6.4 Energy Storage in Capacitors and Inductors

Energy storage in capacitors and inductors is a key concept in electrical circuits. These components act like tiny batteries, storing energy in electric and magnetic fields. Understanding how they work is crucial for designing power supplies, filters, and other essential circuits.

Capacitors store energy when voltage is applied, while inductors store it when current flows. This ability to store and release energy makes them vital in smoothing voltage and current fluctuations. Knowing how to calculate and manage this energy storage is essential for effective circuit design.

Energy storage in capacitors and inductors

Fundamental concepts of energy storage

Top images from around the web for Fundamental concepts of energy storage

• 

  19.5 Capacitors and Dielectrics – College Physics View original

  Is this image relevant?

• 

  Reactance, Inductive and Capacitive | Physics View original

  Is this image relevant?

• 

  Energy Stored in Capacitors | Physics View original

  Is this image relevant?

• 

  19.5 Capacitors and Dielectrics – College Physics View original

  Is this image relevant?

• 

  Reactance, Inductive and Capacitive | Physics View original

  Is this image relevant?

1 of 3

Top images from around the web for Fundamental concepts of energy storage

• 

  19.5 Capacitors and Dielectrics – College Physics View original

  Is this image relevant?

• 

  Reactance, Inductive and Capacitive | Physics View original

  Is this image relevant?

• 

  Energy Stored in Capacitors | Physics View original

  Is this image relevant?

• 

  19.5 Capacitors and Dielectrics – College Physics View original

  Is this image relevant?

• 

  Reactance, Inductive and Capacitive | Physics View original

  Is this image relevant?

1 of 3

• Capacitors store energy in the electric field created between their plates when a voltage applies
• Inductors store energy in the magnetic field generated by current flowing through their coils
• Energy storage becomes a reversible process allowing for storage and release
• Capacitors store energy when charging and release it when discharging
• Inductors store energy as current increases and release it as current decreases
• Amount of energy stored directly relates to capacitance and inductance values
• Energy storage underpins the operation of many electronic circuits and power systems (switched-mode power supplies, energy harvesting devices)
• Crucial for understanding oscillations in LC circuits and energy transfer in electrical systems (radio transmitters, voltage regulators)

Characteristics of energy storage

• Capacitor energy storage depends on applied voltage and capacitance
• Inductor energy storage depends on current flow and inductance
• Both components exhibit non-linear energy storage characteristics
• Capacitors have high energy density but lower power density compared to inductors
• Inductors have lower energy density but higher power density compared to capacitors
• Energy storage capacity affects the component's ability to smooth voltage or current fluctuations (power supply filtering, motor control)
• Temperature and frequency can impact the energy storage capabilities of both components

Equations for energy storage

Derivation of capacitor energy equation

• Energy stored in a capacitor derives from work done to charge it
• Integrate the voltage-charge relationship to obtain the energy equation
• Capacitor energy equation: $E = \frac{1}{2}CV^2$
  • E represents energy in joules (J)
  • C represents capacitance in farads (F)
  • V represents voltage across the capacitor in volts (V)
• Derivation involves calculus, specifically integrating power with respect to time
• Factor of 1/2 arises from the integration process
• Reflects the average energy stored during charging
• Demonstrates quadratic relationship between energy and voltage

Derivation of inductor energy equation

• Energy stored in an inductor derives from work done to establish current
• Integrate the voltage-current relationship to obtain the energy equation
• Inductor energy equation: $E = \frac{1}{2}LI^2$
  • E represents energy in joules (J)
  • L represents inductance in henries (H)
  • I represents current through the inductor in amperes (A)
• Derivation uses calculus, integrating power with respect to time
• Factor of 1/2 emerges from the integration process
• Reflects the average energy stored during current build-up
• Shows quadratic relationship between energy and current

Calculating energy storage

Capacitor energy calculations

• Use the equation $E = \frac{1}{2}CV^2$ for capacitor energy calculations
• Determine capacitance (C) and voltage (V) values before applying the equation
• Consider units of measurement
  • Energy in joules (J)
  • Capacitance in farads (F)
  • Voltage in volts (V)
• Apply unit conversion when necessary (microfarads to farads, kilovolts to volts)
• For multiple capacitors, calculate equivalent capacitance before determining total energy
• Account for dynamic energy changes in AC systems or during transient responses
• Example calculation: 10 µF capacitor charged to 50 V
  • $E = \frac{1}{2} \times (10 \times 10^{-6}) \times 50^2 = 12.5 \times 10^{-3} J = 12.5 mJ$

Inductor energy calculations

• Employ the equation $E = \frac{1}{2}LI^2$ for inductor energy calculations
• Identify inductance (L) and current (I) values before applying the equation
• Consider units of measurement
  • Energy in joules (J)
  • Inductance in henries (H)
  • Current in amperes (A)
• Perform unit conversion when needed (millihenries to henries, milliamperes to amperes)
• For circuits with multiple inductors, determine equivalent inductance before calculating total energy
• Account for dynamic energy changes in AC systems or during transient responses
• Example calculation: 5 mH inductor with 2 A current
  • $E = \frac{1}{2} \times (5 \times 10^{-3}) \times 2^2 = 10 \times 10^{-3} J = 10 mJ$

Energy transfer in LC circuits

LC circuit fundamentals

• LC circuits consist of an inductor (L) and capacitor (C) connected in parallel or series
• Allow for energy oscillation between the two components
• Total energy in an ideal LC circuit remains constant
• Energy alternates between capacitor's electric field and inductor's magnetic field
• Resonant frequency determines oscillation frequency: $f = \frac{1}{2\pi\sqrt{LC}}$
• Energy transfer occurs at twice the frequency of current or voltage oscillations
• Maximum energy in capacitor occurs when inductor current reaches zero, and vice versa

Energy transfer dynamics

• Energy continuously transfers between capacitor and inductor in sinusoidal pattern
• At t=0, capacitor holds maximum energy, inductor energy is zero
• As capacitor discharges, energy transfers to inductor's magnetic field
• When capacitor fully discharges, inductor holds maximum energy
• Process reverses as inductor's field collapses, transferring energy back to capacitor
• Cycle repeats indefinitely in ideal LC circuit (no resistance)
• Real LC circuits experience damping due to resistive losses, causing gradual decrease in total energy
• Understanding energy transfer crucial for analyzing filters, oscillators, and resonant circuit applications (radio tuning circuits, wireless power transfer systems)