RCL Circuit Analysis: Vr, Vl, Vc

In summary, at t=0, the current source will force the current to be 5A, but the inductor will not allow the current to stay at that level for very long. The inductor will start 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.
  • #1
freezer
76
0

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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
Laplace is not for a few more chapters...
 
  • #6
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...
 

FAQ: RCL Circuit Analysis: Vr, Vl, Vc

1. What is an RCL circuit?

An RCL circuit is an electrical circuit that contains a resistor (R), inductor (L), and capacitor (C) connected in series or parallel. These components interact with each other to produce a varying current and voltage.

2. What is the purpose of analyzing Vr, Vl, and Vc in an RCL circuit?

Vr, Vl, and Vc represent the voltages across the resistor, inductor, and capacitor in an RCL circuit, respectively. Analyzing these voltages allows us to understand how each component affects the overall behavior of the circuit and how they interact with each other.

3. How do you calculate Vr, Vl, and Vc in an RCL circuit?

The voltage across a resistor (Vr) can be calculated using Ohm's law (V = IR), where I is the current flowing through the resistor and R is its resistance. The voltage across an inductor (Vl) can be calculated using the equation V = L(di/dt), where L is the inductance of the inductor and di/dt is the rate of change of current. The voltage across a capacitor (Vc) can be calculated using the equation V = Q/C, where Q is the charge stored in the capacitor and C is its capacitance.

4. What is the relationship between Vr, Vl, and Vc in an RCL circuit?

In an RCL circuit, the voltages Vr, Vl, and Vc are related by Kirchhoff's voltage law, which states that the sum of the voltages in a closed loop must be equal to zero. This means that the voltage across the resistor (Vr) is equal to the sum of the voltages across the inductor (Vl) and the capacitor (Vc).

5. How does frequency affect Vr, Vl, and Vc in an RCL circuit?

The voltages Vr, Vl, and Vc in an RCL circuit are affected by the frequency of the input signal. At low frequencies, the capacitor acts as an open circuit and the inductor acts as a short circuit, so most of the voltage is dropped across the resistor (Vr). At high frequencies, the opposite is true and most of the voltage is dropped across the inductor (Vl). At the resonant frequency, the voltage across the capacitor and inductor cancel each other out, resulting in a low voltage across the resistor (Vr).

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