Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / Section C / C11

In sinusoidal steady state, complex impedance is the voltage phasor divided by the current phasor. Its modulus is the amplitude ratio and its argument is the voltage lead. With the $e^{i\omega t}$ convention, $I=dq/dt=i\omega CV$, so $Z_C=1/(i\omega C)$.

The PDF labels $V_1$ across $R_1$, not across the complete driven network. If $I$ is the current through $R_1$, then $V_1=I{R}_1$ and $I=Y_p{V}_2$, whereThe resistor-referenced parallel-resonator response isWriting $b(\omega )=\omega C-1/(\omega L)$,The maximum occurs at ${\omega }_0=1/\sqrt{LC}$, where $|H|=R/R_1$ and the phase is zero. Below resonance the phase is positive, tending to $+\pi /2$; above it the phase is negative, tending to $-\pi /2$. The amplitude tends to zero at both extremes: the inductor shorts the parallel network at low frequency, and the capacitor does so at high frequency. At resonance their reactive currents cancel, leaving only the resistor current.

If instead $V_{\mathrm{in}}$ denotes the voltage across the complete circuit, its different transfer ratio is $V_2/V_{\mathrm{in}}=1/[1+R_1/R+i{R}_{1}b]$, with peak $R/(R+R_1)$. This distinction follows directly from the printed voltage endpoints.

For free parallel RLC voltage decay, measure $V=V_A-V_B$ and branch currents from A to B. ThenThere is no external current, so Kirchhoff's current law gives $I_C+I_R+I_L=0$. Differentiating yieldsFor $V\propto e^{st}$ the characteristic equation is $C{s}^2+s/R+1/L=0$. Its underdamped roots giveThe supplied inequality makes $\omega$ real. A real physical voltage has the general form $V=A{e}^{-\gamma t}\cos (\omega t+\varphi )$, the real part of the printed complex expression. A general complex-valued solution uses both roots, $e^{-\gamma t}(C_{+}{e}^{i\omega t}+C_{-}{e}^{-i\omega t})$. The inductor's initial current is fixed consistently by the undifferentiated current balance; it is not an arbitrary additional voltage-mode constant.