RCL Circuit Analysis: Vr, Vl, Vc

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Discussion Overview

The discussion revolves around the analysis of an RCL circuit driven by a current source, specifically focusing on the voltages across the resistor, inductor, and capacitor over time. Participants are attempting to derive expressions for Vr(t), Vl(t), and Vc(t) based on given parameters and initial conditions.

Discussion Character

  • Homework-related
  • Technical explanation
  • Exploratory

Main Points Raised

  • One participant states the initial conditions and circuit parameters, providing equations for the voltages across the components based on their understanding.
  • Another participant expresses skepticism about the simulator's ability to handle the circuit due to the instantaneous nature of the current source and the inductor's resistance to sudden changes in current.
  • A participant questions the formula used for the capacitor voltage, suggesting an alternative expression that differs from the original calculations.
  • There is a mention of the importance of considering the initial transient response of the reactive components due to the sudden current change at t=0.
  • One participant suggests that using Laplace transforms might simplify the analysis, although another notes that this topic has not yet been covered in their studies.
  • A later reply emphasizes the need to approach the problem using traditional methods, highlighting the driving function of the current as 5cos(50t)U(t), where U(t) is the unit step function.

Areas of Agreement / Disagreement

Participants express differing views on the expected values for Vc(t), with some suggesting it should be zero while others propose different expressions. There is no consensus on the correct approach or the results of the simulation, indicating ongoing disagreement and uncertainty in the analysis.

Contextual Notes

Participants note the potential limitations of the simulator in accurately modeling the circuit's behavior due to the nature of the current source and the initial conditions. The discussion also highlights the complexity of analyzing the circuit without the use of Laplace transforms, which have not yet been introduced in their coursework.

freezer
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Homework Statement



i(t) = 0 for t<0 and 5*cos(50t) for t>=0
Vc(0) = 0

Circuit is current source, 5 ohm resister, 2 Henry inductor, and 3 Farad capacitor in series.

Need to find Vr(t), Vl(t), and Vc(t), voltage across resistor, inductor, and capacitor.

Homework Equations



V= IR for resistor
V= L(di/dt) for inductor
V = 1/c(di/dt) for cap

The Attempt at a Solution


di/dt = -250sin(50t)

Resistor:

Vr(t) = 25*cos(50t)

Inductor:

Vl(t) = 2*-250sin(50t)
=-500sin(50t)

Cap:
Vc(t) = 1/3*-250sin(50t)
= (-250/3)sin(50t)

I ran the simulation on multisim and the numbers do not agree.

The voltage across the cap shows 66.3mV p-p
scope.png

ch1 = node between source and resistor
ch2 = node between resistor and inductor
ch3 = node between inductor and cap
ch4 = current probe 1mV/mA
traces are from top to bottom
 
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Something tells me that the simulator is not going to fair very well handling the circuit as described.

At t=0 the ideal current source is going to want to force the current to be 5A immediately. That's the result of it being a cosine function that kicks in at time t=0. But inductors don't like to change current immediately like that. The simulator might try to generate GV or TV levels of voltages across the inductor for the first instant, causing similarly ridiculous current spikes for the capacitor in the GA or TA range.
 
So my results for Vr and Vl seem reasonable but Vc seems like it should be 0.03sin(50t) so i am not sure i am using the right formula.
 
freezer said:
So my results for Vr and Vl seem reasonable but Vc seems like it should be 0.03sin(50t) so i am not sure i am using the right formula.

Those look reasonable for the steady-state values for those items, but misses the initial transient response for the reactive components that take place because of the instantaneous forced step in current at t=0.

If you've been introduced to Laplace transforms, that might be an easier approach to obtaining the response for the inductor voltage.
 
Laplace is not for a few more chapters...
 
freezer said:
Laplace is not for a few more chapters...

Aurgh. Then I guess you'll have to make do with a careful investigation of the differential equation for the inductor voltage by more traditional methods. The driving function for the current is not just 5cos(50t), but rather 5cos(50t)U(t), where U(t) is the unit step...
 

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