How Do You Solve for i(t) in a Series RLC Circuit with a Given Source?

In summary, the conversation involves solving for the current (i(t)) in a circuit using the KCL equation. The natural response (in(t)) is found through the derivation and is equal to A1+A2.e-5t. The initial conditions (i(0+)=i(0-)=0) are then used to find the final solution, i(t)=in+ip. However, the initial conditions are incorrect and must include the input variable, Vs, to solve accurately.
  • #1
ongxom
26
0

Homework Statement


t2eHpiO.png

R=50Ω
L=5H
C=1/125F
vs=200.[1+u(t)] V

Find i(t)

Homework Equations



The Attempt at a Solution


Write the KCL equation
-vs+Ri+L.i'+1/C.∫idt=0
Do the derivation to get rid of the integral, replace all possibly values, we got:
i''+10i'+25i=1/5.vs' (*)
solve A(s)=s2+10s+25=0, we got the natural response in(t)=A1+A2.e-5t

Replace the inductor with a short circuit, the capacitor with open circuit, so I have ip=0
and the initial condition i(0+)=i(0-)=0. (**)

So the solution should be i(t)=in+ip

I thought I was wrong at (**) and maybe at (*) cause the vs' would be zero and the solution would be zero as well.
 
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  • #2
ongxom said:
1. Replace the inductor with a short circuit, the capacitor with open circuit, so I have ip=0
and the initial condition i(0+)=i(0-)=0. (**)

So the solution should be i(t)=in+ip

I thought I was wrong at (**) and maybe at (*) cause the vs' would be zero and the solution would be zero as well.


i(0+) = i(0-) ≠ 0. Look at the definition of the input. What does vs=200.[1+u(t)] V mean?
 
  • #3
ongxom said:
i''+10i'+25i=1/5.vs' (*)
solve A(s)=s2+10s+25=0, we got the natural response in(t)=A1+A2.e-5t

Replace the inductor with a short circuit, the capacitor with open circuit, so I have ip=0
and the initial condition i(0+)=i(0-)=0. (**)

So the solution should be i(t)=in+ip

I thought I was wrong at (**) and maybe at (*) cause the vs' would be zero and the solution would be zero as well.


Even if Vs = u(t) the current i(t) would not be zero! You are ignoring Vs!
You have to include Vs before trying to fit the initial conditions. Since this is a 2nd order ODE you know there have to be two initial conditions on i.
 

FAQ: How Do You Solve for i(t) in a Series RLC Circuit with a Given Source?

1. What is a Series RLC circuit?

A Series RLC circuit is an electrical circuit that contains a resistor (R), an inductor (L), and a capacitor (C) connected in series. This means that the components are connected end-to-end, with the same current flowing through each component. The circuit can also have a voltage source (S) connected in series with the other components.

2. What is the purpose of a Series RLC circuit?

The purpose of a Series RLC circuit is to control the flow of electrical current in a circuit. The resistor limits the current, the inductor stores energy in the form of a magnetic field, and the capacitor stores energy in the form of an electric field. Together, these components can create a resonant circuit, filter out specific frequencies, or regulate voltage.

3. How does a Series RLC circuit with a source behave?

A Series RLC circuit with a source behaves differently depending on the frequency of the source. At low frequencies, the capacitor acts as an open circuit and the inductor acts as a short circuit, allowing most of the current to flow through the resistor. At high frequencies, the opposite happens and most of the current flows through the inductor and capacitor, minimizing the current through the resistor. At the resonant frequency, the circuit's impedance is minimized, allowing maximum current to flow through the circuit.

4. What is the resonant frequency of a Series RLC circuit with a source?

The resonant frequency of a Series RLC circuit with a source can be calculated using the equation: fr = 1/(2π√(LC)), where fr is the resonant frequency, L is the inductance, and C is the capacitance. At this frequency, the reactance of the inductor and capacitor cancel each other out, resulting in a purely resistive circuit.

5. How does the phase angle change in a Series RLC circuit with a source?

The phase angle in a Series RLC circuit with a source changes depending on the frequency of the source. At low frequencies, the current leads the voltage due to the inductive reactance, resulting in a positive phase angle. At high frequencies, the current lags behind the voltage due to the capacitive reactance, resulting in a negative phase angle. At the resonant frequency, the phase angle is zero, meaning the current and voltage are in phase.

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