6.3 Capacitance, inductance, and transient response

Capacitors and inductors are key components in electrical circuits, storing energy in electric and magnetic fields. They shape how circuits respond to changes, affecting voltage and current over time. Understanding their behavior is crucial for designing and analyzing various electrical systems.

RC and RL circuits show how capacitors and inductors interact with resistors, creating unique voltage and current patterns. These circuits have wide-ranging applications, from power supplies to signal processing, making them essential knowledge for electrical engineers.

Capacitance and Inductance in Circuits

Fundamental Concepts and Definitions

Top images from around the web for Fundamental Concepts and Definitions

• 

  RLC Series AC Circuits | Physics View original

  Is this image relevant?

• 

  Inductance | Physics View original

  Is this image relevant?

• 

  Reactance, Inductive and Capacitive · Physics View original

  Is this image relevant?

• 

  RLC Series AC Circuits | Physics View original

  Is this image relevant?

• 

  Inductance | Physics View original

  Is this image relevant?

1 of 3

Top images from around the web for Fundamental Concepts and Definitions

• 

  RLC Series AC Circuits | Physics View original

  Is this image relevant?

• 

  Inductance | Physics View original

  Is this image relevant?

• 

  Reactance, Inductive and Capacitive · Physics View original

  Is this image relevant?

• 

  RLC Series AC Circuits | Physics View original

  Is this image relevant?

• 

  Inductance | Physics View original

  Is this image relevant?

1 of 3

• Capacitance measures a component's ability to store electric charge $C = \frac{Q}{V}$ where C is capacitance in farads (F), Q is charge in coulombs, and V is voltage
• Inductance quantifies a conductor's opposition to current changes $L = \frac{V}{\frac{dI}{dt}}$ where L is inductance in henries (H), V is induced voltage, and dI/dt is rate of current change
• Capacitors store energy in electric fields between two conductive plates separated by a dielectric material (ceramic, plastic)
• Inductors store energy in magnetic fields, typically using coiled wire around a core (air, ferrite)

Circuit Behavior and Calculations

• DC circuits: capacitors act as open circuits, inductors as short circuits in steady state
• AC circuits: capacitors and inductors exhibit frequency-dependent reactance
• Capacitors in series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ (inverse of parallel resistors)
• Capacitors in parallel: $C_{eq} = C_1 + C_2 + ...$ (similar to parallel resistors)
• Inductors in series: $L_{eq} = L_1 + L_2 + ...$ (similar to series resistors)
• Inductors in parallel: $\frac{1}{L_{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + ...$ (inverse of series resistors)

RC and RL Circuit Behavior

RC Circuit Dynamics

• RC circuits exhibit exponential voltage changes across the capacitor during charging and discharging
• Charging process: gradual accumulation of charge on capacitor plates (electron buildup)
• Discharging process: release of stored charge through the resistor (electron flow)
• Voltage equation for charging: $V_C(t) = V_s(1 - e^{-t/RC})$ where Vs is source voltage
• Voltage equation for discharging: $V_C(t) = V_0e^{-t/RC}$ where V0 is initial capacitor voltage

RL Circuit Dynamics

• RL circuits demonstrate exponential current changes through the inductor during energizing and de-energizing
• Energizing process: current buildup in inductor, creating magnetic field (energy storage)
• De-energizing process: collapse of magnetic field, inducing current flow (energy release)
• Current equation for energizing: $I_L(t) = \frac{V_s}{R}(1 - e^{-Rt/L})$ where Vs is source voltage
• Current equation for de-energizing: $I_L(t) = I_0e^{-Rt/L}$ where I0 is initial inductor current

Time Constants and Transient Response

Time Constant Calculations

• RC circuit time constant: $\tau = RC$ (resistance in ohms, capacitance in farads)
• RL circuit time constant: $\tau = L/R$ (inductance in henries, resistance in ohms)
• One time constant represents 63.2% of total change in circuit response
• Two time constants: 86.5% of total change
• Three time constants: 95% of total change
• Settling time: 4-5 time constants (circuit reaches within 2% of final value)

Transient Response Analysis

• First-order circuit voltage response: $v(t) = V_f + (V_i - V_f)e^{-t/\tau}$
• First-order circuit current response: $i(t) = I_f + (I_i - I_f)e^{-t/\tau}$
• Vi and Ii represent initial values, Vf and If are final (steady-state) values
• Natural response: circuit behavior without external sources (e.g., discharging capacitor)
• Forced response: circuit behavior due to applied sources (e.g., charging capacitor)
• Complete response: sum of natural and forced responses

Energy Storage in Capacitors and Inductors

Energy Calculations and Principles

• Capacitor energy storage: $E = \frac{1}{2}CV^2$ (C in farads, V in volts)
• Inductor energy storage: $E = \frac{1}{2}LI^2$ (L in henries, I in amperes)
• Instantaneous power: $P = VI$ (useful for analyzing energy transfer rates)
• Energy conservation applies during charging/discharging (source, component, resistor)
• Maximum energy storage limited by breakdown voltage (capacitors) and core saturation (inductors)

Practical Applications

• Power supply filtering: smoothing voltage fluctuations (capacitors)
• Energy harvesting: storing small amounts of energy from ambient sources (both)
• Electromagnetic pulse protection: absorbing sudden energy spikes (both)
• Voltage regulation: maintaining stable output voltages (capacitors)
• Motor starting: providing initial current surge (capacitors)
• Inductive heating: generating heat through magnetic field oscillations (inductors)
• Wireless power transfer: transmitting energy through magnetic coupling (inductors)