Lec 06 - Capacitors

Capacitors

Overview

Capacitors are essential energy storage devices used in electrical circuits, integral to various applications from simple timing circuits to complex power supply systems. They store electrical energy in an electric field, formed between two conductive plates separated by an insulating material known as the dielectric. The voltage across a capacitor changes at a rate governed by differential equations, reflecting its charge/discharge properties over time.

Capacitance

• Definition: Capacitance is defined as the amount of electric charge (Q) that a capacitor can store for a given voltage (V) across its plates, indicating its ability to store energy.

• Mathematical Relationship: This relationship is expressed by the formula:

  [ C = \frac{Q}{V} \quad [F] ]where C represents capacitance measured in Farads (F). The higher the capacitance, the greater the charge a capacitor can store per volt.

• Unit of Capacitance: The Farad is a large unit; in practice, you might encounter microfarads (µF) and picofarads (pF) for smaller capacitors.

Capacitor Charging and Discharging

Energy Storage

• Energy Formula: The energy (W) stored in a capacitor depends on both its capacitance and the voltage across it, expressed as:

  [ W = \frac{1}{2} C V^2 \quad [J] ]where W represents energy measured in Joules (J).

• Electric Field Strength: The strength of the electric field (E) between the plates is influenced by the voltage applied and the distance separating the plates:

  [ E = \frac{V}{d} \quad [V/m] ]Here, E is the electric field strength in volts per meter, V is the applied voltage, and d is the separation distance between the plates.

Physical Parameters Affecting Capacitance

The capacitance (C) of a capacitor is influenced by several physical parameters:

• Area of Plates (A): An increase in plate area leads to a higher capacitance due to more surface area available for storing charge.

• Distance Between Plates (d): A greater separation distance diminishes capacitance since it weakens the electric field strength.

• Permittivity of Material (ε): The dielectric material between the plates affects capacitance, with different materials having distinct permittivity values that impact charge storage capabilities.

• Capacitance Formula: The capacitance can be calculated using:

  [ C = \frac{\epsilon A}{d} \quad [F/m] ]

Capacitor Configurations

Series Configuration

• Voltage Distribution: In a series configuration, the total voltage across the capacitors is:

  [ V_C = V_1 + V_2 + ... + V_n ]Each capacitor experiences different voltage drops depending on their capacitance values.

• Charge Consistency: The total charge (Q) remains the same throughout the series:

  [ Q = Q_1 = Q_2 = ... = Q_n ]

• Total Capacitance in Series: The total capacitance can be calculated using:

  [ \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n} ]

Parallel Configuration

• Voltage Alignment: All capacitors in parallel share the same voltage across their terminals:

  [ V = V_1 = V_2 = ... = V_n ]

• Charge Addition: The total charge stored is the sum of individual charges:

  [ Q = Q_1 + Q_2 + ... + Q_n ]

• Total Capacitance in Parallel: The total capacitance is the sum of the individual capacitances:

  [ C = C_1 + C_2 + ... + C_n ]

Circuit Analysis of Capacitors

Charging Phase

• Voltage Equation: During the charging phase, the voltage across the capacitor can be described by:

[ v_C(t) = V_C (1 - e^{-t/\tau}) ]where τ (tau) represents the time constant defined as: [ \tau = RC ]with R being the resistance in the circuit.

• Current Equation: The current during this phase diminishes exponentially as: [ i(t) = \frac{V_C}{R} e^{-t/\tau} ]

Discharging Phase

• Voltage Decay: During the discharging phase, the voltage across the capacitor decreases according to the equation:

  [ v_C(t) = V_C e^{-t/\tau} ]

• Current Equation: The current during discharging can be expressed as: [ i(t) = -\frac{V_C}{R} e^{-t/\tau} ]which indicates that current flows in the opposite direction compared to charging.

Important Conclusions

• Time Constant (τ): Determines how quickly a capacitor charges or discharges, impacting circuit performance. It's essential for timing applications to ensure that the duration for charging/discharging is at least five times the time constant (5τ).

• Frequency Dependency: Capacitors are frequency-dependent components; the critical frequency can be given by:

  [ f_c = \frac{1}{2\pi\tau} ]

• Equivalence Circuits: During certain phases of the operation, capacitors can be represented as short circuits (fully charged) or open circuits (fully discharged), greatly simplifying circuit analysis.

Summary of Key Formulae

Charging:

• Voltage: [ v_C(t) = V_C(1 - e^{-t/\tau}) ]

• Current: [ i(t) = \frac{V_C}{R} e^{-t/\tau} ]

Discharging:

• Voltage: [ v_C(t) = V_C e^{-t/\tau} ]

• Current: [ i(t) = -\frac{V_C}{R} e^{-t/\tau} ]

Time Constant:

• [ \tau = R C ]