Three Phase AC Circuits
Contents
Explain Generation of Three-Phase EMF with Phasor diagram 1
Describe Three-Phase Star & Delta Connections 5
List advantages of Three-Phase systems 6
Derive the expression for Line and Phase voltages in Delta Connection 7
Explain Power in Star Connection 8
Explain Power in Delta Connection 10
Describe Unbalanced Delta connected load 11
Describe Unbalanced Star connected load with 3-wire and 4-wire system 12
Explain Generation of Three-Phase EMF with Phasor diagram
In a three-phase system, three identical voltages or EMFs with the same frequency are generated, each having a phase difference of 120 degrees. These voltages can be created by a three-phase AC generator that incorporates three identical windings positioned 120 degrees apart from each other electrically.
When the windings are fixed in place, and the magnetic field is rotated, as illustrated in figure A, or when the windings remain stationary while the magnetic field is rotated, as depicted in figure B, an EMF is induced in each winding. The magnitude and frequency of these EMFs are identical, but they are phase-shifted by an angle of 120 degrees relative to one another.
Figure A
Consider the above figure, which displays three identical coils labeled as a1a2, b1b2, and c1c2. In this configuration, a1, b1, and c1 represent the starting terminals, while a2, b2, and c2 represent the finishing terminals of the coils. It is crucial to maintain a phase difference of 120 degrees between the starting terminals a1, b1, and c1.
Now, suppose the three coils are aligned on the same axis and are rotated either by keeping the coils stationary and moving the magnetic field or by rotating the coils while keeping the magnetic field fixed in an anticlockwise direction at a rate of (ω) radians per second. As a result of this rotation, three electromotive forces (EMFs) are induced in the respective coils.
Referring to figure C, we can analyze the magnitudes and directions of the induced EMFs as follows:
The coil a1a2 exhibits zero induced EMF initially and then increases in the positive direction, as illustrated by the waveform ea1a2 in figure C.
The coil b1b2 is electrically 120 degrees behind the coil a1a2. The induced EMF in this coil is negative and progressively reaches its maximum negative value, represented by the waveform eb1b2.
Similarly, the coil c1c2 lags behind the coil b1b2 by 120 degrees, or equivalently, it is 240 degrees behind the coil a1a2. The induced EMF in this coil is positive but gradually decreases, as depicted by the waveform ec1c2 in figure C.
Phasor Diagram:
In the phasor diagram shown below, the induced EMFs in the three coils within a three-phase circuit share the same magnitude and frequency. Additionally, they are displaced by an angle of 120 degrees from one another.
The EMFs in three-phase circuits can be represented by the following equations:
In summary, the generation of three-phase EMF involves a three-phase generator with three windings displaced by 120 degrees from each other. As the rotor rotates, it induces voltages in the windings, which can be represented as phasors in a phasor diagram. The phasor diagram helps visualize the magnitude and phase relationships between the three voltages, enabling efficient power transmission in three-phase systems.
Describe Three-Phase Star & Delta Connections
Three-phase electrical systems can be connected in two primary configurations: star (Y) and delta (Δ) connections. These configurations determine how the three phases are interconnected in a three-phase power system.
- Three-Phase Star (Y) Connection:
In a star connection, the three phases are connected at a common point, forming a star-like shape. This common point is known as the neutral or star point (N). Each phase is connected to a separate terminal, labeled as R, Y, and B, representing the three phases.
The key characteristics of a three-phase star connection are as follows:
- The voltage across any phase and the neutral (phase-to-neutral voltage) is lower than the voltage between any two phases (phase-to-phase voltage).
- The phase currents and line currents are different. The line currents are higher than the phase currents by a factor of √3.
- The phase angles between the voltages are 120 degrees.
A star connection is commonly used in low- and medium-voltage distribution systems, where a neutral point is required for grounding and for providing single-phase loads.
- Three-Phase Delta (Δ) Connection:
In a delta connection, the three phases form a closed loop, resembling the shape of a triangle. Each phase is connected to the adjacent phase, creating a continuous path.
The key characteristics of a three-phase delta connection are as follows:
- The phase-to-phase voltage is the same as the phase-to-neutral voltage in a star connection.
- The phase currents and line currents are equal in a delta connection.
- The phase angles between the voltages are 120 degrees, similar to a star connection.
A delta connection is commonly used in high-voltage transmission systems and in applications where a neutral connection is not necessary.
It’s worth noting that these two configurations, star and delta, can be converted from one to another using appropriate connections. This conversion is known as a star-delta transformation or delta-star transformation.
In summary, the star (Y) and delta (Δ) connections are the two main methods of interconnecting three-phase electrical systems. The star connection provides a neutral point, while the delta connection does not. Each configuration has specific applications based on voltage requirements, grounding needs, and system design considerations.
List advantages of Three-Phase systems
There are several advantages of three-phase systems, which include:
- Increased power transmission: Three-phase systems can transmit more power than single-phase systems, as the power transmission is more efficient and the current is distributed evenly.
- Reduced power losses: Three-phase systems have lower power losses compared to single-phase systems, as the current is distributed evenly and the power loss in the conductor is reduced.
- Improved power factor: The power factor of a three-phase system is generally higher than that of a single-phase system, leading to improved power utilisation and reduced power losses.
- Increased motor efficiency: Three-phase motors are more efficient than single-phase motors, as the power is supplied evenly and without fluctuations, leading to improved performance and efficiency.
- Simplified electrical distribution: Three-phase systems are easier to distribute and connect, as they only require three conductors, while single-phase systems require four.
- Increased reliability: Three-phase systems are more reliable than single-phase systems, as any failure in one phase will not result in a complete power outage, but rather a reduction in power supply.
- Lower costs: Three-phase systems are less expensive than single-phase systems, as they are simpler to design, install, and maintain, and require smaller conductors for the same power transmission.
Overall, three-phase systems offer numerous advantages over single-phase systems, making them a popular choice for power transmission and distribution in industrial and commercial applications.
Derive the expression for Line and Phase voltages in Delta Connection
In a delta-connected system, the line voltage is the voltage between any two line wires, while the phase voltage is the voltage between any line wire and the corresponding neutral point or load.
Assuming a balanced three-phase delta-connected system, where each phase has a voltage Vph and an impedance Z, the line voltage Vline is given by:
Vline = √3 Vph
This is because the line voltage is equal to the phase voltage multiplied by the square root of 3, due to the phase shift between the phases in a three-phase system.
On the other hand, the phase voltage Vph is given by:
Vph = Vline / √3
This is because the phase voltage is equal to the line voltage divided by the square root of 3, due to the same phase shift.
So, in a delta-connected system, the relationship between the line voltage and the phase voltage is fixed, and can be expressed as:
Vline = √3 Vph
and
Vph = Vline / √3
Explain Power in Star Connection
The power in a three-phase star (or Y) connection can be calculated using the line voltage and line current. The apparent power in a three-phase system is equal to the product of the line voltage and line current. The formula for apparent power is given by:
S = VL * IL
where S is the apparent power, VL is the line voltage, and IL is the line current.
The real power in a three-phase system is equal to the product of the line voltage and line current times the cosine of the phase angle between the voltage and current waveforms. The formula for real power is given by:
P = VL * IL * cos(Φ)
where P is the real power, VL is the line voltage, IL is the line current, and Φ is the phase angle between the voltage and current waveforms.
The reactive power in a three-phase system is equal to the product of the line voltage and line current times the sine of the phase angle between the voltage and current waveforms. The formula for reactive power is given by:
Q = VL * IL * sin(Φ)
where Q is the reactive power, VL is the line voltage, IL is the line current, and Φ is the phase angle between the voltage and current waveforms.
The power factor in a three-phase system is equal to the cosine of the phase angle between the voltage and current waveforms. The formula for power factor is given by:
PF = cos(Φ)
where PF is the power factor and Φ is the phase angle between the voltage and current waveforms.
Explain the relationship between Line and Phase Voltages Connection and Currents in Delta Connection
In a delta connection, the voltage between each phase and ground (line voltage) is equal to the phase voltage. For example, if the phase voltage between A and B is V, then the line voltage between line 1 and ground is also V. In this connection, the line current is equal to the phase current.
The relationship between line and phase voltages in a delta connection can be expressed as follows:
VAB = VBC = VCA = Vph
Where VAB, VBC, and VCA are the line voltages and Vph is the phase voltage.
Similarly, the relationship between line and phase currents in a delta connection can be expressed as:
I1 = IA
I2 = IB
I3 = IC
Where I1, I2, and I3 are the line currents and IA, IB, and IC are the phase currents.
Explain Power in Delta Connection
The power in a delta connection is calculated in the same way as in a single-phase or three-phase system. The power in a delta connection is the product of voltage and current at each phase. In a delta connection, the line current is equal to the phase current, and the line voltage is equal to the phase voltage, so the power in a delta connection can be calculated using either line or phase quantities.
The power in a delta connection can be expressed as follows:
P = Vph * Iph
Where P is the power, Vph is the phase voltage, and Iph is the phase current.
The power factor in a delta connection is defined as the ratio of real power to apparent power, and it is expressed as a scalar value between 0 and 1. The power factor in a delta connection is the same for all the phases. The real power in a delta connection is equal to the product of the phase voltage, phase current, and the cosine of the phase angle between the voltage and current. The apparent power in a delta connection is equal to the product of the phase voltage and phase current.
Explain Two-Wattmeter method for Balanced Three-Phase load for measurement of Power and Power factor
The Two-Wattmeter method is a technique used to measure the power and power factor of a balanced three-phase load. It is commonly employed in situations where accurate power measurement is required for balanced systems, such as in industrial power distribution or motor control applications.
The Two-Wattmeter method involves the use of two wattmeters connected to the load in a specific configuration. The wattmeters are typically true power measurement devices that can measure both active and reactive power.
Here’s a step-by-step explanation of how the Two-Wattmeter method works:
- Connection Configuration:
The two wattmeters are connected in the system such that one wattmeter measures the power in one phase (W₁), and the other wattmeter measures the power in the combination of the remaining two phases (W₂).
- Phase-to-Phase Voltages:
Measure the phase-to-phase voltages (V_ab, V_bc, V_ca) of the load using appropriate voltage measurement instruments. These voltages are required for accurate power calculations.
- Current Measurement:
Measure the line currents (I_a, I_b, I_c) of each phase using suitable current measurement devices, such as current transformers.
- Power Calculations:
Using the measured voltages and currents, the power readings of the two wattmeters can be used to calculate the total power and power factor of the balanced three-phase load.
- The power measured by the first wattmeter (W₁) represents the power in the individual phase.
- The power measured by the second wattmeter (W₂) represents the power in the combination of the other two phases.
The total power (P_total) of the balanced load can be calculated as:
P_total = W₁ + W₂
The power factor (PF) of the balanced load can be determined using the power measurements as well:
PF = P_total / (V_ab * I_a)
By employing the Two-Wattmeter method, the power and power factor of a balanced three-phase load can be accurately determined. This method takes into account the individual phase power and the power in the combination of the remaining two phases, enabling comprehensive power measurement and analysis.
Describe Unbalanced Delta connected load
An unbalanced delta-connected load refers to a three-phase electrical load in which the impedances or loads connected between the three phases are unequal. This imbalance can occur due to variations in resistances, reactances, or both in the individual components of the load.
In an unbalanced delta configuration, the three phases form a closed loop, resembling a triangle. Each phase is connected to the adjacent phase, creating a continuous path for the flow of current.
The key characteristics of an unbalanced delta-connected load are as follows:
- Imbalance in Currents: Due to the unequal impedances of the loads, the phase currents flowing through each phase of the delta load are unequal. This results in different magnitudes of current flowing through each phase.
- Voltage Imbalance: The unequal currents in an unbalanced delta load lead to an imbalance in the voltages across the load. This is because the voltage drop across each load is proportional to the current flowing through it. Therefore, the voltage levels across the phases will be different.
- Unbalanced Power Distribution: In an unbalanced delta load, the power distribution among the three phases is unequal. The power consumed by each phase depends on its respective impedance or load.
- Unbalanced Power Factor: The power factor of an unbalanced delta-connected load can vary between the individual phases due to the unequal impedances. This means that the load may have different leading or lagging power factors for each phase.
It’s important to note that an unbalanced delta load can cause various issues, including excessive heating in certain components, unbalanced voltage levels, and power quality problems. Therefore, it’s necessary to carefully analyze and address these imbalances to ensure the safe and efficient operation of the electrical system.
Balancing techniques such as load redistribution, impedance correction, or phase swapping can be employed to mitigate the effects of an unbalanced delta-connected load and restore balanced conditions in the system.
Describe Unbalanced Star connected load with 3-wire and 4-wire system
When it comes to an unbalanced star-connected load, we can consider two scenarios: a 3-wire system and a 4-wire system. Let’s explore each of them:
- Unbalanced Star Connected Load in a 3-Wire System:
In a 3-wire system, the star-connected load consists of three individual impedances or loads, typically denoted as Z₁, Z₂, and Z₃, connected between the three phases (R, Y, B) and a common neutral point (N).
An unbalanced star-connected load means that the three impedances or loads are unequal. This can happen due to variations in resistances, reactances, or both. As a result, the load draws different currents from each phase, leading to an imbalance in the system.
Due to this imbalance, the neutral current is not zero, even in the absence of a neutral connection. The neutral current is the vector sum of the three phase currents and flows in the neutral wire (if present) or through the earth in a grounded system.
- Unbalanced Star Connected Load in a 4-Wire System:
In a 4-wire system, an additional neutral wire is included along with the three phase wires. The neutral wire provides a dedicated path for the return currents of the loads. This system is often used in commercial and industrial applications to support both single-phase and three-phase loads.
In the case of an unbalanced star-connected load in a 4-wire system, the neutral wire carries the unbalanced currents resulting from the unequal impedances of the loads. The neutral current is the algebraic sum of the phase currents and can be significantly higher than in a balanced load scenario.
To avoid potential issues in the system, such as overheating of conductors or voltage fluctuations, it is crucial to size the neutral conductor appropriately based on the expected unbalance in the load.
Overall, in both the 3-wire and 4-wire systems, an unbalanced star-connected load leads to unequal currents in the phases and a non-zero neutral current. Proper analysis and consideration of these imbalances are necessary to ensure the safe and efficient operation of the electrical system.