Power System - Section 1
Basic Electrical Elements, Single-Phase Transmission and Three-Phase Transmission
This section covers the basic concepts required for understanding electrical power systems. It begins with the fundamental electrical elements—resistor, inductor, and capacitor—and their energy-storage properties. It then introduces single-phase AC power, including active, reactive, and apparent power and power factor. Finally, it explains balanced three-phase systems and the important equations used to calculate three-phase active, reactive, and apparent power.
Q1. What are the basic electrical elements in power system?
A1. Electrical circuits are mainly represented using three passive elements: resistance R, inductance L, and capacitance C. Each element behaves differently when electrical energy is supplied.
1.1 Resistor
A resistor opposes the flow of current and converts electrical energy into heat.
A resistor follows Ohm's Law: V = I.R,
Where,
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V = voltage in volts
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I = current in amperes
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R = resistance in ohms
Power consumed by a resistor: P = V.I
Using Ohm's law: P = I².R = V²/R
The electrical energy converted into heat during time is: W = I².R.t
1.2 Inductor
An inductor stores electrical energy in the form of a magnetic field.
The energy stored in an inductor is: WL = 0.5*L.I²
where, L = inductance in Henries
The inductive reactance is: XL = ω.L
Since, ω = 2*π.f. So, XL = 2*π.f.L
For an ideal inductor, current lags voltage by 90°.
1.3 Capacitor
A capacitor stores electrical energy in an electric field.
The energy stored in a capacitor is: WC = 0.5*C.V²
where, C = capacitance in Farads
The capacitive reactance is: XC = 1/ω.C = 1/2*π.f.C
For an ideal capacitor, current leads voltage by 90°.
Q2. What are the elements of Single-Phase Transmission system?
A2. A single-phase AC system consists of an alternating voltage and current. When the load is reactive, voltage and current are separated by a phase angle ϕ.
2.1 Sinusoidal Voltage & Current
The instantaneous voltage can be represented as: v = Vm.sin ω.t
The current can be written as: i = Im.sin(ω.t - ϕ)
where:
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Vm = maximum voltage
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Im = maximum current
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ω = angular frequency
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t = time
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ϕ = phase angle
2.2 Instantaneous Power
Instantaneous power is: p = v.i
For sinusoidal voltage and current: p = V.I.[cosϕ - cos(2ωt - ϕ)]
where and are RMS values.
The instantaneous power varies with time, while its average value represents the active power.
2.3 Active Power
Active power is the average power actually consumed by the load.
Relation: P = V.I.cosϕ
Unit: watt (W)
Examples include power converted into:
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Heat
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Mechanical work
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Light
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Useful output
2.4 Reactive Power
Reactive power is associated with energy exchange between the source and reactive elements such as inductors and capacitors.
Relation: Q = V.I.sinϕ
Unit: volt-ampere reactive (VAr)
For an inductive load, reactive power is positive.
For a capacitive load, reactive power is negative.
2.5 Apparent Power
Apparent power is: S = V.I
Unit: volt-ampere (VA)
The relationship between active, reactive, and apparent power is: | S | = √(P² + Q²)
or S² = (P² + Q²)
2.6 Power Factor
Power factor indicates the relationship between active and apparent power.
Power Factor = cosϕ, where cosϕ = P/S.
Types of power factor:
Unity power factor: ϕ = 0°, cosϕ = 1. Occurs for a purely resistive load.
Lagging power factor: Occurs mainly with inductive loads.
Leading power factor: Occurs mainly with capacitive loads.
2.7 Reactive Power Compensation
Inductive loads require reactive power. Capacitors can be connected to compensate for part of this reactive power.
Qnet = QL - QC
This can reduce the reactive power supplied by the source and consequently reduce the current required for a given active-power demand.
Q3. What are the elements of Three-Phase Transmission system?
A3. A three-phase system contains three sinusoidal voltages of equal magnitude and frequency, separated from one another by 120°.
For a balanced system:
va = Vm.sin ω.t
vb = Vm.sin (ω.t - 120°)
vc = Vm.sin (ω.t + 120°)
The same 120° phase displacement exists between the corresponding phase currents.
3.1 Three-Phase Active Power
Using phase quantities: P3ph = 3*Vp.Ip.cosϕ
Using line quantities: P3ph = √3*VL.IL.cosϕ
where:
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Vp = phase voltage
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Ip = phase current
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VL = line voltage
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IL = line current
3.2 Three-Phase Reactive Power
Using phase quantities: Q3ph = 3*Vp.Ip.sinϕ
Using line quantities: Q3ph = √3*VL.IL.sinϕ
3.3 Three-Phase Apparent Power
Using phase quantities: S3ph = 3*Vp.Ip
Using line quantities: S3ph = √3*VL.IL
The relationship remains: | S3ph | = √(P3ph² + Q3ph²)
3.4 Important Point About Balanced Three-Phase Power
For a balanced three-phase system, the total instantaneous power is constant.
This is an important advantage of three-phase power systems and is one reason they are widely used for generation, transmission, and industrial loads.
Practice MCQ
1. Which electrical element stores energy in a magnetic field?
A. Resistor
B. Inductor
C. Capacitor
D. Diode
2. The energy stored in an inductor is:
A. L.I²
B. 0.5*L.I²
C. 0.5*C.V²
D. I².R
3. In a purely inductive circuit, the current:
A. Leads voltage by 90°
B. Lags voltage by 90°
C. Is in phase with voltage
D. Leads voltage by 180°
4. The active power in a single-phase AC circuit is:
A. V.I
B. V.I.sinϕ
C. V.I.cosϕ
D. V/I
5. The unit of reactive power is:
A. W
B. VA
C. VAr
D. Wh
6. The power factor of a purely resistive load is:
A. 0
B. 0.5
C. 1
D. -1
7. In a balanced three-phase system, the phase displacement between the three phase voltages is:
A. 60°
B. 90°
C. 120°
D. 180°
8. The active power of a balanced three-phase system using line quantities is:
A. VL.IL.cosϕ
B. 3*VL.IL.cosϕ
C. √3*VL.IL.cosϕ
D. VL.IL.sinϕ
9. A capacitive load generally has:
A. Lagging power factor
B. Leading power factor
C. Unity power factor only
D. Zero power factor only
10. The apparent power in a single-phase AC circuit is:
A. V.I
B. V.I.cosϕ
C. V.I.sinϕ
D. I²R
Answer: 1. B, 2. B, 3. B, 4. C, 5. C, 6. C, 7. C, 8. C, 9. B, 10. A.
Practice Problems
Problem 1 — Resistor Power
A 20 ohm resistor is connected to a 100V DC supply. Calculate the current and power consumed.
Problem 2 — Energy Stored in an Inductor
An inductor of 0.5 H carries a current of 4 A. Calculate the energy stored.
Problem 3 — Single-Phase Active and Reactive Power
A single-phase load consumes 10 A from a 230 V supply at a power factor of 0.8 lagging. Calculate active and reactive power.
Problem 4 — Three-Phase Active Power
A balanced three-phase load is supplied by a 400 V line-to-line voltage and draws 20 A line current at 0.8 power factor lagging. Find the active power.
Problem 5 — Three-Phase Reactive Power
A three-phase load operates at 415 V, draws 30 A, and has a 0.9 lagging power factor. Calculate the reactive power.