2nd order circuits

Today we started class of with learning about 2^nd order circuit. For second order differential equation we need additional boundary values.

V(0)=v(0-)

Capacitor voltage and inductor current are variables that cannot change abruptly.

Source free series RLC Circuit: the circuit is being excited by the energy initially stored in the capacitor and inductor.

Source free circuits do not have voltage and current. 

Series RLC Circuit Step Response: In this lab we modeled and tested a series RLC second order circuit by analyzing step response of a ciruit and comparing with expected values on the dampling ration and natural frequency.

The initial value of i is given as I0 across the inductor

Pre lab: we wrote differential equation relating Vout and Vin for the RLC circuit.

We tested step response of a given circuit. Then we compared the measured response with expected or calculated values based on the damping ration and natural frequency of the circuit.  

The roots s₁ and s₂ are called natural frequencies, measured in nepers per second (Np/s), because they are associated with the natural response of the circuit; ω ₀ is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second (rad/s); and α is the neper frequency or the damping factor, expressed in nepers per second.

1.  If α > ω ₀ , we have the overdamped case.

2.  If α = ω ₀ , we have the critically damped case.

3.  If α < ω ₀ , we have the underdamped case.

The critically damped case is the borderline between the underdamped and overdamped cases and it decays the fastest. And overdamped case takes longer to settle.

Damping is the gradual loss of the initial stored energy, as evidenced by the continuous decrease in the amplitude of the response. The damping effect is due to the presence of resistance R. The damping factor α determines the rate at which the response is damped.

Following is the expected graph of RLC series circuit step response. It is Underdamped circuit and
( α < ω 0 )

Following is the graph of underdamped circuit we captured on waveform. 

Percent Error: we calculated 19.5% error as calculated in above few pictures. The percent error might be caused from the capacitor and inductor that we used are non ideal. The inductor has non negligible resistance. 

Above is the output graph fromt the capacitor.
This graph yields a damping formula of V=0.43e^(-2340t), meaning that our experimental alpha is 2340, which is about 0.4% of our theoretical alpha value of 550000. 

Summary: 

Possible source of error is from our input variables, the way we measured the circuit, or maybe did wrong calculation for theoretical value.