- #1
yugeci
- 61
- 0
Homework Statement
Hi there. I have a simple RC circuit with a battery voltage of 10 V, R = 1 Ω, C = 1 F and a switch. I want to use the Explicit Euler (forward divided difference) to solve the equation and check for stability, rather than using a ODE. I am finding the equation for when the capacitor is fully charged and then the battery removed, standard situation etc.
Homework Equations
C. dv/dt + V(t)/R = 0 (loop equation for the circuit)
The Attempt at a Solution
[/B]
The first thing I attempted to do was rewrite / rearrange the circuit in the form v(tnew) = v(told)(1 + a.Δt)n then check what value of t the equation would be unstable for.
dv/dt + V(t)/RC = 0
Rewriting dv/dt as (Vf - Vi)/Δt,
(Vf - Vi)/t + V(t)/RC = 0
... leads to
Vf = Vi - Δt/RC
Vf = Vi(1 - Δt/RC) ----- equation 1
So a = -1/RC = -1, giving
v(t) = Vi(1 - Δt)
v(t) = 10(1 - Δt)
For v(t) to be unstable, |1 - Δt| > 1 ∴ t should be 2sec or more.
However, logically speaking shouldn't the equation be unstable at values of t > 1 sec. Because if you plug Δt = 1 sec, the final voltage is always 0. Where have I made a mistake?