)=I \end{gathered}$$

Hence, We get require expression i.e.$I=I_{max}\cos \left(\mathrm{Ï•}\right)$

We need to find out that,

Average power is,$P_{source}=P_{max}{\cos }^2\left(\mathrm{Ï•}\right)$

Let peak power is, $P_{source}$

And current is,$i$

And voltage is, $V$

We know that,

$P_{source}=iV$

Now, let phase difference in current is, $\left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)$

And in voltage is, $\mathrm{Ó\neg }t$

Therefore, current is, $i=I\cos \left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)$

And voltage is, $V=V_0\cos \left(\mathrm{Ó\neg }t\right)$

And power is,

$$\begin{gathered} P_{source}=\left(I\cos \left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)\right)\left(V_0\cos \left(\mathrm{Ó\neg }t\right)\right) \\ =I{V}_0\cos \left(\mathrm{Ó\neg }t\right)\cos \left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right) \\ =I{V}_0\cos \left(\mathrm{Ï•}\right){\cos }^2\left(\mathrm{Ó\neg }t\right)+I{V}_0\sin \left(\mathrm{Ï•}\right)\sin \left(\mathrm{Ó\neg }t\right)\cos \left(\mathrm{Ó\neg }t\right) \\ =\frac{1}{2}I{V}_0\cos \left(\mathrm{Ï•}\right) \\ =I_{rms}{V}_{rms}\cos \left(\mathrm{Ï•}\right) \\ =P_{max}{\cos }^2\left(\mathrm{Ï•}\right) \end{gathered}$$

Here, $V_{rms}$is rms voltage

And $I_{rms}$is rms current

And Maximum Power is, $P_{max}=\frac{1}{2}{I}_{max}{V}_0$

Hence, we get required expression.