# Single-Phase Circuits in PE Power Exam

Welcome to my blog on single-phase circuits in the PE Power Exam. According to the guidelines provided by NCEES®, the single-phase circuit is an important topic you must not overlook for PE Power exam preparation.

Aspiring power engineers preparing for the PE Power Exam must understand single-phase circuits as they form a fundamental part of Power systems.

This blog will delve into the key concepts, analysis techniques, and practical applications of single-phase circuits, equipping you with the knowledge and skills to confidently tackle this critical exam topic. Let’s dive in and discuss this in detail.

## Fundamentals of Single-Phase Circuits

To begin with, single-phase circuits in the PE Power exam, let’s start with some fundamental concepts. The single-phase circuits are electrical circuits that use a single alternating current (AC) waveform.

These circuits are commonly used in residential and small commercial applications. In a single-phase circuit, the voltage and current vary sinusoidally over time and have the same frequency.

### Components Used in Single-Phase Circuits

The critical components of power that you must know and form the structure of single-phase circuits in the PE Power exam include:

• Voltage source – A single-phase circuit typically receives power from a single-phase AC generator or utility grid.
• Transformer – Different types of transformers are used to step up or down the voltage level in a single-phase circuit, depending on the application.
• Circuit breakers and fuses – These protective devices interrupt the circuit in case of overcurrent or short circuits.
• Switches – Switches control the flow of current in the circuit.
• Loads – Loads in single-phase circuits can be resistive, inductive, or capacitive, depending on the nature of the device being powered.

### Phasor Diagrams in Single-Phase Circuits

Phasor diagrams are graphical representations that analyze voltage and current behavior in single-phase circuits. The following aspects are commonly used to illustrate and solve phasor diagrams in different forms (vector, trigonometric, and Euler’s forms):

• Magnitude – A phasor’s magnitude represents the sinusoidal waveform’s amplitude, typically denoted by uppercase letters such as V or I.
• Angle – The angle of a phasor represents the phase shift or the angular displacement of the waveform from a reference point, typically denoted by lowercase letters such as φ or θ.

Reference axis – A reference axis is used as a point of comparison for the phasors. It represents the zero phase or zero degrees.

Identifying differences between single-phase and three-phase circuits is crucial for further studies. Learn critical differences between Single-phase and three-phase circuits.

In a single-phase circuit, both voltage and current waveforms have the same frequency and vary sinusoidally over time. The voltage waveform leads or lags the current waveform depending on the type of load (resistive, inductive, or capacitive) and the power factor of the circuit. The relationship between voltage and current can be determined using Ohm’s Law or phasor diagrams.

For instance, In a single-phase circuit, the voltage waveform has an amplitude of 120 V and a frequency of 60 Hz. The current waveform has an amplitude of 5 A and lags the voltage waveform by 30 degrees.

To draw the voltage and current waveforms, we start with the reference axis, which represents time. We can assume the reference axis is horizontal.

Draw the voltage waveform:

• The voltage waveform has an amplitude of 120 V. From the reference axis, mark a point at 120 V above the axis.
• Since it is a sinusoidal waveform, the voltage will oscillate above and below the axis. Draw a sinusoidal curve that passes through the marked point, oscillating above and below the reference axis.

Draw the current waveform:

• The current waveform lags the voltage waveform by 30 degrees. From the reference axis, rotate 30 degrees clockwise.
• The current waveform has an amplitude of 5 A. Mark a point at 5 A below the axis from the rotated reference axis.
• Draw a sinusoidal curve that passes through the marked point, oscillating above and below the reference axis.

The resulting waveform will show the voltage waveform leading the current waveform by 30 degrees as the current waveform lags the voltage waveform.

Explore our detailed study guide on phasor diagrams in the PE Power exam.

### Calculating Power in Single-Phase Circuits

Power in a single-phase circuit can be calculated using one of the following formulas, depending on the type of load:

Power (P) = Voltage (V) × Current (I) × Power Factor (PF)

Power (P) = Voltage (V) × Current (I) × Power Factor (PF) × Power Factor Angle (θ)

Consider a single-phase circuit having a voltage of 230 V, a current of 10 A, and a power factor of 0.8 lagging.

To calculate the power using the formula for resistive load:

P = V × I × PF

P = 230 V × 10 A × 0.8

P = 1,840 W

The power in the circuit is 1,840 watts.

*If the load is inductive or capacitive, an additional term of cos(θ) or sin(θ) needs to be multiplied to account for the power factor angle (θ).

## Analysis Techniques for Single-Phase Circuits

Analysis of single-phase circuits in the PE Power exam ranges from some fundamental techniques to some advanced methodologies. Let’s explore this in more detail.

### Ohm’s Law

Ohm’s Law relates the voltage (V), current (I), and resistance (R) in a circuit: V = I × R. It applies to resistive components in single-phase circuits. Impedance (Z) extends Ohm’s Law to include reactive components (inductors and capacitors) apart from resistors. It is a complex quantity that incorporates resistance and reactance. V = I × Z gives the relationship.

Explore in further detail and learn how to calculate Impedance.

*Ohm’s Law determines a resistive component’s voltage, current, or resistance. Impedance calculation and formula are used to calculate the voltage and current in circuits with reactive components, such as inductive or capacitive loads.

### Kirchhoff’s Laws in Single-Phase Circuits

Kruchhoff’s laws are also used to solve single-phase circuits in the PE Power exam. They are categorized into two different types:

• Kirchhoff’s Current Law (KCL) states that the sum of currents entering a node equals the sum of currents leaving the node. It analyzes the current distribution and nodal voltages in complex single-phase circuits.
• Kirchhoff’s Voltage Law (KVL) states that the sum of voltages around any closed loop in a circuit is zero. It analyzes voltage drops and loop currents in complex single-phase circuits.

### Mesh and Nodal Analysis

Mesh Analysis and Nodal Analysis are commonly used techniques in circuit analysis for the single-phase circuits in the PE Power exam. Their main objective is to determine the circuit’s unknown voltages and currents.

#### Mesh Analysis – Methodology

• Identify the essential meshes – Identify and label the essential meshes or loops in the circuit. A mesh is a closed path that doesn’t contain any other closed paths within it.
• Assign loop currents – Assign a clockwise or counterclockwise loop current to each mesh. Label each loop current (e.g., I1, I2, I3, etc.).
• Apply Kirchhoff’s Voltage Law (KVL) – Write a KVL equation for each mesh, considering the voltage drops across the mesh’s resistors, capacitors, and inductors. Express the voltage drops in terms of the assigned loop currents.
• Solve the simultaneous equations – Solve the resulting system of equations to determine the values of the loop currents.
• Calculate other voltages and currents – Once the loop currents are known, you can calculate the voltages and currents of other elements in the circuit using Ohm’s Law or other relevant formulas.

#### Nodal Analysis – Methodology

• Identify essential nodes – Identify and label the essential nodes in the circuit. A node is a point in the circuit where multiple elements are connected.
• Assign node voltages – Assign unknown node voltages to each node, except for the reference node (usually chosen as the ground).
• Apply Kirchhoff’s Current Law (KCL) – Write a KCL equation for each essential node, expressing the sum of currents entering the node equal to the sum of currents leaving the node. Use Ohm’s Law or relevant formulas to express the currents regarding the node voltages.
• Solve the simultaneous equations – Solve the resulting system of equations to determine the values of the node voltages.
• Calculate other voltages and currents – Once the node voltages are known, you can calculate the voltages and currents of other elements in the circuit using Ohm’s Law or other relevant formulas.

Consider a single-phase circuit with the following parameters:

• Voltage source: V = 120 V
• Resistors: R1 = 10 Ω, R2 = 20 Ω, R3 = 30 Ω
• Current source: I = 5 A

Let’s apply Mesh Analysis to analyze the current in the given single-phase circuit. To do so:

Assign loop currents:

Let I1 be the current flowing through the left mesh (counterclockwise) and I2 be the current flowing through the proper mesh (clockwise).

Write KVL equations:

• Loop 1: -V + R1 × I1 + (R1 + R2) × (I1 – I2) = 0
• Loop 2: (R1 + R2) × (I2 – I1) + R3 × I2 = 0

Simplify the equations:

-120 + 10I1 + 30(I1 – I2) = 0

30(I2 – I1) + 30I2 = 0

Solve the equations (simultaneous solution):

-40I1 + 30I2 = 120 …(1)

30I1 – 30I2 = 0 …(2)

Solving equations (1) and (2) simultaneously gives:

• I1 = 1.5 A
• I2 = 1.5 A

Calculate other currents and voltages:

• Current through R1: I1 = 1.5 A
• Current through R2: I1 – I2 = 1.5 A – 1.5 A = 0 A
• Current through R3: I2 = 1.5 A
• Voltage across R1: V1 = R1 × I1 = 10 Ω × 1.5 A = 15 V
• Voltage across R2: V2 = R2 × (I1 – I2) = 20 Ω × 0 A = 0 V
• Voltage across R3: V3 = R3 × I2 = 30 Ω × 1.5 A = 45 V

Therefore, the currents and voltages in the given single-phase circuit are as follows:

• I1 = 1.5 A
• I2 = 1.5 A
• V1 = 15 V
• V2 = 0 V
• V3 = 45 V

*The choice between the two techniques depends on the complexity and structure of the circuit. Mesh Analysis is more suitable for circuits with multiple loops, while Nodal Analysis is preferable for circuits with multiple nodes.

### Superposition Theorem in Single-Phase Circuits

The Superposition Theorem is a powerful analysis technique used in single-phase circuits to determine the individual contributions of multiple independent sources (voltage sources or current sources) to the overall response of the circuit.

It states that the response of a linear circuit to multiple sources can be calculated by considering the effects of each source separately while setting the other sources to zero.

### Superposition Theorem – Methodology

• Identify independent sources – Identify all the circuit’s independent sources, including voltage and current.
• Turn off all but one independent source – Consider one independent source at a time and turn all other sources off (or deactivate) by replacing voltage sources with short circuits and current sources with open circuits.
• Analyze the circuit – With only one independent source active, analyze the circuit using standard techniques such as Ohm’s Law, Kirchhoff’s Laws, or mesh/nodal analysis. Determine the voltages and currents in the circuit.
• Repeat for each independent source – Repeat steps 2 and 3 for each independent source, considering one source at a time while turning off the others.
• Superposition principle – The final response of the circuit is obtained by summing the individual responses calculated in step 3 for each independent source. Consider the algebraic signs and directions of the voltages and currents calculated during the analysis.

*Superposition applies to linear circuits where the response is directly proportional to the input. It assumes that the sources and circuit elements are linear and doesn’t consider nonlinear components such as diodes or transistors.

### Use-Cases of Superposition Theorem

The Superposition Theorem is particularly useful in analyzing complex circuits with multiple sources and components. It provides the following benefits:

• Simplifies circuit analysis – By considering the effects of each source individually, complex circuits can be broken down into simpler sub-circuits, making the analysis more manageable.
• Allows for easy identification of source contributions – The theorem enables determining how each source influences the voltages and currents in the circuit, facilitating fault finding or optimization.
• Works well for circuits with linear components – Superposition is effective in circuits consisting mainly of linear components such as resistors, capacitors, and inductors.
• Complements other analysis techniques – Superposition can be combined with other techniques, such as mesh or nodal analysis, to solve complex circuits more efficiently.

Consider a single-phase circuit with the following parameters:

• Voltage source: V = 120 V
• Resistors: R1 = 10 Ω, R2 = 20 Ω
• Current source: I = 5 A

To calculate the contribution of the voltage source:

1. Disconnect the current source (replace it with an open circuit).
2. Calculate the voltage across each resistor using voltage division:
• Voltage across R1 = V × (R1 / (R1 + R2)) = 120 V × (10 Ω / (10 Ω + 20 Ω)) = 40 V
• Voltage across R2 = V × (R2 / (R1 + R2)) = 120 V × (20 Ω / (10 Ω + 20 Ω)) = 80 V
1. Calculate the contribution of the current source:

*Disconnect the voltage source (short-circuit it).

1. Calculate the current through each resistor using current division:
• Current through R1 = I × (R2 / (R1 + R2)) = 5 A × (20 Ω / (10 Ω + 20 Ω)) = 2 A
• Current through R2 = I × (R1 / (R1 + R2)) = 5 A × (10 Ω / (10 Ω + 20 Ω)) = 1 A
1. Calculate the total voltage and total current:

The total voltage across R1 and R2 is the sum of the contributions from the voltage source and the current source:

• Voltage across R1 = 40 V + 0 V = 40 V
• Voltage across R2 = 80 V + 0 V = 80 V

The total current through R1 and R2 is the sum of the contributions from the voltage source and the current source:

• Current through R1 = 2 A + 0 A = 2 A
• Current through R2 = 1 A + 0 A = 1 A

Therefore, the voltages and currents in the given single-phase circuit are as follows:

• Voltage across R1 = 40 V
• Voltage across R2 = 80 V
• Current through R1 = 2 A
• Current through R2 = 1 A

## Single-Phase Circuit Analysis Methods

Let’s look at some fundamental analysis methods and their key concepts used for the analysis of single-phase circuits in the PE Power exam.

### Series and Parallel Connections

• In series connections, components are connected sequentially, sharing the same current. The total resistance in a series connection is the sum of individual resistances, while the total voltage is the sum of individual voltage drops.

*In series connections, voltage division occurs based on the ratio of resistances. The voltage drop across each component is proportional to its resistance.

• In parallel connections, components are connected in parallel, sharing the same voltage. The reciprocal of the total resistance in a parallel connection is the sum of the reciprocals of individual resistances, while the total current is the sum of individual currents.

*In parallel connections, current division occurs based on the inverse ratio of resistances. The current flowing through each component is inversely proportional to its resistance.

### Thevenin and Norton Equivalent Circuits

• Thevenin Equivalent Circuit represents a complex circuit as an equivalent voltage source in series with an equivalent resistance. It simplifies the circuit analysis by reducing it to a single voltage source and a resistor.
• Norton Equivalent Circuit represents a complex circuit as an equivalent current source parallel to an equivalent resistance. It provides an alternative representation to Thevenin’s theorem.

Explore in more detail about Thevenin and Norton Equivalent Circuits.

### Impedance Matching

Impedance matching involves adjusting a load’s impedance to match the source’s impedance in a single-phase circuit. It ensures maximum power transfer from the source to the load. Matching the impedance reduces the reflections and losses in the circuit, resulting in efficient power transfer.

For instance, a source with an internal impedance of 20 Ω delivers power to a load with a resistance of 50 Ω. Let’s find the load impedance that results in maximum power transfer.

To achieve maximum power transfer, the load impedance should be the complex conjugate of the source impedance.

Calculate the complex conjugate of the source impedance:

The complex conjugate of 20 Ω is 20 – j0 Ω.

The load impedance should be the complex conjugate of the source impedance, which is 20 – j0 Ω.

Hence, maximum power transfer is achieved by matching the load impedance to the complex conjugate of the source impedance (20 – j0 Ω). In this case, the load impedance is purely resistive, indicating that the load is purely resistive, which results in efficient power transfer from the source to the load.

## Power Factor Correction in Single-Phase Circuits

Power Factor Correction is the process of improving the power factor of a single-phase circuit. The power factor is the ratio between real (active) and apparent (total) power.

A low power factor indicates inefficient power utilization, leading to increased energy consumption, higher electricity bills, and additional strain on electrical distribution systems.

In single-phase circuits, power factor correction improves power by reducing the reactive power component and increasing the displacement power factor. Mathematically, the power factor (PF) can be expressed as the cosine of the angle (θ) between the voltage and current phasors:

PF = cos(θ)

Where:

• PF is the power factor
• θ is the phase angle between the voltage and current waveforms

The power factor correction aims to minimize the angle θ and bring it closer to zero, resulting in a higher power factor. The power factor correction methods typically involve adding reactive components, such as capacitors, to the circuit.

Reactive power (Q) is the component of power that oscillates between the source and the load without being consumed by the load. It is proportional to the sine of the angle θ:

Q = Apparent Power (S) × sin(θ)

Adding capacitors in parallel to the load can compensate for the reactive power component. Capacitors act as reactive power generators, supplying the reactive power required by the load. As a result, the angle θ decreases, leading to an improved power factor.

Adding capacitors reduces the overall reactive power and increases the displacement power factor, bringing it closer to unity (1.0). This reduces the reactive power flow between the source and the load, minimizing energy losses and increasing the efficiency of the circuit.

*Power factor correction should be implemented carefully to avoid overcorrection, which can result in leading power factor conditions. Overcorrection may lead to issues in the electrical system, such as increased voltages and harmonic distortion.

### Power Factor Correction Methods

The following power factor correction methods are widely used in the Power industry.

• Capacitor Banking – Adding a capacitor bank parallel to the load compensates for reactive power, improving the power factor.
• Synchronous Condenser – A synchronous condenser is a rotating machine connected to the circuit helps provide reactive power and improve the power factor.
• Static Var Compensator (SVC) – An SVC consists of power electronic devices that dynamically inject or absorb reactive power to maintain a desirable power factor.
• Active Power Factor Correction (APFC) – Utilizing active power electronics to control the power factor by continuously adjusting the reactive power injection.
• Harmonic Filters – Implementing harmonic filters in the circuit helps suppress harmonic currents, reducing power factor distortions caused by nonlinear loads.

### Use-Cases and Applications of Power Factor Correction

Industrial Applications – Power factor correction is commonly employed in industrial facilities with large motor-driven equipment, such as motors, pumps, and compressors.

Commercial Buildings – Power factor correction is utilized in commercial buildings to improve energy efficiency and reduce electricity costs.

Power Distribution Systems – Implementing power factor correction techniques at the utility level helps improve system stability and reduce losses.

Renewable Energy Integration – Power factor correction is essential in integrating renewable energy sources, such as wind turbines and solar panels, into the grid.

Data Centers – Power factor correction is crucial in data centers to maintain efficient operation and optimize energy consumption.

Let’s calculate and analyze Power Factor in Single-Phase Circuit. Consider In a single-phase circuit, the voltage is 230 V, and the current is 10 A. The power factor angle (θ) is 45 degrees lagging.

To Calculate the power factor:

Power factor (PF) = cos(θ) = cos(45 degrees) = 0.707

Determine the apparent power (S):

Apparent power (S) = Voltage (V) × Current (I) = 230 V × 10 A = 2300 VA

Determine the real power (P):

Real power (P) = Apparent power (S) × Power factor (PF) = 2300 VA × 0.707 = 1625 W

The power factor of the circuit is 0.707, indicating that the circuit is operating at a lagging power factor. The real power consumed is 1625 W, which is the adequate power utilized by the load.

Explore power factor calculations in detail to create a firm grip on this crucial topic

## Safety Considerations in Single-Phase Circuits

The following safety considerations are instrumental in ensuring the safe and secure handling and operations of single-phase circuits in the PE Power exam and in real life:

### Electrical Insulation and Protection

Ensure proper insulation of conductors and equipment in single-phase circuits to prevent accidental contact with live parts. Inspect and avoid using equipment with frayed cords, damaged insulation, or broken plugs.

Utilize appropriate protective measures such as insulating gloves, goggles, and other personal protective equipment (PPE) when working on branch circuits or handling electrical components.

### Proper Circuit De-Energization

Before working on receptacles or electrical equipment in single-phase circuits, always de-energize the circuit by switching off the mains. Consider posting clear signs on the service panel to prevent accidental activation of the main switch.

Before repairs or maintenance, use electrical testers to verify the circuit is de-energized. Test all wires, including those within the service panel and any exposed or hanging wires, to ensure they are free from electrical current.

### Grounding and Bonding

Implement appropriate grounding and bonding techniques in single-phase systems to provide a safe path for fault currents, ensuring the protection of individuals and equipment. Follow the grounding and bonding guidelines stipulated by the electrical code specific to your region.

Learn more about Grounding Resistance Testing Methods and discover how to implement the Equally-spaced 4-pin Method, Unequally spaced 4-pin Method, and Driven Rod Method for ground testing.

### Equipment Safety and Warning Signs

Be aware of electrical hazards in single-phase circuits, such as exposed energized parts and unguarded equipment that may unexpectedly become energized. Adhere to safety rules and warning signs, including those indicating “Shock Risk,” as the applicable electrical code prescribes.

Regularly inspect and maintain electrical equipment to ensure it remains in safe operating condition. Promptly address any signs of damage, wear, or potential hazards.

### Work Environment and Equipment

Avoid working with electricity in wet conditions or with wet hands, as it enhances electrical conductivity and increases the risk of electric shock.

Refrain from using aluminum or steel ladders while working on receptacles at heights, as they can conduct electricity. Instead, utilize ladders made of non-conductive materials such as bamboo, wood, or fiberglass.

### Compliance With Electrical Codes

You must acknowledge and understand electrical codes and regulations applicable to your country. Stay updated on any revisions or amendments to ensure compliance and adherence to the recommended safety practices.

Read our detailed guide on codes and standards in the PE Power exam.

### GFCI Testing

Conduct regular checks on Ground Fault Circuit Interrupters (GFCIs) within single-phase circuits. GFCIs are designed to protect against electrical shock hazards and should be tested monthly to ensure proper functioning. Follow the manufacturer’s instructions for testing procedures.

Remember, electrical work can be hazardous. Electrical safety and inspection should be done by qualified professionals who adhere to established safety protocols, codes, and standards.

For information and study preparation for single-phase circuits in the PE Power exam, look at the following techniques for troubleshooting and maintaining single-phase systems.

### Visual Inspection

Perform regular visual inspections of the single-phase circuit components, including wiring, connections, switches, and outlets. Look for signs of wear, damage, loose connections, or any abnormality that may indicate potential issues.

### Test Equipment

Measurement and Instrumentation can make or break your circuit maintenance and troubleshooting drive. Utilize appropriate test equipment, such as multimeters, to measure circuit voltage, current, and resistance.

Test the functionality of the outlets, switches, and other electrical devices to identify malfunctions.

### Systematic Approach

Adopt a systematic approach when troubleshooting single-phase circuits. Begin with the most apparent or accessible components and work towards more complex or hard-to-reach areas.

Follow a logical flow, checking each component and connection along the circuit path to identify the source of the problem.

### Circuit Breakers and Fuses

Check and reset circuit breakers or replace blown fuses when a circuit or specific electrical device fails to function. Ensure the amperage rating of the circuit breaker or fuse matches the electrical load to prevent overloading.

### Loose Connections

Inspect and tighten all electrical connections, including wire terminals, screws, and connectors. Loose connections can cause intermittent power loss or overheating.

### Wire Insulation

Insulation testing in the PE Power exam is also an important topic to prepare for. Examine the insulation of wires for signs of damage, fraying, or deterioration. Replace any wires with compromised insulation to prevent electrical hazards.

Check the load distribution across phases in the circuit to ensure balanced operation. Unbalanced loads can lead to overheating, voltage fluctuations, and inefficient power consumption.

### Regular Maintenance

Establish a routine maintenance schedule for single-phase circuits. Clean equipment, remove dust or debris and inspect for any signs of wear or deterioration.

For a more detailed and in-depth study of safety and protection in the circuits, read our detailed guide on surge protection in the PE Power exam.

### Conclusion

Understanding single-phase circuits in the PE Power Exam is essential, per the guidance provided by NCEES®. With the right resources and study materials, you can confidently approach this important exam topic and increase your chances of success. Read our detailed study guide on three-phase circuits in the PE Power exam to explore how three-phase circuits work.

Study for FE is here to support your journey. As a trusted platform, we offer a range of resources, including practice exams, study guides, and video tutorials, designed to help you grasp the intricacies of single-phase circuits and excel in your PE Power Exam preparation. Explore Study for FE and confidently embark on your path to PE licensure. 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.