For the half$-$wave rectifier circuit illustrated in Figure $3$ assume that there is a $0\mathrm{.}7$ V drop across the diode when it is forward biased.

Figure $4$: Full$-$wave Bridge Rectifier

Figure $2$: Half$-$wave Rectifier

Figure $3$: Output of the Half$-$wave Rectifier

a$.)$ If the input $v_{\text{I}}$ is a $1$ kHz sinusoid with an amplitude of $2\mathrm{.}5$ V and $R=10k,$ Longrightarrow sketch the input $v_{\text{I}}$ and the output $v_O$ on a single graph, labeling important points on the time and voltage axes for a couple of cycles of the sinusoid$.$

b$.)$ Longrightarrow Compute or estimate the average value of $v_O$ over one cycle of the input.

c$.)$ Keep the same circuit but increase the $1$ kHz input to an amplitude of $10$ V $,$ Longrightarrow sketch the input $v_{\text{I}}$ and the output $v_O$ on a single graph, labeling important points on the time and voltage axes for a couple of cycles of the sinusoid$.$

d$.)$ Longrightarrow Describe the similarities and differences between the two cases.

$2.$ For the peak detector circuit illustrated in Figure $5$ assume that there is a $0\mathrm{.}7$ V drop across the diode when it is forward biased. Let $R=10k,$ and $C=1F.$

a$.)$ Longrightarrow Compare the $RC$ time constant and the period $T$ of the sinusoid$.$

b$.)$ If the input $v_{\text{I}}$ is a $1$ kHz sinusoid with an amplitude of $2\mathrm{.}5V,$Longrightarrow roughly sketch the input $v_{\text{I}}$ and the output $v_O$ on a single graph for a couple of cycles of the sinusoid$.$

c$.)$ Keep the same circuit but increase the $1$ kHz input to an amplitude of $10$ V $,$ Longrightarrow roughly sketch the input $v_{\text{I}}$ and the output $v_O$ on a single graph for a couple of cycles of the sinusoid$.$

d$.)$ Longrightarrow Describe the similarities and differences between the two cases.

$3.$ For the full$-$wave bridge rectifier circuit illustrated in Figure $4$ assume that there is a $0\mathrm{.}7$ V drop across the diodes when they are forward biased.

a$.)$ If the input $v_S$ is a $1$ kHz sinusoid with an amplitude of $2\mathrm{.}5$ V and $R=10k,$ Longrightarrow sketch the input $v_S$ and the output $v_O$ on a single graph, labeling important points on the time and voltage axes for a couple of cycles of the sinusoid$.$

b$.)$ Keep the same circuit but increase the $1$ kHz input to an amplitude of $10$ V $,$ Longrightarrow sketch the input $v_S$ and the output $v_O$ on a single graph, labeling important points on the time and voltage axes for a couple of cycles of the sinusoid$.$

c$.)$ Longrightarrow Describe the similarities and differences between the two cases.

$4.$ For the precision half$-$wave rectifier circuit illustrated in Figure $8$ assume that $R=2\mathrm{.}2k$ and that the op$-$amp is powered by $+-3V$ and that the diode has a $0\mathrm{.}7$ V drop across it when it is forward biased.

a$.)$ For an input $v_{\text{I}}$ between $-5$ and $5VDC,$Longrightarrow plot $v_O$ and $v_A$ as a function of $v_{\text{I}}$ on the same graph, labeling important points on the axes.

b$.)$ Assume a $1$ kHz sinusoidal input with a peak$-$to$-$peak voltage of $10V.$Longrightarrow Plot $v_{\text{I}},v_O,$ and $v_A$ versus time on the same graph, labeling important points on the time and voltage axes for a couple of cycles of the sinusoid$.$

c$.)$ Longrightarrow Describe how this rectifier behaves differently from the other ones you have analyzed previously without the op$-$amp.

Figure $5$: Peak Detector, aka a half$-$wave rectifier with "smoothing" capacitor

Figure $8$: Precision Half$-$wave Rectifier. Please draw all sketch