Capacitors in PE Power

Capacitors in PE Power are one of the most important exam topics. But why? The reason is their notable usage and importance in regulating and improving the Power circuits. Capacitors in PE Power involve studying their types, behavior, and uses in AC and DC circuits. 

This detailed study guide on Capacitors in PE Power will help you cover this topic in complete detail as per the NCEES® exam guidelines and roadmap. Let’s start with the fundamentals.

Capacitors and Their Importance in Power Circuits

A capacitor is a passive electronic component that stores electrical energy in an electric field. It consists of two conductors separated by an insulator, known as a dielectric.

The capacity of a capacitor to store charge is measured in farads (F). It is determined by the physical characteristics of the capacitor, including the area of the plates, the separation distance between the plates, and the dielectric material used.

Capacitors are used in circuits for various reasons. Let’s discuss a few important uses in a nutshell.

  • Harmonic Mitigation with Capacitors: Capacitors are used in power systems to mitigate harmonics by creating resonant circuits that filter out specific harmonic frequencies. This is achieved by tuning the capacitor and inductor combinations to resonate at unwanted harmonic frequencies, thereby reducing their presence in the power system.
  • Capacitors and Voltage Fluctuations: Capacitors help stabilize voltage fluctuations in power systems by providing reactive power compensation. When connected to a power network, capacitors can absorb or release reactive power, which helps maintain a more consistent voltage level, especially in systems with fluctuating loads or significant inductive components.
  • Capacitors and Line Loss Reduction: By providing reactive power locally, capacitors reduce the need to transport reactive power over long distances in power lines, thus reducing line losses. This improves the efficiency of power transmission and distribution networks, as it decreases I²R losses (where I is current and R is resistance) in the conductors

Fundamental Equation of a Capacitor

Fundamental Equation of a Capacitor

The basic equation that describes a capacitor and its behavior is:

Q=C×V

Where:

  • Q is the charge stored in the capacitor, measured in coulombs (C).
  • C is the capacitance of the capacitor, measured in farads (F).
  • V is the voltage across the capacitor, measured in volts (V).

It shows that the charge stored by a capacitor is directly proportional to the voltage applied across it.

Construction and Operation of a Parallel Plate Capacitor

Construction and Operation of a Parallel Plate Capacitor

A parallel plate capacitor is constructed by placing two conductive plates parallel and separated by a dielectric material. The plates can be made of aluminum or copper, and the dielectric can be air, glass, plastic, or ceramic.

When a voltage is applied across the plates, an electric field is established between them, causing a positive charge to accumulate on one plate and a negative charge on the other. This stored charge creates an electric field in the dielectric, storing energy.

Capacitance of a Parallel Plate Capacitor

Capacitance of a Parallel Plate Capacitor

For a parallel plate capacitor, the formula for capacitance can be rewritten as:

C=εA/d

Where:

  • ε is the permittivity of the dielectric material between the plates.
  • A is the area of one of the plates, measured in square meters (m²). 
  • d is the plate separation distance, measured in meters (m).

*We are using SI units for simplicity.

The permittivity ε is a measure of how much electric field flux (electric field lines) the material can accommodate and is defined as:

ε=ε_r * ε_0

Where ε_0 is the permittivity of free space (approximately 8.85×10^-12 F/m). ε_r is the material’s relative permittivity (or dielectric constant), a dimensionless number (scaler constant that depends on the nature or type of dielectric medium).

Impact of Air and Dielectric Medium

A dielectric material between the plates of a capacitor increases its capacitance compared to when the space is filled with air or a vacuum. This is because the dielectric material reduces the electric field within the capacitor, allowing it to store more charge at a given voltage.

Charge Storage and Coulomb’s Law

Coulomb’s Law is fundamental in understanding the forces between charges in a capacitor. It states that the force (F) between two point charges is directly proportional to the product of the magnitudes of the charges (q_1 and q_2) and inversely proportional to the square of the distance (r) between them:

F=k (q_1)(q_2)/r²

Where:

k is Coulomb’s constant (8.9875×10^9 N m²/C²).

Deriving Formulas for Point Charge and Test Charge

Let’s derive the electric field (E) from a point charge in air and a dielectric medium.

Since E=F/q

*Electric intensity is defined as force per unit charge.

Let’s put q_1=q_2=q

The force from Columb’s law can be now written as:

F=Kq²/r²

Where, 

  • k=1/(4πε_0), for air
  • 1/(4πε), for dielectric medium

For air, the electric field E at a distance r from a point charge q is given by (after substituting the value of force):

E= 1/(4πε_0) * q/r²

For a dielectric medium, the equation becomes:

E=1/4πε * q/r²

Where ε is the permittivity of the dielectric medium given as ε=ε_r * ε_0, and ε_r is the material’s relative permittivity (or dielectric constant).

Types of Capacitors

Types of Capacitors

Capacitors come in various types and classifications, each suited for specific applications and characteristics. Here’s a detailed overview of the different kinds and classifications of capacitors:

1. Electrolytic Capacitors

Aluminum Electrolytic Capacitors: They are known for their high capacitance-to-volume ratio; these capacitors use an aluminum oxide film and an electrolytic solution. They are polarized, meaning they must be connected with the correct polarity. Commonly used in power supply filtering applications.

Tantalum Electrolytic Capacitors: They are smaller and more stable than aluminum types; they have a lower risk of leakage and are more reliable. Tantalum capacitors are also polarized and are used in space-constrained applications like mobile phones and laptops.

2. Ceramic Capacitors

Multilayer Ceramic Capacitors (MLCCs): They are composed of alternating layers of metal and ceramic and offer a compact and non-polarized size. Used in a wide range of applications, from high-frequency to general electronic circuits.

Disc Ceramic Capacitors: They are often used for noise suppression and are non-polarized. They are suitable for relatively low capacitance requirements.

3. Film Capacitors

Polyester Film Capacitors: They are known for their stability and low cost. They are used in applications where reliability and stability are important, such as power supplies and audio circuits.

Polypropylene Film Capacitors: They have a higher temperature tolerance and lower dielectric absorption than polyester. They’re used in high-frequency applications like resonant converters.

Polystyrene Film Capacitors: They are known for their excellent stability and low leakage but are heat-sensitive. Used in precision timing and filtering circuits.

Polyethylene Terephthalate (PET) Capacitors: They offer good dielectric strength and are used in general-purpose applications.

Polytetrafluoroethylene (PTFE) Capacitors: They are known for their high-temperature resistance and stability. Used in high-frequency and high-temperature applications.

4. Supercapacitors

Electric Double-Layer Capacitors (EDLCs): They are also known as supercapacitors or ultracapacitors; they store charge in an electric double layer and offer high capacitance values. Used in applications requiring rapid charge and discharge cycles, such as in regenerative braking systems in vehicles.

Pseudocapacitors: They store charge chemically and have higher energy density than EDLCs. They are used in similar applications as EDLCs.

5. Specialized Capacitors

Silver Mica Capacitors: They are known for their stability, accuracy, and low loss at high frequencies. Used in RF applications.

Air Variable Capacitors: They are typically used in radio tuning circuits; they allow for changing the capacitance by adjusting the distance between the plates.

Trimming Capacitors: They are used for fine circuit adjustments, often in oscillator or RF applications.

Power Factor Correction Capacitors: They are designed to improve the power factor in electrical power systems. They generally have larger capacitance values to handle the reactive power in AC systems and withstand the higher voltages typically found in power distribution systems.

Surge Capacitors: They are designed to protect electrical equipment from voltage spikes and surges. They can withstand high transient voltages to absorb and then temporarily release excess energy from surges.

Classification by Dielectric Material

Capacitors can also be classified based on the dielectric material used:

  • Air: Used in variable capacitors and some high-frequency applications.
  • Ceramic: Common in general-purpose applications, offering a range of dielectric constants.
  • Plastic Film: Stable and reliable, and used in various electronic circuits.
  • Electrolytic: High capacitance values, and used in power supply circuits.
  • Tantalum: Stable, reliable, and used in miniaturized electronics.
  • Paper: Historically significant but less common today.

Classification by Application

  • General Purpose: For various electronic circuits where specific characteristics are not critical.
  • Power Electronics: High voltage and current handling, such as in power supplies and inverters.
  • Precision: For timing, filtering, and precision applications.
  • High-Frequency: For RF, communication, and high-speed digital circuits.
  • Temperature Compensated: Used in circuits where capacitance stability over temperature is critical.

Capacitors in AC and DC Circuits

Capacitors behave differently in AC and DC circuits due to the different wave nature of both circuits. In DC circuits, capacitors charge up to the supply voltage and then block further current, acting as an open circuit.

In AC circuits, capacitors continually charge and discharge with the alternating voltage, leading to a phase shift. 

Let’s discuss this in further detail.

Capacitor Behavior in DC Circuits

The following workflow can help you understand how Capacitors behave in DC settings.

Step 01 – Connection to DC Source

When a capacitor is connected to a DC source, the initial condition is akin to a short circuit, as there’s no charge on either plate.

Step 02 – Charging Phase

Step 02 A – Step Electron Movement: Electrons are pushed away from the battery’s negative terminal and accumulate on the capacitor’s adjacent plate, leaving a positive charge on the opposite plate.

Step 02 B – Electric Field Development: This charge accumulation creates an electric field across the dielectric, opposing further electron flow.

Step 02 C – Voltage Build-Up: The voltage across the capacitor plates builds up, gradually approaching the voltage of the DC source.

Step 03 – Reaching Steady State

Step 03 A – Charge Equilibrium: Eventually, the voltage across the capacitor equals the DC source voltage, and the flow of charge stops. The capacitor is now fully charged.

Step 03 B – Open Circuit Condition: At this point, the capacitor acts as an open circuit, blocking further DC flow.

Step 04 – Discharging Phase

Step 04 A – Circuit Change: If the capacitor is disconnected from the source and connected to a conductive path (like a resistor), it begins discharging.

Step 04 B – Current Flow: The stored charge on the plates causes a current to flow through the circuit, decreasing as the capacitor discharges.

Things to Remember

Time Constant (τ)

The time constant of a capacitor in a DC circuit, typically when connected with a resistor, is the time it takes for the voltage across the capacitor to either charge up to about 63% of its maximum value or discharge to about 37% of its initial value.

It’s calculated as τ=RC (where R is resistance, C is capacitance).

Capacitor Behavior in AC Circuits

Step 01 – Connection to AC Source

Unlike DC, AC voltage continuously changes from positive to negative. This alternating characteristic leads to a different behavior of capacitors in AC circuits.

Step 02 – Continuous Charging and Discharging

Step 02 A – Phase Change Effect: The capacitor continuously charges and discharges as the AC voltage changes. This leads to a flow of current in the circuit.

Step 02 B – Cyclic Process: The process repeats with every cycle of AC voltage.

Step 03 – Phase Difference

Step 03 A – Voltage and Current Relationship: In a purely capacitive AC circuit, the current leads the voltage by 90 degrees. This phase difference is crucial for timing and filtering applications.

Step 03 B – Cause of Phase Shift: The phase shift occurs because the current responds immediately to voltage changes (charging or discharging). In contrast, the voltage across the capacitor takes time to build up or decrease.

Things to Remember

Capacitive Reactance (X_C)

It is given by X_C = 1/2πfC, where 

Where f is the frequency (the reactance decreases with increasing frequency).

Current Flow at High Frequency: At high frequencies, X_C is low, so more current flows through the capacitor.

Current Flow at Low Frequency: At low frequencies, X_C is high, restricting current flow.

Impedance (Z) in AC Circuits

Impedance combines resistance and reactance, providing a complete picture of the capacitor’s behavior in AC circuits. This is the total opposition of a circuit and is given by  Z=Under Root (R²+X_C²). In purely capacitive circuits, Z=X_C

Energy Stored in a Capacitor

Energy in a Capacitor

To understand how energy is stored in a capacitor, let’s start from the basic concept of work done in moving a charge in an electric field and then derive the energy stored in a capacitor formula.

Step 1: Basic Concept of Work Done in Moving a Charge within the Electric Field

Work is defined as the force applied to an object times the distance over which it’s applied. In the context of electric fields and charges, work is done when a charge is moved within an electric field.

The work done (W) in moving a small charge (dq) from one plate of the capacitor to the other is given by:

W=F×d

Where F is the force and d is the distance.

Step 2: Force on a Charge – Coulomb’s Law

According to Coulomb’s Law, the force (F) experienced by a charge (q) in an electric field (E) is:

F=qE

Step 3: Electric Field in a Capacitor

In a parallel plate capacitor, the electric field (E) is uniform and is given by:

E=V/d

Where V is the voltage across the capacitor, and d is the distance between the plates.

Substituting this into the expression for force, we get:

F=qV/d

Step 4: Work Done in Moving a Charge in a Capacitor

Now, substituting the expression for F into the work formula, the work done in moving a small charge dq across the capacitor is:

dW=q(V/d)×d

Which simplifies to:

dW=qV

Step 5: Relation Between Charge and Voltage in a Capacitor

From the basic capacitor formula Q=CV, where Q is the charge, C is the capacitance, and V is the voltage, we can express V as:

V=Q/C

Step 6: Substituting into the Work Formula

Substitute the expression for V in the work done formula:

dW=q(Q/C)

Step 7: Integrating to Find Total Work Done

The total work done in moving the charge from one plate to the other (which is the energy stored in the capacitor) is found by integrating this expression as the charge Q goes from 0 to its final value:

W=∫[Lower Limit = 0, Upper Limit = Q] q(1/C)dq

This integral evaluates to:

W= (q²/2C) | [Lower Limit = 0, Upper Limit = Q] 

Applying limits (keeping C constant), we will get:

W= (1/2C)[Q²-0²]

W=½ Q²/C

Step 8: Expressing Energy in Terms of Voltage

Since 

Q=CV, we can substitute this into the expression for energy to get it in terms of voltage:

W= ½ (CV)²/C

Simplifying, we find the renowned formula for the energy stored in a capacitor:

W= 1/2CV²

Conclusion

This is it for capacitors in PE Power. You now have rich knowledge of capacitors’ behavior and working along with different parameters of calculating the energy, voltage, and capacitance as inhibited by capacitors in PE Power.

If you are gearing up for PE Power exam preparation, don’t forget to check out valuable study resources, guides, and tailored preparation courses at Study for FE – your go-to place for all things FE-related.

wasim-smal

Licensed Professional Engineer in Texas (PE), Florida (PE) and Ontario (P. Eng) with consulting experience in design, commissioning and plant engineering for clients in Energy, Mining and Infrastructure.