⚡️College Physics III – Thermodynamics, Electricity, and Magnetism Unit 8 – Capacitance in Electrical Systems

Key Concepts and Definitions

• Capacitance measures a capacitor's ability to store electric charge, denoted by the symbol $C$ and measured in farads ($F$)
• Electric field intensity $E$ represents the force per unit charge experienced by a test charge in an electric field, measured in newtons per coulomb ($N/C$) or volts per meter ($V/m$)
• Electric potential difference $\Delta V$ is the work done per unit charge to move a positive test charge from one point to another in an electric field, measured in volts ($V$)
• Dielectric constant $\kappa$ (also known as relative permittivity) compares the permittivity of a material to that of vacuum, influencing the capacitance of a capacitor
• Dielectric strength refers to the maximum electric field a dielectric material can withstand before breakdown occurs, measured in volts per meter ($V/m$)
• Parallel plate capacitor consists of two parallel conducting plates separated by a dielectric material or vacuum, with capacitance given by $C = \frac{\kappa \varepsilon_0 A}{d}$
• Capacitive reactance $X_C$ represents the opposition to the flow of alternating current (AC) in a capacitor, measured in ohms ($\Omega$) and calculated as $X_C = \frac{1}{2\pi fC}$

Capacitors: Structure and Function

• Capacitors store electric charge and energy in an electric field between two conducting plates
• Consist of two conducting plates separated by an insulating material called a dielectric
• Plates accumulate equal and opposite charges when connected to a voltage source, creating an electric field between them
• Capacitance depends on the area of the plates, the distance between them, and the dielectric constant of the insulating material
• Act as open circuits to direct current (DC) and allow alternating current (AC) to pass through, making them useful for filtering and signal processing
• Prevent sudden changes in voltage across their terminals, providing a smoothing effect in power supplies
• Used in various applications such as energy storage, signal filtering, and voltage regulation

Types of Capacitors

• Ceramic capacitors use ceramic dielectric materials, offer high dielectric constants, and are suitable for high-frequency applications
• Electrolytic capacitors (aluminum and tantalum) provide high capacitance values in a compact size but are polarized and have lower voltage ratings
  • Aluminum electrolytic capacitors are commonly used in power supplies and filtering applications
  • Tantalum capacitors offer higher capacitance density and better temperature stability compared to aluminum electrolytic capacitors
• Film capacitors (polyester, polypropylene, and polystyrene) have low losses and are suitable for high-frequency and high-voltage applications
• Mica capacitors offer high stability, low losses, and high voltage ratings, making them suitable for high-frequency and high-voltage applications
• Variable capacitors (air or vacuum dielectric) allow for adjustable capacitance, used in radio and television tuning circuits
• Supercapacitors (also known as ultracapacitors) store large amounts of energy and have high power density, used in energy storage and power backup systems

Capacitance Calculations

• Capacitance $C$ is the ratio of the charge $Q$ stored on each plate to the potential difference $\Delta V$ between the plates, given by $C = \frac{Q}{\Delta V}$
• For a parallel plate capacitor, capacitance is calculated using the formula $C = \frac{\kappa \varepsilon_0 A}{d}$, where $\kappa$ is the dielectric constant, $\varepsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} F/m$), $A$ is the area of the plates, and $d$ is the distance between the plates
• Capacitors in series have an equivalent capacitance $C_{eq}$ given by $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$
  • Voltage across each capacitor in series is inversely proportional to its capacitance
  • Total voltage across the series combination equals the sum of the voltages across individual capacitors
• Capacitors in parallel have an equivalent capacitance $C_{eq}$ given by $C_{eq} = C_1 + C_2 + ... + C_n$
  • Voltage across each capacitor in parallel is the same
  • Charge stored in the parallel combination equals the sum of the charges stored in individual capacitors
• Capacitive reactance $X_C$ in an AC circuit is calculated using the formula $X_C = \frac{1}{2\pi fC}$, where $f$ is the frequency of the AC signal

Capacitors in Circuits

• Act as open circuits to DC, preventing the flow of steady-state current
• Allow AC to pass through, with the amount of current flow dependent on the capacitance and frequency
• Introduce a phase shift between voltage and current in AC circuits, with current leading voltage by 90 degrees
• In RC circuits, the time constant $\tau = RC$ determines the charging and discharging behavior of the capacitor
  • Capacitor charges to 63.2% of its final value in one time constant during the charging process
  • Capacitor discharges to 36.8% of its initial value in one time constant during the discharging process
• Used in various applications such as filtering, coupling, decoupling, and energy storage
• Capacitive filters (low-pass, high-pass, and band-pass) can be designed using capacitors and resistors to control the frequency response of a circuit
• Coupling capacitors allow AC signals to pass while blocking DC, used in audio and signal processing circuits

Energy Storage in Capacitors

• Energy stored in a capacitor is given by $U = \frac{1}{2}CV^2$, where $U$ is the energy in joules ($J$), $C$ is the capacitance in farads ($F$), and $V$ is the voltage across the capacitor in volts ($V$)
• Energy density of a capacitor is the amount of energy stored per unit volume, expressed in joules per cubic meter ($J/m^3$)
• Capacitors can release stored energy quickly, making them suitable for applications requiring high power density
• Supercapacitors have higher energy density compared to conventional capacitors, bridging the gap between capacitors and batteries
• Capacitors are used in power supplies, uninterruptible power systems (UPS), and energy harvesting devices to store and release energy as needed
• Charging and discharging cycles of capacitors can be used to generate high-voltage pulses in applications such as flash photography and pulsed lasers

Dielectrics and Their Effects

• Dielectric materials are insulators placed between the plates of a capacitor to increase its capacitance
• Dielectric constant $\kappa$ (relative permittivity) represents the factor by which the capacitance increases when a dielectric is inserted between the plates
• Polarization of dielectric molecules in the presence of an electric field contributes to the increased capacitance
• Dielectric strength is the maximum electric field a dielectric can withstand before breakdown occurs, an important factor in capacitor design and selection
• Dielectric loss (dissipation factor) represents the energy lost as heat in the dielectric material when subjected to an alternating electric field
• Common dielectric materials include ceramic, polymer films (polyester, polypropylene, and polystyrene), mica, and electrolytes (in electrolytic capacitors)
• Dielectric materials with high dielectric constants (high-k dielectrics) are used in advanced capacitor designs to achieve higher capacitance values in smaller packages

Real-World Applications

• Power supply smoothing and filtering: Capacitors are used to reduce ripple and noise in DC power supplies, ensuring a stable output voltage
• AC coupling and DC blocking: Coupling capacitors allow AC signals to pass while blocking DC components, used in audio and signal processing circuits
• Energy storage and power backup: Capacitors, particularly supercapacitors, are used in uninterruptible power supplies (UPS), energy harvesting systems, and regenerative braking systems in electric vehicles
• Tuned circuits and resonance: Capacitors are used in combination with inductors to create tuned circuits for frequency selection and filtering in radio and television systems
• Motor starting and power factor correction: Capacitors provide reactive power compensation and improve power factor in AC motor starting and power distribution systems
• Touchscreens and sensors: Capacitive sensing is used in touchscreens, proximity sensors, and moisture detectors, relying on the change in capacitance caused by human touch or the presence of objects
• Pulsed power applications: Capacitors are used to generate high-voltage, high-current pulses in applications such as flash photography, pulsed lasers, and electromagnetic forming
• Electromagnetic interference (EMI) suppression: Capacitors are used in EMI filters to suppress high-frequency noise and interference in electronic systems