Chapter 5 B L Theraja Cylindrical Capacitor.pptx

A cylindrical capacitor is a specific type of capacitor characterized by its cylindrical structure. It consists of two coaxial (aligned along the same axis) cylinders or conductors, one inside the other, separated by a dielectric material.

Here are the key components and characteristics of a cylindrical capacitor:

1. **Structure**: It comprises an inner cylinder and an outer cylinder, both arranged along the same axis. The space between these cylinders is filled with a dielectric material that prevents direct electrical contact between the cylinders.

2. **Dielectric Medium**: The dielectric material, which could be air, vacuum, or other non-conductive substances, helps in maintaining a potential difference between the cylinders without allowing the flow of current between them.

3. **Capacitance Factors**: The capacitance of a cylindrical capacitor is influenced by several factors, including the radii of the cylinders, their lengths, and the properties of the dielectric material between them. Formulas exist to calculate the capacitance based on these parameters.

4. **Applications**: Cylindrical capacitors find applications in various fields such as electronics, power systems, and telecommunications due to their relatively high capacitance compared to other capacitor designs. They are utilized where efficient energy storage in a compact form is required.

5. **Energy Storage**: During charging, energy from an external source is expended to charge the capacitor. This energy gets stored within the electrostatic field formed in the dielectric material. Upon discharge, this stored energy is released.

6. **Functions**: Cylindrical capacitors serve multiple functions, including energy storage, signal processing in circuits, filtering, and regulation of electrical energy in power transmission systems.

In summary, cylindrical capacitors are a specific design of capacitors consisting of coaxial cylinders separated by a dielectric medium. Their structure and properties make them valuable in various technological applications where efficient energy storage and manipulation of electrical signals are required.

1. Cylindrical Capacitor Basic Structure : A cylindrical capacitor comprises two coaxial cylinders, typically an inner and an outer cylinder, aligned along the same axis. Dielectric Medium: It involves an insulating material, the dielectric, placed between the cylinders to prevent direct electrical contact. This dielectric can vary from air to specific non-conductive substances. Capacitance Factors: The capacitance of a cylindrical capacitor depends on the radii of the cylinders, their lengths, and the permittivity of the dielectric material. Formulas exist to calculate capacitance based on these parameters. Applications: Cylindrical capacitors are used widely in electronics, power systems, and telecommunications. They're valued for their relatively high capacitance compared to other capacitor designs. Practical Use: They serve various functions, including energy storage, signal processing in circuits, filtering, and regulation of electrical energy in power transmission systems.

2. A single-core cable or cylindrical capacitor consisting two co-axial cylinders of radii a and b metre . Let the charge per metre length of the cable on the outer surface of the inner cylinder be + Q coulomb and on the inner surface of the outer cylinder be −Q coulomb. Let εr be the relative permittivity of the medium between the two cylinders. The outer cylinder is earthed. let us find the value of electric intensity at any point distant x metres from the axis of the inner cylinder. Total flux coming out radially from the curved surface of this imaginary cylinder is Q coulomb. Area of the curved surface = 2 π x × 1 = 2 π x m2 .

3. Potential Gradient in a Cylindrical Capacitor

4. Capacitance Between Two Parallel Wires This case is of practical importance in overhead transmission lines. The simplest system is 2-wire system (either d.c. or a.c.). In the case of a.c. system, if the transmission line is long and voltage high, the charging current drawn by the line due to the capacitance between conductors is appreciable and affects its performance considerably.  let d = distance between centres of the wires A and B  r = radius of each wire (≤d)  Q = charge in coulomb/metre of each wire Now, let us consider electric intensity at any point P between conductors A and B. Electric intensity at P* due to charge + Q coulomb/metre on A is 

5. Capacitors in Series C1 , C2 , C3 = Capacitances of three capacitors V1 , V2 , V3 = p.ds. across three capacitors. V = applied voltage across combination C = combined or equivalent or joining capacitance.  In series combination, charge on all capacitors is the same but p.d. across each is different. 

6. Capacitors in Parallel I n this case, p.d. across each is the same but charge on each is different. ∴ Q = Q1 + Q2 + Q3 or CV = C1V + C2V + C3V or C = C1 + C2 + C3 For such a combination, dV /dt is the same for all capacitors.

7. Cylindrical Capacitor with Compound Dielectric 

8. Insulation Resistance of a Cable Capacitor

9. Current Flow in Cable Capacitor: Current primarily flows along the axis of the core in a cable capacitor, serving its intended purpose. Leakage of Current: However, some current inevitably leaks, and this leakage occurs radially, perpendicular to the intended current flow. Insulation Resistance: The resistance presented to this radial leakage current is termed as the insulation resistance of the cable. Relation to Cable Length: With longer cable lengths, there's an increase in leakage due to the radial current flow, resulting in reduced insulation resistance. In essence, as cable length grows, more current leaks, leading to a decrease in insulation resistance. Inverse Proportionality: The relationship between insulation resistance and cable length is inversely proportional. Longer cables exhibit lower insulation resistance due to increased leakage. Differentiation from Conductor Resistance: It's crucial not to confuse insulation resistance with conductor resistance. Insulation resistance pertains to the resistance against radial leakage and varies with cable length, whereas conductor resistance, related to the conductor material, is directly proportional to the cable length.

10. Energy Stored in a Capacitor Initial Energy Expenditure : Charging a capacitor requires energy from an external source. This energy is stored within the electrostatic field that forms in the dielectric medium. Energy Storage and Discharge : The energy expended during charging gets stored in the electrostatic field. Upon discharge, this stored energy is released as the field collapses. Work During Charging : Initially, when the capacitor is uncharged, minimal work is needed to transfer charge between the plates. However, as more charge is added, overcoming the repulsive force between the accumulated charges demands additional work. Calculation of Energy Spent : The energy spent in charging a capacitor of capacitance C to a voltage V can be determined. At any point during charging, if the potential difference across the plates is v and the next charge increment transferred is 'dq', the work done is involved in moving 'dq' charge from one plate to the other.

11. Force of Attraction Between Oppositely-charged Plates Two parallel conducting plates A and B carrying constant charges of + Q and −Q coulombs respectively. Let the force of attraction between the two be F newtons. If one of the plates is pulled apart by distance dx, then work done is = F × dx joules ...(i)  Since the plate charges remain constant, no electrical energy comes into the arrangement during the movement dx.  ∴ Work done = change in stored energy

12. Tutorial-5.2

13. Current-Voltage Relationships in a Capacitor

14. Time Constant

15. Discharging of a Capacitor when S is shifted to b, C is discharged through R. It will be seen that the discharging current flows in a direction opposite to that the charging current. Hence, if the direction of the charging current is taken positive, then that of the discharging current will be taken as negative. To begin with, the discharge current is maximum but then decreases exponentially till it ceases when capacitor is fully discharged.

16. Transient Relations During Capacitor Charging Cycle Steady-State Transition: When a circuit moves from one stable state to another, it goes through a transient state. This transient state is brief compared to the overall duration. Initial and Final Conditions: The initial state is the starting stable condition, and the final state is the target stable condition of the circuit. Transient Phase: The transient state occurs between the initial and final conditions. It represents the period during which changes in current and voltage occur in the circuit. Example with RC Circuit: Consider an RC circuit where initially, the switch isn't connected (at neither a nor b). This is the initial steady state with no current flow and consequently no voltage drops across components. Switch Position Change: When the switch moves to position a, current flows through resistor R, initiating changes in voltage across R and the capacitor C. During this phase, transient voltages develop until they reach their stable, final values. Duration of Transient Condition: The time frame in which current and voltage variations occur and stabilize is termed as the transient condition in the circuit. In essence, this process describes how a circuit, such as an RC circuit, undergoes changes in current and voltage as it transitions from an initial stable state to a final stable state, passing through a brief transient state in between.

17. Transient Relations During Capacitor Discharging Cycle

18. Charging and Discharging of a capacitor with Initial Charge  Initial Capacitor Potential: In this scenario , the capacitor starts with an initial potential (Vo) that is lower than the applied battery voltage (V). Opposition to Applied Voltage: The initial potential (Vo) acts in opposition to the battery voltage (V) during the charging process. Effect on Rate of Rise: illustrates that the initial rate of rise of the capacitor voltage ( vc  ) is somewhat slower when the capacitor begins with an initial potential (V) compared to when it starts uncharged. Rise from V  to Final V: The capacitor's voltage rises from the initial value (V0 ) to the final value (V) over one time constant. Calculation of Initial Rate of Rise: The initial rate of rise of the capacitor voltage ( vc  ) from an initial value of V0  to the final value of V in one time constant can be determined based on the specific circuit characteristics and the relationship between the initial and final voltages.

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