⚡Electrical Circuits and Systems I Unit 6 – Capacitors and Inductors in Circuit Analysis

Key Concepts and Definitions

• Capacitance $(C)$ represents the ability of a capacitor to store electric charge, measured in farads $(F)$
• Inductance $(L)$ represents the ability of an inductor to store energy in a magnetic field, measured in henries $(H)$
• Time constant $(\tau)$ characterizes the response of a capacitor or inductor in a circuit, equal to $RC$ for capacitors and $L/R$ for inductors
• Transient response refers to the behavior of a circuit during the time it takes to reach steady-state after a change in the circuit
• Impedance $(Z)$ represents the total opposition to current flow in an AC circuit, consisting of resistance and reactance
  • Capacitive reactance $(X_C)$ opposes changes in voltage, calculated as $X_C = 1/(2\pi fC)$
  • Inductive reactance $(X_L)$ opposes changes in current, calculated as $X_L = 2\pi fL$
• Resonance occurs when the capacitive and inductive reactances are equal, resulting in a purely resistive circuit
• Quality factor $(Q)$ measures the ratio of energy stored to energy dissipated per cycle in a resonant circuit

Capacitor Basics and Behavior

• Capacitors consist of two conductive plates separated by an insulating material called a dielectric
• The capacitance of a parallel plate capacitor is directly proportional to the area of the plates and inversely proportional to the distance between them
• Capacitors oppose changes in voltage across their terminals, causing a delay in voltage changes
• The current through a capacitor is proportional to the rate of change of the voltage across it, given by $i_C = C \frac{dv_C}{dt}$
• Capacitors in series have an equivalent capacitance that is less than any individual capacitance, calculated as $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$
• Capacitors in parallel have an equivalent capacitance that is the sum of the individual capacitances, calculated as $C_{eq} = C_1 + C_2 + ... + C_n$
• The voltage across a capacitor cannot change instantaneously, as it takes time to charge or discharge the capacitor

Inductor Fundamentals

• Inductors are passive components that store energy in a magnetic field when current flows through them
• The inductance of an inductor is directly proportional to the number of turns, the cross-sectional area, and the permeability of the core material, and inversely proportional to the length of the coil
• Inductors oppose changes in current through them, causing a delay in current changes
• The voltage across an inductor is proportional to the rate of change of the current through it, given by $v_L = L \frac{di_L}{dt}$
• Inductors in series have an equivalent inductance that is the sum of the individual inductances, calculated as $L_{eq} = L_1 + L_2 + ... + L_n$
• Inductors in parallel have an equivalent inductance that is less than any individual inductance, calculated as $\frac{1}{L_{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + ... + \frac{1}{L_n}$
• The current through an inductor cannot change instantaneously, as it takes time to build up or collapse the magnetic field

Charging and Discharging Processes

• Capacitor charging occurs when a voltage is applied across its terminals, causing current to flow and charge to accumulate on the plates
  • The voltage across the capacitor increases exponentially during charging, approaching the applied voltage
  • The current through the capacitor decreases exponentially during charging, starting at a maximum value and approaching zero
• Capacitor discharging occurs when a charged capacitor is connected to a load, allowing the stored charge to flow out of the capacitor
  • The voltage across the capacitor decreases exponentially during discharging, starting at the initial voltage and approaching zero
  • The current through the capacitor decreases exponentially during discharging, starting at a maximum value and approaching zero
• Inductor energizing occurs when a current is applied to an inductor, causing a magnetic field to build up around the coil
  • The current through the inductor increases exponentially during energizing, approaching the steady-state current
  • The voltage across the inductor decreases exponentially during energizing, starting at a maximum value and approaching zero
• Inductor de-energizing occurs when the current through an inductor is interrupted, causing the magnetic field to collapse
  • The current through the inductor decreases exponentially during de-energizing, starting at the initial current and approaching zero
  • The voltage across the inductor increases exponentially during de-energizing, starting at zero and reaching a maximum value before decreasing

Time Constants and Transient Analysis

• The time constant $(\tau)$ determines the rate at which a capacitor charges or discharges, or an inductor energizes or de-energizes
  • For a capacitor, $\tau = RC$, where $R$ is the resistance in series with the capacitor
  • For an inductor, $\tau = L/R$, where $R$ is the resistance in series with the inductor
• The time constant represents the time it takes for the voltage or current to reach approximately 63.2% of its final value during charging or discharging
• After one time constant, the capacitor voltage or inductor current reaches 63.2% of its final value
• After five time constants, the capacitor voltage or inductor current reaches approximately 99.3% of its final value, considered the steady-state value
• Transient analysis involves examining the behavior of a circuit during the time it takes to reach steady-state after a change in the circuit
  • This can include analyzing the voltage and current waveforms during capacitor charging/discharging or inductor energizing/de-energizing
• The transient response of a circuit can be described using differential equations, which can be solved using techniques such as Laplace transforms or numerical methods

Energy Storage in Capacitors and Inductors

• Capacitors store energy in the electric field between their plates, with the energy given by $E_C = \frac{1}{2}CV^2$
  • The energy stored is proportional to the capacitance and the square of the voltage across the capacitor
• Inductors store energy in the magnetic field around the coil, with the energy given by $E_L = \frac{1}{2}LI^2$
  • The energy stored is proportional to the inductance and the square of the current through the inductor
• The energy stored in a capacitor or inductor can be released back into the circuit when the component discharges or de-energizes
• The power dissipated in a capacitor or inductor is zero under steady-state conditions, as the voltage and current are 90° out of phase
• In a resonant circuit, energy is continuously transferred between the capacitor and inductor, with the total energy remaining constant if there are no losses
• The maximum energy storage capacity of a capacitor or inductor is limited by factors such as the breakdown voltage of the dielectric, the saturation of the magnetic core, and the power dissipation of the component

AC Circuit Analysis with Capacitors and Inductors

• In AC circuits, capacitors and inductors introduce frequency-dependent impedances that affect the circuit's behavior
• Capacitive reactance $(X_C)$ decreases with increasing frequency, as the capacitor allows more current to flow at higher frequencies
  • At high frequencies, a capacitor acts like a short circuit, allowing AC current to pass through easily
  • At low frequencies, a capacitor acts like an open circuit, blocking AC current
• Inductive reactance $(X_L)$ increases with increasing frequency, as the inductor opposes changes in current more strongly at higher frequencies
  • At high frequencies, an inductor acts like an open circuit, blocking AC current
  • At low frequencies, an inductor acts like a short circuit, allowing AC current to pass through easily
• The total impedance $(Z)$ in an AC circuit with capacitors and inductors is the vector sum of the resistance and the reactances, given by $Z = \sqrt{R^2 + (X_L - X_C)^2}$
• The phase angle $(\phi)$ between the voltage and current in an AC circuit with capacitors and inductors is determined by the relative magnitudes of the resistance and reactances, given by $\phi = \tan^{-1}(\frac{X_L - X_C}{R})$
• Resonance occurs in an AC circuit when the capacitive and inductive reactances are equal, resulting in a purely resistive circuit with a minimum impedance and maximum current

Practical Applications and Circuit Examples

• Power factor correction: Capacitors are used to improve the power factor in AC circuits by compensating for the inductive load, reducing the phase angle between voltage and current
• Filters: Capacitors and inductors are used in various filter circuits to selectively pass or block certain frequencies
  • Low-pass filters (LPFs) allow low frequencies to pass through while attenuating high frequencies, using a series inductor and a shunt capacitor
  • High-pass filters (HPFs) allow high frequencies to pass through while attenuating low frequencies, using a series capacitor and a shunt inductor
  • Band-pass filters (BPFs) allow a specific range of frequencies to pass through while attenuating frequencies outside that range, using a combination of LPFs and HPFs
• Resonant circuits: Capacitors and inductors are used in resonant circuits to create oscillators, amplifiers, and tuned filters
  • LC tank circuits are used in oscillators to generate sinusoidal signals at a specific resonant frequency
  • Tuned amplifiers use resonant circuits to selectively amplify signals at a desired frequency while rejecting signals at other frequencies
• Switching power supplies: Inductors are used in switching power supplies to store energy and smooth the output voltage, while capacitors are used to filter the output and reduce ripple
• Electromagnetic interference (EMI) suppression: Capacitors and inductors are used in EMI suppression circuits to reduce the emission and susceptibility of electronic devices to electromagnetic interference
  • Decoupling capacitors are placed close to integrated circuits to provide a local, low-impedance source of power and to filter out high-frequency noise
  • EMI filters use a combination of capacitors and inductors to attenuate high-frequency noise and prevent it from entering or leaving the device