Circuits

Consider the DC circuit in the diagram below:

graphnewbox graphtempnewdimen = =by 1by 2 by .5exby 1.083in to 0pt$\:$-$\:$ =.5exby 0.833in to 0pt$\:$V_s$\:$ =by -1by 2 by .5exby 0.583in to 0pt$\:$+$\:$ =by 1by 2 by .5exby 0.167in to 0pt$\:$R₁$\:$ =by -1by 2 by .5exby 0.083in to 0pt$\:$I₁$\:$ =.5exby 0.833in to 0pt$\:$R₂$\:$ =.5exby 0.400in to 0pt$\:$I₂$\:$ =.5exby 0.833in to 0pt$\:$R₃$\:$ =by -1by 2 by .5exby 0.583in to 0pt$\:$+$\:$ =.5exby 0.833in to 0pt$\:$V₃$\:$ =by 1by 2 by .5exby 1.083in to 0pt$\:$-$\:$ =.5exby 0.400in to 0pt$\:$I₃$\:$ depth1.603in width0pt height 0pt

1.

Determine a general expression for the current I₁ and the voltage V₃ as a function of the input voltage and the resistor values.

2.

Give numeric values for I₁ and V₃ if V_s=5V, $R_1=1K\Omega$, $R_2=R_3=2K\Omega$.

3.

Now consider the following AC circuit:

graphnewbox graphtempnewdimen = =by 1by 2 by .5exby 1.083in to 0pt$\:$-$\:$ =.5exby 0.833in to 0pt$\:$v_s(t)$\:$ =by -1by 2 by .5exby 0.583in to 0pt$\:$+$\:$ =by 1by 2 by .5exby 0.167in to 0pt$\:$R₁$\:$ =by -1by 2 by .5exby 0.083in to 0pt$\:$i₁(t)$\:$ =.5exby 0.833in to 0pt$\:$R₂$\:$ =.5exby 0.400in to 0pt$\:$i₂(t)$\:$ =.5exby 0.833in to 0pt$\:$C$\:$ =by -1by 2 by .5exby 0.783in to 0pt$\:$+$\:$ =.5exby 0.833in to 0pt$\:$v_c(t)$\:$ =by 1by 2 by .5exby 0.883in to 0pt$\:$-$\:$ =.5exby 0.533in to 0pt$\:$i₃(t)$\:$ depth1.603in width0pt height 0pt

Assume that the input signal is a complex exponential, $v_s(t) = \exp(j 2\pi ft)$. Then, the voltage v_c(t) can be expressed in the form $v_c(t)= A \exp(j 2\pi \hat{f}t + \phi)$. Determine the paramters A, $\hat{f}$, and $\phi$.

4.

What is the frequency response of this circuit?