Short- and Long-term Thinking

The questions below are due on Tuesday November 12, 2024; 10:00:00 PM.
 
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Consider the circuit below, whose input I(t) is given by I_0\cos(\omega t):

1) Sinusoidal Steady State

When the system has reached steady state, find expressions for i_{\rm R}(t), v_{\rm R}(t), i_{\rm L}(t), and v_{\rm L}(t). Enter your answers as Python expressions below. Feel free to use mathematical constants like pi or e or j, or functions like sqrt or ln or atan, or values from the circuit like L or R or omega or I_0.

i_{\rm R}(t) = A\cos(\omega_0 t + \phi_0), where:
 
  

  

  

i_{\rm L}(t) = A\cos(\omega_0 t + \phi_0), where:
 
  

  

  

v_{\rm R}(t) = A\cos(\omega_0 t + \phi_0), where:
 
  

  

  

v_{\rm L}(t) = A\cos(\omega_0 t + \phi_0), where:
 
  

  

  

2) Transient Reminders

Let's now assume that the input current was instead given by I(t) = \begin{cases} 0 & \text{if}~t < 0\\ I_0\cos(\omega t) & \text{if}~t\geq 0 \end{cases}

This won't actually change the steady-state values from the first part (as t\to \infty), but let's think a little bit about what happens in the interim. Determine the following:

What is i_{\rm L}(0_+)?

What is i_{\rm R}(0_+)?

At what rate is i_{\rm L}(t) changing at t=0_+?

At what rate is v_{\rm L}(t) changing at t=0_+?