Problem $5$

Consider the circuit shown in the diagram. The values of the components are: $\backslash ($ R$=\backslash )\backslash (4\backslash$Omega$,$ L$=4\backslash$mathrm$\{$H$\},$ C$=1\backslash$mathrm$\{$~F$\}\backslash ).$ The angular frequency of the AC source is $\backslash (\backslash$omega$=1\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash ).$

a$)$ Calculate the reactance of the capacitor and the reactance of the inductor at this frequency.

b$)$ Draw the phasor diagram for this circuit using the axes below. The current phasor is along the $\backslash ($ x $\backslash )-$axis as shown. DRAW and LABEL the voltage phasors for the resistor, capacitor, and the inductor. Make sure their relative magnitudes agree with your answers to part $($a$).$ Be sure to also DRAW and LABEL the AC source voltage phasor.

c$)$ What is the total impedance of the circuit at this angular frequency?

d$)$ Assuming the angular frequency of the AC source can be varied, what angular frequency would the total impedance of this circuit be a minimum?