Chapter 4. Capacitors and Coils
Recommended Article : Circuit Theory
1. Capacitors
2. Coils
1. Capacitors
⑴ Definition : Component that accumulates charge to store energy
① Capacitor can be considered as two metal plates with an area almost infinite compared to the distance between them.
② Capacitor is represented by two wires of equal length.
Figure 1. Symbol of Capacitor
⑵ Principle : Accumulation of charge
Figure 2. Accumulation of charge
① Before charging : No charge is accumulated on A and B.
② During charging : Electrons move from A to B.
③ After charging
○ When the voltage between A and B reaches V, no further charge transfer occurs.
○ Capacitor retains energy even when the switch is open.
⑶ Terminal characteristics
① Mathematical representation
Figure 3. Terminal characteristics and sign of capacitor
② Proof : Proof for Q = CΔV for two conductors
○ Assumption 1: Two conductors A and B with voltages VA (t) and VB (t) are connected by a wire.
○ Assumption 2: Initial charges on A and B and VA (t) - VB (t) are both 0.
○ (Note) Since any two conductors always start in Assumption 2, this assumption is reasonable.
○ Current flows from B to A through the wire, causing displacement current from A to B : Displacement current is described by Maxwell’s equations.
○ Applying integration by parts gives
○ C : = g(t) ⇒ Q = CΔV
○ (Note) In reality, ∫ f ‘(t) (VA - VB) ∝ VA - VB does not hold, so Q = CΔV doesn’t hold for real capacitors.
○ (Note) Knowing Q ＞ 0 and ΔV ＞ 0 implies C ＞ 0.
③ Proof : Proof for Q = CΔV for n conductors
○ Assumption 1: n+1 conductors with voltages vp for each conductor with respect to a reference conductor.
○ Assumption 2: All voltages except vp and vq are 0.
○ Can use conclusion of Case 1.
○ If voltage between conductors is electrically linear, it can be expressed as follows.
④ Proof : Terminal characteristics for two conductors with wide plates facing each other
○ General equation
○ Case 1: When the distance of the capacitor keeps changing : Used for creating speed sensors based on C variation.
○ Case 2: Capacitors treated in circuit theory have fixed electrodes, so the second term is always 0.
⑷ Stored energy
⑸ Capacitance in different scenarios
① Series combination
② Parallel combination
③ Capacitance of a spherical capacitor
Figure 4. Capacitance of a spherical capacitor
⑹ Relative permittivity and breakdown voltage
① Relative permittivity (Dielectric Constant)
Table. 1. Relative permittivity
② Breakdown voltage (Dielectric Strength)
○ Capacitor experiences a stress proportional to 0.5εE2
○ Dielectrics have a limit to the mechanical stress they can endure under electrical stress.
○ Note: 1 mil = 1/1000 inch = 0.0254 mm
Table. 2. Examples of breakdown voltage
⑺ Real capacitors : Leakage current exists.
① Assume no free electron flow until breakdown voltage is reached.
② In reality, impurities in dielectric or forces within dielectric cause free electron flow.
③ Leakage current is explained using the following circuit model
Figure 5. Leakage current model
⒜ represents charging process, ⒝ represents discharging process
④ Leakage current is very small, but considerable in Electrolytic type Capacitors
⑻ Types of capacitors
① Type 1: Fixed Capacitor
② Type 1-1: Mica
○ Resistant to temperature changes and suitable for high-voltage applications.
○ Very small leakage current (Rleakage is approximately 1000 MΩ)
○ Used from a few pF to 200 pF, voltage around 100 V
○ Temperature coefficient : -20 ppm/℃ ~ +100 ppm/℃
Figure 6. Leakage current model
③ Type 1-2: Ceramic Capacitor
○ Available in types with Silver electrodes and Metal electrodes.
○ Very small leakage current (Rleakage is approximately 1000 MΩ)
○ Used in both DC and AC circuits
○ Used from a few pF to 2,000 pF, voltage around 5,000 V
Figure 7. Leakage current model
④ Type 1-3: Electrolytic Capacitor
○ Symbol
Figure 8. Symbol of Electrolytic Capacitor
○ Most commonly used in applications ranging from mF to thousands of mF
○ Insulating in one direction, conducting in the other
○ Used in DC and short-term AC applications
○ Large leakage current (Rleakage is approximately 1 MΩ), low breakdown voltage
○ Typically used from μF to thousands of μF, operating voltage around 500 V
Figure 9. Leakage current model
⑤ Type 1-4: Tantalum Capacitor
○ Available in solid type and wet-slug type
○ 1st. High-purity tantalum powder is compacted into rectangular or cylindrical shapes
○ 2nd. Anode lead wires are inserted into the structure
○ 3rd. Structure is baked in a high-temperature vacuum to create a porous structure
○ 4th. Porous structure increases surface area per unit volume
○ 5th. Thin manganese dioxide (MnO2) layer forms on porous material when dipped in acid solution
○ 6th. Electrolyte is added between MnO2 layer and cathode to create Solid tantalum Capacitor
○ 7th. Adding wet acid turns it into a wet-slug tantalum Capacitor
Figure 10. Tantalum Capacitor
⑥ Type 1-5: Polyester-film Capacitor
○ Two metal films are separated by insulation (e.g., Mylar)
○ Large ones have printed data for static capacitance and operating voltage on the surface
○ Small ones use color coding
○ Black band is printed on the outer metal film near the lead connection
○ Lead near this band must be connected to low voltage
○ Very small leakage current (Rleakage is approximately 1000 MΩ), used in DC and AC
○ Axial lead type : Used from 0.1 μF to 18 μF, operating voltage up to 630 V
○ Radial lead type : Used from 0.01 μF to 10 μF, operating voltage up to 1,000 V
Figure 11. Polyester-film Capacitor
⑦ Type 2: Variable Capacitor
○ Symbol
Figure 12. Symbol of Variable Capacitor
○ Type 1: Different electrode area structure
Figure 13. Structure of Variable Capacitor
○ Consists of a semicircular fixed metal plate and a rotatable metal plate
○ Principle : Dial rotation → Changing area of opposing plates → Change in capacitance
○ Type 2: Different electrode spacing structure
○ Feature : Insulator is air. Typically below 300 pF.
⑼ Marking Schemes
① Color Coding
○ Add -00 after the voltage value.
○ 8 : 0.01, 9 : 0.1 signify.
Figure 14. Structure of a Variable Capacitor
② Standard values are used like resistors.
③ Provide the values of capacitance, allowable tolerance, and if necessary, the maximum operating voltage.
④ The size of the capacitor represents the capacitance value (small units in pF, large ones in μF).
⑤ M : ±20 %, K : ±10 %, J : ±5 %, F : ±1 %
⑽ Utilization of Capacitors
① Computer Keyboard
Figure 15. Structure of a Computer Keyboard
○ 1st. Pressing the keyswitch.
○ 2nd. Decreased gap between capacitor’s metal plates increases electrical capacitance.
○ 3rd. Since charge is constant, increased electrical capacitance leads to higher voltage.
○ 4th. Computer recognizes voltage increase, resulting in character input.
② Touch Screen
Figure 16. Structure of a Touch Screen
○ 1st. Applying voltage to the glass charges the surface.
○ 2nd. Touching the top glass surface with a finger attracts stored electrons to the contact point, changing surface charge.
○ 3rd. Since electrical capacitance is constant, changes in charge result in voltage variation.
○ 4th. Sensor detects voltage changes.
③ Condenser Microphone
Figure 17. Structure of a Condenser Microphone
○ 1st. Vibrating metal plate due to sound.
○ 2nd. Changing gap between two metal plates alters electrical capacitance.
○ 3rd. Since charge is constant, capacitance change leads to voltage variation.
○ 4th. Voltage change is converted into an electrical signal.
○ 1st. Capacitor charged before taking a photo.
○ 2nd. Pressing the flash switch discharges ignition capacitor.
○ 3rd. Current flows in flash tube, generating light.
⑤ Automated Defibrillator
○ 1st. Powering on the automated defibrillator charges the capacitor.
○ 2nd. Pressing the switch discharges briefly, sending a strong current to stimulate the heart.
2. Coil (Inductor)
⑴ Definition : Component that stores energy through a magnetic field.
Figure 18. Structure of a Coil
⑵ Terminal Characteristics
① Mathematical Expression
Figure 19. Terminal Characteristics of a Coil
② Proof : Self-inductance
○ First, consider the situation below.
Figure 20. Situation for Proof of Self-inductance
○ By Maxwell’s Equations and Green’s Theorem, the following equation holds.
○ Green’s Theorem applies to arbitrary surfaces, considering a circular plane like the one in the figure.
○ n vector is defined outward from the ground.
○ Naturally, the orientation of the surface’s boundary is counterclockwise as shown.
○ (Note) The direction in Green’s Theorem is important. Let’s use the right hand often!
○ The coil is a solenoid.
○ N : Number of turns
○ ℓ : Solenoid length
○ L : Proportionality constant between magnetic flux ΦB and current i.
○ Case 1: Assume the coil is right-handed according to the current direction.
○ View the situation from the direction where the current flows out.
Figure 21. Assuming the Coil is Right-Handed according to Current Direction
○ Magnetic field and surface normal are parallel.
○ Case 2: Assume the coil is left-handed according to the current direction.
○ View the situation from the direction where the current flows in.
Figure 22. Assuming the Coil is Left-Handed according to Current Direction
○ Magnetic field and surface normal are parallel.
○ Conclusion: The forms of the equation are the same whether it’s right-handed or left-handed.
○ Extension of Conclusion: Now, consider the coil with N layers and measure the potential difference between both ends.
③ Proof : Extension of ② Proof
○ General Expression: You can write the following equation for any conductor.
○ Case 1: If L varies with time:
○ The 2nd term arises from the coil’s movement, relevant in electromechanical applications.
○ Can also be used as a sensor.
○ Case 2: If L is constant: Generally, coils dealt with in circuit theory are fixed, so the 2nd term is assumed to be 0.
④ Understanding the Signs
○ Case 1: If i increases, induced current flows in the direction to hinder i’s increase due to inertia.
Figure 23. Scenario for Increasing i
○ Case 2: If i decreases, induced current flows in the direction to hinder i’s decrease due to inertia.
Figure 24. Scenario for Decreasing i
⑶ Stored Energy
⑷ Various Cases of Inductive Coefficients
① Series Composition
② Parallel Composition
⑸ Characteristics of a Coil
① Notation of a Coil
Figure 25. Notation of a Coil
② Standard values for coils (5, 10%) are like resistors.
③ Types of Coils
Figure 26. Types of Coils
⑹ Real Coils
① Model actual coils as shapes containing resistors and capacitors.
② Ignore capacitors in the model for practical applications.
③ Rl is around several Ω to hundreds of Ω; if the coil conductor is thin and long, resistance increases.
Figure 27. Modeling of Actual Coil
Input: 2016.01.12 11:17
Modified: 2018.12.13 18:08