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 w

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

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niczorrrr 4 years 2021-07-24T04:12:44+00:00 1 Answers 12 views 0

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  1. Sigrid
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    2021-07-24T04:14:42+00:00
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    Answer:

    a) t = 4.6\tau

    b) L = 0.0187 \: H

    Explanation:

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

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

    Where τ is the time constant and is given by

    \tau = \frac{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^{\frac{-t}{\tau} })\\0.99 = (1 - e^{\frac{-t}{\tau} })\\1 - 0.99 = e^{\frac{-t}{\tau}}\\ln(0.01) = ln(e^{\frac{-t}{\tau}})\\-4.6 = \frac{-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\frac{L}{R}\\t = 4.6\frac{L}{R}\\0.0012 = 4.6\frac{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.

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