Cylindrical and Spherical Capacitors

Series and Parallel Combinations - Cylindrical Capacitor: Introduction

• Capacitors are devices that store electrical energy in the form of electric charge.
• They consist of two conductive plates separated by an insulating material, also known as the dielectric.
• In this lesson, we will explore cylindrical capacitors and their series and parallel combinations.
• Cylindrical capacitors have a cylindrical shape with a central conductor rod and an outer cylindrical shell.
• The space between the rod and the shell is filled with a dielectric material.
• The capacitance of a cylindrical capacitor is determined by its geometry and the properties of the dielectric material.

Capacitance of a Cylindrical Capacitor

• The capacitance of a cylindrical capacitor is given by the formula: where:
  • C is the capacitance,
  • ε₀ is the permittivity of free space,
  • L is the length of the capacitor,
  • R₁ and R₂ are the inner and outer radii of the capacitor.
• The logarithmic term in the formula accounts for the cylindrical shape of the capacitor and its effect on the electric field.

Series Combination of Cylindrical Capacitors

• When cylindrical capacitors are connected in series, the total capacitance can be calculated as follows: where:
  • C_total is the total capacitance,
  • C₁, C₂, C₃, …, C_n are the individual capacitances of the series capacitors.
• In a series combination, the total capacitance is less than the capacitance of any individual capacitor.
• Series combinations are used to create capacitors with smaller capacitance values.

Parallel Combination of Cylindrical Capacitors

• When cylindrical capacitors are connected in parallel, the total capacitance can be calculated as follows: where:
  • C_total is the total capacitance,
  • C₁, C₂, C₃, …, C_n are the individual capacitances of the parallel capacitors.
• In a parallel combination, the total capacitance is equal to the sum of the individual capacitances.
• Parallel combinations are used to create capacitors with larger capacitance values.

Example: Series Combination of Cylindrical Capacitors

• Let’s consider a series combination of three cylindrical capacitors with capacitances C₁, C₂, and C₃.
• The total capacitance of the series combination can be calculated using the formula:
• Substituting the respective values, we can find the total capacitance.

Example: Parallel Combination of Cylindrical Capacitors

• Let’s consider a parallel combination of three cylindrical capacitors with capacitances C₁, C₂, and C₃.
• The total capacitance of the parallel combination can be calculated using the formula:
• Substituting the respective values, we can find the total capacitance.

Applications of Cylindrical Capacitors

• Cylindrical capacitors have various applications in electrical circuits and systems.
• They are commonly used in high-power systems, such as power factor correction circuits and motor starters.
• Cylindrical capacitors also find applications in energy storage systems, such as electric vehicles and renewable energy systems.
• They are used in electronic devices like amplifiers and filters to stabilize signals and improve performance.
• Cylindrical capacitors are an essential component in many electronic devices and systems, ensuring proper functioning and performance.

11. Dielectric Material in Cylindrical Capacitors

• The dielectric material used in cylindrical capacitors affects their capacitance.
• Dielectrics with higher permittivity (ε_r) result in higher capacitance.
• Common dielectric materials include air (ε_r = 1), paper, mica, and various types of plastics.

12. Dielectric Strength and Breakdown

• Dielectric strength refers to the maximum electric field a dielectric material can withstand without breaking down.
• When the electric field exceeds the dielectric strength, the dielectric material undergoes electrical breakdown, resulting in a loss of insulation properties.
• Different dielectric materials have different dielectric strengths, and it is essential to choose a dielectric with adequate strength for a given application.

13. Effect of Dielectric on Electric Field

• The presence of a dielectric material affects the electric field in a cylindrical capacitor.
• The electric field is weaker inside the dielectric, reducing the voltage difference between the plates.
• This results in an increase in capacitance since capacitance is inversely proportional to the voltage difference.

14. Energy Stored in Cylindrical Capacitors

• Cylindrical capacitors store energy in the electric field between the plates.
• The energy stored (U) in a capacitor can be calculated using the formula: where:
  • U is the energy stored,
  • C is the capacitance,
  • V is the voltage across the capacitor.

15. Example: Calculation of Energy Stored

• Let’s consider a cylindrical capacitor with a capacitance of 10 μF and a voltage of 100 V.
• The energy stored in the capacitor can be calculated using the formula:
• Solve this equation to find the energy stored in the capacitor.

16. Charging and Discharging of Cylindrical Capacitors

• When a cylindrical capacitor is connected to a voltage source, it charges up as the electric field builds up between the plates.
• The time it takes for a capacitor to charge or discharge depends on its capacitance and the resistance in the circuit.
• Charging and discharging processes in capacitors are important in various electrical and electronic applications.

17. Time Constant of a Capacitor

• The time constant (τ) of a capacitor is a measure of the time it takes for the voltage across the capacitor to reach approximately 63.2% of its final value during the charging or discharging process.
• The time constant can be calculated using the formula: where:
  • τ is the time constant,
  • R is the resistance in the circuit,
  • C is the capacitance.

18. Example: Calculation of Time Constant

• Let’s consider a cylindrical capacitor with a capacitance of 10 μF and a resistance of 1 kΩ in the charging circuit.
• The time constant of the circuit can be calculated using the formula:
• Solve this equation to find the time constant of the circuit.

19. Voltage-Time Graph for Charging/Discharging

• The voltage across a capacitor during the charging or discharging process follows an exponential curve.
• The voltage increases or decreases exponentially with time, reaching its final value after approximately 5 time constants.
• This voltage-time behavior is crucial in understanding the behavior of capacitors in various circuits.

20. Applications of Series and Parallel Combinations

• Series and parallel combinations of cylindrical capacitors have various applications in practical circuits.
• Series combinations help achieve smaller effective capacitance values, which are necessary in certain electronic circuits.
• Parallel combinations are used to create larger effective capacitance values, e.g., for power factor correction in electrical systems.

21. Application: Power Factor Correction

• Power factor correction is a technique used to improve the power factor of electrical systems, resulting in optimal power usage.
• Cylindrical capacitors connected in parallel can be used to correct the power factor by compensating for the reactive power in inductive loads.
• By selecting the appropriate capacitance value, the reactive power can be counteracted, reducing the overall power consumption.
• This helps in improving the efficiency of the electrical system and reducing electricity costs associated with low power factor.

22. Application: Motor Starters

• In motor systems, cylindrical capacitors are commonly used in motor starters to provide additional starting torque and reduce the strain on the motor during startup.
• The capacitor is connected in parallel with the motor’s winding, creating a phase difference between the voltage and current.
• This phase difference helps in generating a rotating magnetic field needed for starting the motor smoothly and reducing the initial current surge.
• Motor starters with capacitors are widely used in various industrial and residential applications.

23. Example: Power Factor Correction

• Let’s consider an inductive load with a power factor of 0.6.
• We want to improve the power factor to a target value of 0.9 using cylindrical capacitors.
• The required capacitance can be calculated using the formula: where:
  • C is the required capacitance,
  • Q is the reactive power,
  • ω is the angular frequency,
  • V is the voltage.
• Substituting the respective values, we can find the required capacitance to correct the power factor.

24. Example: Motor Starter Capacitance

• Let’s consider a motor with a starting torque requirement of 100 Nm.
• The motor starter uses a cylindrical capacitor connected in parallel with the motor winding.
• The required capacitance can be calculated using the formula: where:
  • C is the required capacitance,
  • T is the starting torque,
  • ω is the angular frequency,
  • V₀ is the initial voltage.
• Substituting the respective values, we can find the required capacitance for the motor starter.

25. Difference Between Cylindrical and Spherical Capacitors

• While cylindrical capacitors have a cylindrical shape with a central conductor rod and an outer cylindrical shell, spherical capacitors have concentric spherical conductors separated by a dielectric material.
• The capacitance of a spherical capacitor is given by the formula: where:
  • C is the capacitance,
  • ε₀ is the permittivity of free space,
  • R₁ and R₂ are the radii of the inner and outer conductors.
• The geometry and electric field distribution in spherical capacitors are different from cylindrical capacitors, leading to different capacitance formulas.

26. Series Combination of Spherical Capacitors

• When spherical capacitors are connected in series, the total capacitance is calculated using the formula: where:
  • C_total is the total capacitance,
  • C₁, C₂, C₃, …, C_n are the individual capacitances of the series capacitors.
• The total capacitance of a series combination is inversely proportional to the sum of the reciprocals of the individual capacitances.

27. Parallel Combination of Spherical Capacitors

• When spherical capacitors are connected in parallel, the total capacitance is calculated using the formula: where:
  • C_total is the total capacitance,
  • C₁, C₂, C₃, …, C_n are the individual capacitances of the parallel capacitors.
• The total capacitance of a parallel combination is equal to the sum of the individual capacitances.

28. Comparison: Series vs. Parallel Combinations

• In series combinations, the total capacitance is always less than the smallest individual capacitance.
• In parallel combinations, the total capacitance is always greater than the largest individual capacitance.
• Series combinations are used to reduce capacitance, while parallel combinations are used to increase capacitance.
• The choice of series or parallel combination depends on the specific requirements of the circuit or application.

29. Real-Life Examples of Capacitor Combinations

• Capacitor combinations, whether series or parallel, are widely used in various electrical and electronic devices.
• Series combinations are used in electronic circuits to create low-pass filters, where they can attenuate high-frequency signals.
• Parallel combinations are used in audio systems to create high-pass filters, allowing only high-frequency signals to pass through.
• These capacitor combinations are crucial in shaping the frequency response and performance of electrical and electronic systems.

30. Summary and Revision

• In this lesson, we have learned about cylindrical capacitors and their capacitance formula.
• We explored series and parallel combinations of cylindrical capacitors and their applications.
• We also discussed the difference between cylindrical and spherical capacitors.
• The concepts of power factor correction and motor starters using cylindrical capacitors were explained.
• Finally, we compared series and parallel combinations of both cylindrical and spherical capacitors.
• Understanding these concepts and their applications is essential in the study of capacitors and their practical uses in electrical systems.