$3\mathrm{.}1$ Pre$-$Lab Activities

Consider the series RLC circuit illustrated in Figure $1.$ Note that $R_L$ is not an actual component, but it represents the internal resistance of the inductor. Longrightarrow Predict the value of $\frac{V_{\text{o}\text{u}\text{t}}}{V_{in}}$ at very high and very low $($close to zero$)$ frequencies.

For the series RLC circuit illustrated in Figure $1,$ let $R_0=2200,R_L=R_S=0,L=0\mathrm{.}5H,$ and $C=47nF.$

a$.)$ Longrightarrow Calculate the resonant frequency, ${}_0.$

b$.)$ Longrightarrow Calculate the maximum value of the magnitude of $H(j)$ at this frequency.

c$.)$ Longrightarrow Calculate the the half$-$power frequencies ${}_{LO},{}_{HI},$ and the bandwidth BW in rads$/$sec$.$

d$.)$ Longrightarrow Convert the values of ${}_0,{}_{LO},{}_{HI},$ and the bandwidth BW to Hertz.

e$.)$ Now assume $R_L=200.$Longrightarrow Using equation $9$ and Python, Matlab, etc., plot the magnitude of the frequency response $H(j)$ between $0$ and $4$ kHz $.$ Longrightarrow Compare the maximum value to the ideal result.

For the series RLC circuit illustrated in Figure $1,$ change the value of $R_0$ to $220,$ and let $R_L=R_S=0,L=0\mathrm{.}5H,$ and $C=47nF$ as before.

a$.)$ Longrightarrow Calculate the resonant frequency, ${}_0.$

b$.)$ Longrightarrow Calculate the maximum value of the magnitude of $H(j)$ at this frequency.

c$.)$ Longrightarrow Calculate the the half$-$power frequencies ${}_{LO},{}_{HI},$ and the bandwidth BW in rads$/$sec$.$

d$.)$ Longrightarrow Convert the values of ${}_0,{}_{LO},{}_{HI},$ and the bandwidth BW to Hertz.

e$.)$ Now assume $R_L=200.$Longrightarrow Using equation $9$ and Python, Matlab, etc., plot the magnitude of the frequency response $H(j)$ between $0$ and $4$ kHz $.$ Longrightarrow Compare the maximum value to the ideal result.

f$.)$ Longrightarrow Compare this result to what you obtained with $R_0=2200$ and $R_L=200.$

Figure $1$: RLC Circuit $($source and internal inductive resistances also shown$)$