Question

A battery with emf e m f is connected in series with an inductance L and a resistance R. (a) Assuming the current has reached steady state when it is at 99% of its maximum value, how long does it take to reach steady state, assuming the initial current is zero? (Use any variable or symbol stated above as necessary. To represent e m f, use E.) t99% = (b) If an emergency power circuit needs to reach steady state within 1.2 ms of turning on and the circuit has a total resistance of 72 Ω, what values of the total inductance of the circuit are needed to satisfy the requirement? (Give the maximum value.) H

Answer 1

a)
`t = 4.6\tau`

b)
`L = 0.0187 \: H`

Step-by-step explanation:

The current flowing in a R-L series circuit is given by

`I = I_(0) (1 - e^{(-t)/(\tau) })`

Where τ is the time constant and is given by

`\tau = (L)/(R)`

Where L is the inductance and R is the resistance

Assuming the current has reached steady state when it is at 99% of its maximum value,

`0.99I_(0) = I_(0) (1 - e^{(-t)/(\tau) })\\0.99 = (1 - e^{(-t)/(\tau) })\\1 - 0.99 = e^{(-t)/(\tau)}\\ln(0.01) = ln(e^{(-t)/(\tau)})\\-4.6 = (-t)/(\tau)\\t = 4.6\tau\\`

Therefore, it would take t = 4.6τ to reach the steady state.

(b) If an emergency power circuit needs to reach steady state within 1.2 ms of turning on and the circuit has a total resistance of 72 Ω, what values of the total inductance of the circuit are needed to satisfy the requirement?

`t = 4.6(L)/(R)\\t = 4.6(L)/(R)\\0.0012 = 4.6(L)/(72)\\0.0864 = 4.6 L\\L = 0.0864/4.6\\L = 0.0187 \: H`

Therefore, an inductance of 0.0187 H is needed to satisfy the requirement.