91Ó°ÊÓ

The LR time constant, symbolized as \( \tau \), is a pivotal concept in analyzing how circuits with inductors and resistors behave over time. It represents the time it takes for the current to reach approximately 63.2% of its maximum value after a voltage is suddenly applied across an RLC circuit.

To calculate the time constant, you use the formula \( \tau = \frac{L}{R} \), where \( L \) is the inductance in henrys, and \( R \) is the resistance in ohms. In our case, with \( L = 1.25 \, \text{mH} \) and \( R = 50.0 \, \Omega \) the time constant is \( \tau = \frac{1.25 \times 10^{-3} \, \text{H}}{50.0 \, \Omega} \) which equals \( 0.025 \, \text{ms} \).

Understanding the time constant is crucial because it helps predict how fast the current will rise to its peak value. A smaller time constant means the circuit reaches steady state more quickly.

Inductor current growth in an RLC circuit is not instantaneous; instead, it follows an exponential pattern. When you close the switch in our exercise, the current does not instantly reach its maximum value. Instead, it gradually increases according to the formula \( I(t) = I_{\text{max}}(1 - e^{-t/\tau}) \), where \( I(t) \) is the current at time \( t \) and \( I_{\text{max}} \) is the maximum possible current once the circuit is at steady state.

For our exercise, with a circuit voltage of \( V = 35.0 \, \text{V} \) and resistance of \( R = 50.0 \, \Omega \) the maximum current \( I_{\text{max}} \) is calculated by \( V/R \), giving us \( 0.7 \, \text{A} \). To find when the current is half of its maximum, we set \( I(t) \) to \( 0.5 \times I_{\text{max}} \) and solve for \( t \) resulting in \( t = -\tau \ln(0.5) \) which approximates to \( 0.018 \, \text{ms} \) for our circuit.

Inductors store energy in the form of a magnetic field when a current flows through them. The energy stored in an inductor can be expressed using the equation \( U(t) = 0.5L[I(t)]^2 \), where \( U(t) \) is the energy at time \( t \) and \( L \) is the inductance. In relation to our original problem, the maximum stored energy \( U_{\text{max}} \) is \( 0.5L[I_{\text{max}}]^2 \).

To find out when the energy is at half its maximum value, we set \( U(t) \) to \( 0.5 \times U_{\text{max}} \) and solve for the time \( t \) which follows an exponential increase similar to the current. Following the calculation, we derive that \( t = -\frac{\tau}{2} \ln(0.5) \) which results in approximately \( 0.013 \, \text{ms} \) for our example.

The exponential transient response of an RLC circuit refers to how the variables such as current and energy change over time from the moment a voltage is applied or changed. This response can be characterized by an exponential curve which initially rises or falls steeply, then levels off as it approaches a steady state.

The formulas such as \( I(t) = I_{\text{max}}(1 - e^{-t/\tau}) \) for current and \( U(t) = 0.5L[I(t)]^2 \) for energy display this transient behavior. When facing problems like ours, it is important to understand that all transient phenomena will follow this pattern, and that the key to solving for particular time points involves being comfortable with manipulating exponential functions.

In both cases of current and energy for our circuit, we dealt with steps that involved the natural logarithm function \( \ln \) and the time constant \( \tau \) to solve for the time at which the system reaches a specific percentage of its maximum stable value.