Engineering How Is Alpha Calculated in the Discretization of an RC Circuit?

[Feodalherren]

Homework Statement

[/B]
Derive a discrete formula for an RC circuit for Vab[k] using the forward difference approximation. It should be of the form Vab[k + 1] = α Vab[k], and depend only on T, R, and C. For T = 0.076 s, R = 3 kΩ, and C = 10 mF, what is α?

T is the period.
The current is going from a to b.

Homework Equations

Ohms law
Forward difference formula

The Attempt at a Solution

I can't find the equation editor...

dVab(t)/dt = (1/c)I(t)

discretisized:

dVab[k]/dt = I[k]/c

which is approximately equal to

(Vab[k+1] - Vab) / T

solving for Vab[k+1]

Vab[k+1] = (T/C)I[k] + Vab[k]

From Ohm's law for a simple RC circuit we can find that

I = Vab/R

thus

Vab[k+1] = (T/RC)Vab[k] + Vab[k]

Vab[k+1] = ((T/RC)+1) Vab[k]

it follows that

α = (T/RC)+1)

which yields a result of approximately 1.0025

however, the solutions manual claims that the answer is 0.9975 which is (1-(T/RC))

 

[Feodalherren]

I think I solved it. I swapped
(Vab[k+1] - Vab) / T

to
(-Vab[k+1] + Vab) / T

because of the orientation of the current.

 

[rude man]

Derive a discrete formula for an RC circuit for Vab[k]...

There is only ONE RC circuit in the universe? How about showing us a schematic drawing ...
The problem wording suggests that the output voltage is decaying in which case you know alpha has to be < 1.