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Question 7.21: The coil of a certain relay is operated by a 12-V battery. I...

The coil of a certain relay is operated by a 12-V battery. If the coil has a resistance of 150 Ω and an inductance of 30 mH and the current needed to pull in is 50 mA, calculate the relay delay time.

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The current through the coil is given by

i(t)=i(\infty)+[i(0)-i(\infty)] e^{-t / \tau}

where

i(0)=0, \quad i(\infty)=\frac{12}{150}=80 mA
\tau=\frac{L}{R}=\frac{30 \times 10^{-3}}{150}=0.2 ms

Thus,

i(t)=80\left[1-e^{-t / \tau}\right] mA

If i\left(t_{d}\right)=50 mA, then

50=80\left[1-e^{-t_{d} / \tau}\right] \quad \Longrightarrow \quad \frac{5}{8}=1-e^{-t_{d} / \tau}

or

e^{-t_{d} / \tau}=\frac{3}{8} \quad \Longrightarrow \quad e^{t_{d} / \tau}=\frac{8}{3}

By taking the natural logarithm of both sides, we get

t_{d}=\tau \ln \frac{8}{3}=0.2 \ln \frac{8}{3} ms =0.1962 ms

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