Find vc(t) for t>0 in the circuit in Fig. P 7.104. |...

Name: Problem 104, Chapter 7: Basic Engineering Circuit Analysis
Uploaded: 2023-10-22T19:03:29-08:00
Channel: Victor Salazar
Description: Find vc(t) for t>0 in the circuit in Fig. P 7.104.

Key Concepts

Solving First-Order Differential Equations

The behavior of an RC circuit is governed by a first-order differential equation that arises from applying Kirchhoff's voltage law. Solving this differential equation involves integrating factors or standard solution techniques for first-order linear equations to obtain an expression for the capacitor voltage as a function of time. This solution typically takes the form of an exponential function that characterizes the transient process in the circuit.

Initial and Steady-State Conditions

Analyzing the response of an RC circuit requires consideration of the initial conditions (the state of the capacitor just before the change) and the final steady-state conditions (the state of the circuit after a long time has passed). These conditions are essential for solving the differential equations, using methods such as the superposition of natural and forced responses, to accurately determine the voltage across the capacitor at any time.

RC Circuit Transient Response

In circuits containing a resistor and a capacitor (RC circuits), the transient response refers to the behavior of the system as it transitions from one steady state to another when a change, such as a switch operation or a sudden application of a voltage source, is introduced. The capacitor does not charge or discharge instantaneously; instead, its voltage changes gradually, following an exponential behavior that is derived from the underlying differential equation governing the circuit.

Time Constant

The time constant, typically denoted by ? (tau), is a key parameter in first?order RC circuits that quantifies the rate at which the circuit responds to changes. It is calculated as the product of the resistance and the capacitance (? = RC). The time constant indicates the time required for the capacitor voltage to reach approximately 63.2% of its final value during charging, or to decay to about 36.8% during discharging.

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