The inductor L in the circuit shown in Figure P5. 16 is the...

The inductor $L$ in the circuit shown in Figure P5. 16 is the coil of a relay. When the current through the coil is equal to or greater than +2 mA the relay functions. Assume steady-state conditions at $t<0$. If $$ \begin{aligned} & V_s=12 \mathrm{~V} \\ & L=10.9 \mathrm{mH} \quad R_1=3.1 \mathrm{k} \Omega \end{aligned} $$ determine $R_2$ so that the relay functions at $t=2.3 \mathrm{~s}$.

The inductor $L$ in the circuit shown in Figure P5. 16 is the coil of a relay. When the current through the coil is equal to or greater than +2 mA the relay functions. Assume steady-state conditions at $t<0$. If

$$
\begin{aligned}
& V_s=12 \mathrm{~V} \\
& L=10.9 \mathrm{mH} \quad R_1=3.1 \mathrm{k} \Omega
\end{aligned}
$$

determine $R_2$ so that the relay functions at $t=2.3 \mathrm{~s}$.

Key Concepts

Steady-State versus Transient Conditions

Understanding the difference between steady-state and transient conditions is essential in circuit analysis. Steady-state refers to the condition where all transient effects have died out and the circuit variables remain constant over time, whereas transient conditions describe the period immediately following a change in the circuit, during which variables evolve exponentially toward their new steady values.

RL Circuit Transient Analysis

This concept involves the study of how current and voltage change over time in a circuit containing resistors and inductors when the circuit conditions change. The analysis is based on differential equations that describe the behavior of inductors, whose current cannot change instantaneously, leading to an exponential response during the transient state.

Time Constant in RL Circuits

The time constant, often denoted by ? and calculated as the ratio of inductance to resistance (? = L/R), is a crucial parameter in determining how quickly an RL circuit responds to changes. It characterizes the rate at which current builds up or decays, with the circuit typically reaching about 63% of its final value after one time constant.

Relay Operation in Circuits

Relays are electromechanical switches that require a certain threshold current through their coil to activate. The analysis of relay circuits often involves ensuring that the transient and steady-state current conditions are sufficient to trigger the relay, making it important to understand both the time-dependent behavior of the coil and the underlying circuit design principles.

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