Capacitor Charging and Behavior: Understanding Time Constants, Reactance, and More

Understanding Capacitor Behavior in DC and AC Circuits

This comprehensive guide explores key concepts related to capacitors, including:

• Charging Time: Discover how long it takes a capacitor to charge to a certain percentage of its final charge when connected to a DC voltage source through a resistor.* Voltage and Current Characteristics: Understand the shape of the voltage curve across a resistor in a series RC circuit when connected to a DC source.* Capacitive Reactance: Learn how to calculate the opposition a capacitor offers to alternating current (AC) at a specific frequency.* Phase Relationship: Explore the phase relationship between voltage and current when a sinusoidal AC voltage is applied to a capacitor. * Switched Capacitors: Discover how switched capacitors can emulate the behavior of larger capacitors.

Let's dive into the details:

Q7: When a capacitor is connected through a resistor to a DC voltage source, the charge on the capacitor will reach 50% of its final charge in:

a. less than one time constant

The time constant (τ = RC) determines the charging rate of a capacitor. It takes approximately one time constant for the capacitor to charge to about 63.2% of its final value. Therefore, it reaches 50% charge in less than one time constant.

Q8: When a capacitor is connected through a series resistor and switch to a DC voltage source, the voltage across the resistor after the switch is closed has the shape of:

c. a falling exponential

Initially, the capacitor acts like a short circuit, and the voltage across the resistor is equal to the source voltage. As the capacitor charges, the voltage across it increases, causing the voltage across the resistor to decrease exponentially.

Q9: The capacitive reactance of a 100 mF capacitor at 60 Hz is approximately:

c. 37.7 Ω

Capacitive reactance (Xc) is calculated using the formula: Xc = 1 / (2πfC), where f is the frequency and C is the capacitance.

Q10: If a sine wave from a function generator is applied to a capacitor, the current will:

b. lag voltage by 45°

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees. However, in practical circuits, there is always some resistance, which reduces the phase difference to less than 90 degrees.

Q11: A switched capacitor emulates a:

b. larger capacitor

By rapidly charging and discharging a smaller capacitor using switches, it is possible to simulate the behavior of a larger capacitor with a higher capacitance value.

This information provides a fundamental understanding of capacitor behavior in both DC and AC circuits.