LC-Circuit Problem: Find I as Function of Time

  • Thread starter atlantic
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In summary, an LC-circuit with L = 64mH and C = 121nF has an initial charge of 10μC on the capacitor and an initial current of 0.3A in the inductor. To find the current in the inductor as a function of time, the differential equations for the network can be used with the given initial conditions to solve for the current over time. Alternatively, the Laplace transform can be used to solve the problem.
  • #1
atlantic
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An LC-circuit has L = 64mH, C = 121nF.
At the time t=0, the charge on the condensator is 10μC and the current in the inductor is 0.3A. What is the current in the inductor as a function of time?

With:[itex]q = Q_mcos(\omega_0t+\phi)[/itex] we get that: [itex]I = Q_m\omega_0cos(\omega_0t+\phi + \pi/2)[/itex] (because I = dq/dt), where [itex]\omega_0 = 1/(√LC)[/itex]

I thought that the initial conditions would mean that I have to solve:
[itex]q(0) = 10*10^{-6} =Q_mcos(\phi) [/itex] and [itex]I(0) = 0.3 = Q_m\omega_0cos(\phi + \pi/2)[/itex]. But these equations have no solution(!)
 
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  • #2
Here's what I do with a problem like this:

1. write the differential equation without regard to initial conditions.
2. s-transform the equation term-by-term, paying attention to
f'(t) <--> sF(s) - f(0+) and f''(t) <--> s2F(s) - sf(0+) - f'(0+).

The you just invert the ensuing transfer function back to the time domain, and you get all the i.c.'s included.

(If you haven't had the Laplace transform yet I don't know what to tell you.)
 
  • #3
No, I don't know Laplace:uhh:
 
  • #4
OK, then you have to use standard differential equations for the network, and solve in the traditional diff e manner including initial conditions.

I didn't follow you argument but I would write

i = -CdV/dt (i is + if flowing out of C)
V = Ldi/dt (i is + if flowing into L)

So i = -C(d/dt)Ldi/dt = -LCd2i/dt2 or

LCd2i/dt2 + i = 0

I.C. 1: i(0) = 0.3A
I.C. 2: di/dt(0+) = V0/L but V0 = Q0/C = 1e-5/C so di/dt(0+) = Q0/LC

So now just solve the 2nd order diff eq. with those two initial conditions.
 
  • #5
in the real domain.

I would first verify the given initial conditions and equations to ensure their accuracy. If they are indeed accurate, I would then use the given values for L and C to calculate the value of \omega_0. From there, I would use the given equations to solve for Q_m and \phi, which would then allow me to determine the current in the inductor as a function of time using the equation I = Q_m\omega_0cos(\omega_0t+\phi + \pi/2). It is possible that the given initial conditions and equations may have a solution in the complex domain, in which case I would need to use complex numbers to solve for Q_m and \phi before determining the current in the inductor as a function of time. Additionally, I would also consider the effects of resistance in the circuit, which may impact the accuracy of the calculated values and the resulting current function.
 

What is an LC circuit?

An LC circuit is an electric circuit consisting of an inductor and a capacitor connected in series or parallel. It is also known as a resonant circuit because it can store and release energy at a specific frequency.

What is the problem of finding I as a function of time in an LC circuit?

The problem of finding I as a function of time in an LC circuit is a common problem in physics and engineering. It involves determining the current (I) flowing through the circuit at different points in time, taking into account the values of the inductance (L), capacitance (C), and initial conditions.

How is the current (I) related to time in an LC circuit?

In an LC circuit, the current (I) is related to time through the equation I(t) = I0cos(ωt), where I0 is the initial current and ω is the angular frequency of the circuit. This equation shows that the current oscillates sinusoidally with time.

What factors affect the current in an LC circuit?

The current in an LC circuit is affected by several factors, including the inductance (L), capacitance (C), initial conditions, and external factors such as resistance and external voltage sources. Changes in any of these factors can significantly impact the current in the circuit.

How can I solve the LC circuit problem and find I as a function of time?

To solve the LC circuit problem and find I as a function of time, you can use the equation I(t) = I0cos(ωt) and plug in the values of the inductance (L), capacitance (C), and initial conditions. You can also use mathematical tools such as differential equations and Laplace transforms to solve more complex problems. Additionally, there are online calculators and simulation software available to help with solving LC circuit problems.

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