In Fig. 27.26, take ℰ0=12 V, R=2.7 Ω, and L=20 H . Initially the switch is in position B and the

step 1 Okay, so we know the basic equation for RL circuit, RLC series circuit, current is just EMF over

In Fig. 27.26, take ℰ0=12 V, R=2.7 Ω, and L=20 H . Initially...

In Fig. $27.26,$ take $\mathcal{E}_{0}=12 \mathrm{V}, R=2.7 \Omega,$ and $L=20 \mathrm{H} .$ Initially the switch is in position $B$ and there's no current anywhere. At $t=0$ the switch is thrown to position $A,$ and at $t=10 \mathrm{s}$ it's returned to $B$. Find the inductor current at (a) $t=5.0 \mathrm{s}$ and (b) $t=15 \mathrm{s}.$

Key Concepts

Time Constant

The time constant in an LR circuit, defined as ? = L/R, is a measure of how quickly the circuit responds to changes. It represents the time required for the current to reach approximately 63% of its final value during a rising transient or to decay to about 37% during a falling transient. This parameter is fundamental in predicting the dynamic behavior of inductive circuits.

Inductor Current Continuity

A fundamental property of inductors is that their current must remain continuous through any switching event. This means that the current just before the switching moment is the same as immediately afterwards. This principle is critical for setting up the initial conditions required to solve the transient response in circuits containing inductors.

Switching Dynamics

Switching dynamics refer to the changes in circuit behavior when a switch alters the circuit’s topology. In circuits with inductors, switching events are managed by the principle that the inductor current cannot change abruptly. Analyzing these dynamics involves considering both the initial conditions just before switching and the new circuit configuration immediately after the switch is thrown.

LR Circuit Analysis

This concept involves studying circuits that include both an inductor and a resistor. In these circuits, the interplay between the inductor’s tendency to resist changes in current and the resistor’s dissipative effects leads to a transient response that is mathematically modeled by exponential functions. Analyzing LR circuits is essential for understanding how current builds up or decays over time when a voltage is applied or removed.

Exponential Transient Response

When a switch in an LR circuit is altered, the current does not change instantaneously but follows an exponential behavior. This response can be characterized as an exponential rise towards a steady-state value when a voltage is applied or an exponential decay when the circuit configuration forces the current to decrease. Understanding this behavior is key to analyzing how circuits evolve from one steady state to another.

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