Note: In this problem we have chosen numbers for the part parameters to make it easier to compute an answer :-). By the way, it is also hard to arrange zero resistance, except with superconducting materials at very low temperatures.
In the circuit shown below L=35.0H and C=11.58mF.
The current source puts out an impulse of area A=2/π=0.637 Coulomb at time t=9.0s.
At t=0 the state is: vC(0)=0.0V and iL(0)=1.0A.
The equation governing the evolution of the inductor current in this circuit is
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Note: In this problem we have chosen numbers for the part parameters to make it easier to compute an answer :-). By the way, it is also hard to arrange zero resistance, except with superconducting materials at very low temperatures.
In the circuit shown below L=35.0H and C=11.58mF .
For the remaining parts of the question, you should round the above value off to two decimal places to make calculations easier.
At the initial time what is the total energy, in Joules, stored in the circuit?
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At the time just before the impulse happens t=9.0s− what is the total energy, in Joules, stored in the circuit?
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At the time just before the impulse happens what is the current iL(9.0s−) , in Amperes, through the inductor?
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At the time just before the impulse happens what is the voltage vC(9.0s−) , in Volts, across the capacitor?
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At the time just after the impulse happens what is the current iL(9.0s+) , in Amperes, through the inductor?
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At the time just after the impulse happens what is the voltage vC(9.0s+) , in Volts, across the capacitor?
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At the time just after the impulse happens what is the total energy, in Joules, stored in the circuit?
The differential equation associated with the input voltage vi(t) and the output voltage vo(t) of this system can be written in the following form:
A⋅d2vodt2+B⋅dvodt+C⋅vo=vi
Where the coefficients A, B, and C can all be written in terms of the component values L, C and R
(a) What is the algebraic expression for coefficient A associated with the d2vodt2 term?
unanswered
(b) What is the algebraic expression for coefficient B associated with the dvodt term?
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(c) What is the algebraic expression for coefficient C associated with the vo term?
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(d)The circuit would ring when excited with a step response vi=u(t) . Assuming that there is negligible damping in the circuit, what is the undamped natural frequency (in Hertz) in terms of the component values?
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(e) The ringing will be damped by the factor e−αt . What is the expression for α in terms of the component values?
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(f) What is the expression for the "Quality Factor" Q of this circuit, in terms of the component values?
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(g) Suppose we need to suppress the ringing. We could change the value of R to make this circuit critically damped (Hint: make the Q = 0.5). What is the expression for this critical R, in terms of L and/or C?
The equation governing the evolution of the inductor current in this circuit is given by:
L diL/dt + 1/C ∫iL dt = Aδ(t - t0)
where:
- L is the inductance (35.0H in this case)
- C is the capacitance (11.58mF in this case)
- diL/dt is the rate of change of the inductor current with respect to time
- ∫iL dt is the integral of the inductor current with respect to time
- A is the area of the impulse (0.637 Coulomb in this case)
- δ(t - t0) is the Dirac delta function, which represents an impulse of area A at time t0 (9.0s in this case)
To solve this equation, you can follow these steps:
1. Differentiate the equation with respect to time to eliminate the integral term.
L d^2iL/dt^2 + 1/C diL/dt = A dδ(t - t0)/dt
2. Since the impulse function is non-zero only at t = t0, the derivative of the impulse function with respect to time is zero except at t = t0.
L d^2iL/dt^2 + 1/C diL/dt = A δ'(t - t0)
3. Integrate both sides of the equation with respect to time to find the solution for the inductor current.
∫[L d^2iL/dt^2 + 1/C diL/dt] dt = ∫A δ'(t - t0) dt
Using the properties of the Dirac delta function, the integral of the derivative of the delta function can be written as:
L diL/dt + 1/C iL = A
4. Solve this differential equation for iL(t).
You can solve this equation using various methods for solving linear differential equations, such as separation of variables, integrating factors, or Laplace transforms. The specific method depends on your preference and familiarity with different techniques.
Once you obtain the solution for iL(t), you can analyze the behavior of the inductor current in this circuit at different times.