How to solve the differential equation for driven series RLC circuit?

In summary, the conversation discusses a driven series RLC circuit and the associated differential equation for its possible solution. It is given that one possible solution is Q(t) = Q_0 cos(ωt - φ), and the attempt at a solution involves substituting this into the differential equation and using trigonometric identities to try and solve for Q_0 and φ. However, it is mentioned that there is no way to eliminate t and φ and solve for Q_0 or cancel Q_0 and t and solve for φ. The equations must hold for any value of t.
  • #1
omoplata
327
2

Homework Statement



It is the driven series RLC circuit. It is given in the following images.
zqTR1Ra.png
KUpllgB.png


It is from the section 12.3 in this note.

Homework Equations



The differential equation, as given by 12.3.3, is ##L \frac{d^2 Q}{d t^2} + R \frac{d Q}{d t} + \frac{Q}{C} = V_0 \sin{(\omega t)}##.

It says that one possible solution is ##Q(t) = Q_0 \cos{(\omega t - \phi)}##, where ##Q_0 = \frac{V_0}{\omega \sqrt{R^2 + (\omega L - 1 / \omega C)^2}}## and ##\tan \phi = \frac{1}{R} \left( \omega L - \frac{1}{\omega C} \right)##.

The Attempt at a Solution



So, assuming ##Q(t) = Q_0 \cos{(\omega t - \phi)}##, I get$$\frac{d Q}{d t} = -Q_0 \omega \sin{(\omega t - \phi)}\\\frac{d^2 Q}{d t^2} = -Q_0 \omega^2 \cos{(\omega t - \phi)}$$
Substituting,$$-L Q_0 \omega^2 \cos{(\omega t - \phi)} - R Q_0 \omega \sin{(\omega t - \phi)} + \frac{Q_0}{C} \cos{(\omega t - \phi)} = V_0 \sin{(\omega t - \phi)}$$

I cannot figure out show to eliminate ##t## and ##\phi## and solve this for ##Q_0##.

I try applying$$\cos{(\omega t - \phi)} = \cos{(\omega t)} \cos \phi + \sin{(\omega t)} \sin \phi\\\sin{(\omega t - \phi)} = \sin{(\omega t)} \cos \phi - \cos{(\omega t)} \sin \phi$$

But still there's no way cancel ##t## and ##\phi## and solve for ##Q_0##, or cancel ##Q_0## and ##t## and solve for ##\phi##.
 
Last edited:
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  • #2
The equations should hold for any value of t. Choose wisely :wink: :smile:
 

1. How do I determine the initial conditions for the differential equation?

The initial conditions for the differential equation can be determined by analyzing the circuit at t = 0. This includes determining the initial voltage and current values for each component in the circuit.

2. What is the general form of the differential equation for a driven series RLC circuit?

The general form of the differential equation for a driven series RLC circuit is d^2i/dt^2 + (R/L)*di/dt + (1/(LC))*i = (1/L)*V_in(t), where i is the current, R is the resistance, L is the inductance, C is the capacitance, and V_in(t) is the input voltage.

3. How do I solve the differential equation for a driven series RLC circuit?

The differential equation can be solved using various methods such as the Laplace transform, the method of undetermined coefficients, or the method of variation of parameters. The method used will depend on the complexity of the circuit and the available resources.

4. What are the key factors that affect the solution of the differential equation for a driven series RLC circuit?

The key factors that affect the solution of the differential equation include the values of the resistance, inductance, and capacitance, as well as the input voltage. These parameters determine the behavior of the circuit and can greatly impact the solution of the differential equation.

5. How does the solution of the differential equation for a driven series RLC circuit relate to the circuit's behavior?

The solution of the differential equation represents the current flowing through the circuit as a function of time. This can provide information about the behavior of the circuit, such as the transient response, steady-state response, and resonance frequency. It can also help determine the stability and performance of the circuit.

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