Figure 25.40 on the next page shows the voltage across a...

Figure 25.40 on the next page shows the voltage across a capacitor that's charging through a $4700-\Omega$ resistor in the circuit of Fig. 25.18. Use the graph to determine (a) the battery voltage, (b) the time constant, and (c) the capacitance.
(FIGURE CANT COPY)

Key Concepts

RC Charging Equation

The RC charging equation, V(t) = V?(1 - exp(-t/(RC))), describes how the voltage across a capacitor in an RC circuit increases over time when connected to a constant voltage source. This equation provides a mathematical model for the transient behavior of the circuit, allowing one to relate the time t, the battery voltage V?, the resistance R, and the capacitance C.

Time Constant

The time constant, represented by ? (tau) and defined as ? = RC, is a crucial parameter in RC circuits. It quantifies the time required for the capacitor to charge up to approximately 63.2% of its final voltage. This value is fundamental in assessing the speed of the circuit's response to changes and is commonly extracted from graph analysis of the charging curve.

Capacitance

Capacitance is a measure of a capacitor's capacity to store electrical charge per unit voltage. In the context of an RC circuit, knowing the time constant and the resistance allows for the calculation of capacitance using the formula C = ?/R. This relationship is essential for understanding the energy storage and timing behavior in the circuit.

Graphical Analysis of Transient Response

Graphical analysis involves interpreting the voltage versus time plot of a charging capacitor. By identifying the asymptotic voltage level (which indicates the battery voltage) and measuring the time at which the capacitor's voltage reaches 63.2% of this level (indicating the time constant), one can extract key circuit parameters. This analytical method is fundamental in determining circuit behavior without needing to solve the differential equations directly.

Transcript

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