If, in Fig. 34-2, R=20 Ω, L=0.30 H, and ℰ=90 V, what will...

If, in Fig. $34-2, R=20 \Omega, L=0.30 \mathrm{H}$, and $\mathscr{E}=90 \mathrm{~V}$, what will be the current in the circuit $0.050 \mathrm{~s}$ after the switch is closed? We are going to use the exponential equation for $i$ given on p. 374 . The time constant for this circuit is $L / R=0.015 \mathrm{~s}$, and $i_{\infty}=\mathscr{/} R=4.5 \mathrm{~A}$. Then $$ i=i_{\infty}\left(1-e^{-t / L / R}\right)=(4.5 \mathrm{~A})\left(1-e^{-3.33}\right)=(4.5 \mathrm{~A})(1-0.0357)=4.3 \mathrm{~A} $$

If, in Fig. $34-2, R=20 \Omega, L=0.30 \mathrm{H}$, and $\mathscr{E}=90 \mathrm{~V}$, what will be the current in the circuit $0.050 \mathrm{~s}$ after the switch is closed?
We are going to use the exponential equation for $i$ given on p. 374 .
The time constant for this circuit is $L / R=0.015 \mathrm{~s}$, and $i_{\infty}=\mathscr{/} R=4.5 \mathrm{~A}$. Then
$$
i=i_{\infty}\left(1-e^{-t / L / R}\right)=(4.5 \mathrm{~A})\left(1-e^{-3.33}\right)=(4.5 \mathrm{~A})(1-0.0357)=4.3 \mathrm{~A}
$$

Key Concepts

Steady-State Current

The steady-state current, represented as i?, is the maximum current that flows through the circuit once all transient effects have decayed. In an RL circuit, this is determined by Ohm's law as i? = V/R, where V is the applied voltage. This current is reached as time approaches infinity, after the exponential term in the transient response has effectively diminished.

Exponential Growth in Transient Analysis

The transient response of an RL circuit is modeled by an exponential function, where the current increases over time as i = i?(1 - e^(-t/?)). This equation captures the gradual transition from zero current at t = 0 to the steady-state current, encapsulating the dynamic behavior due to the inductor's opposition to changes in current.

RL Circuit

An RL circuit is an electric circuit that contains a resistor (R) and an inductor (L). In such circuits, the inductor resists changes in current, causing the current to increase gradually rather than instantaneously. This property is fundamental in understanding how circuits respond when a voltage is applied, particularly during the turn-on transient period.

Time Constant

The time constant, denoted as ? and calculated as L/R, is a measure of how quickly the current in an RL circuit reaches its steady state. It characterizes the rate of exponential growth of the current, indicating that after a duration of one time constant, the current reaches approximately 63.2% of its final value.

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