Measurement of Power, Energy, and Power Factor - GATE Notes & Videos for Electrical

Electric Power Measurements

DC Electric Circuits

To characterise a DC electrical bipole (single two-terminal element) we measure the current, I, flowing through it and the voltage, V, across its terminals. These measurements are made with a DC ammeter and a DC voltmeter. The instantaneous electrical power associated with the bipole is

P = V × I

Using a consistent sign convention, P is positive when the element absorbs power and negative when it supplies power. In DC steady state this product gives the constant power delivered to or taken from the element.

Single-Phase AC Circuits

For sinusoidal single-phase alternating voltage and current, instantaneous quantities vary in time. Let

v(t) = Vm cos(ωt)

i(t) = Im cos(ωt - φ)

Then the instantaneous power is

p(t) = v(t) i(t) = Vm Im cos(ωt) cos(ωt - φ)

Using trigonometric identities, this expands to

p(t) = (Vm Im / 2) cos φ + (Vm Im / 2) cos(2ωt - φ)

The first term is the average (real) power and the second is an alternating term at twice the supply frequency whose average over a cycle is zero. Using RMS values, Vrms = Vm/√2 and Irms = Im/√2, the average (real) power becomes

P = Vrms × Irms × cos φ

Here cos φ is the power factor angle, the phase difference between the fundamental components of voltage and current.

For reference, the instantaneous power expression may be written as

p(t) = 2 V I cos ωt · cos(ωt - φ)

and the average power (over one period) is

where V and I denote the effective (RMS) values of voltage and current respectively.

Three-Phase Power Measurement using Two Wattmeters

The two-wattmeter method is widely used to measure power in a three-phase, three-wire system. The connection uses two wattmeters connected to measure two line voltages and the respective line currents; the algebraic sum of their readings gives the total active power even for an unbalanced load.

In a star (wye) connection, for the two wattmeters denoted W₁ and W₂ the indicated powers are

• P₁ = V_AB I_A cos Φ_AB-A
• P₂ = V_CB I_C cos Φ_CB-C

Here Φ_CB-C is the phase difference between V_CB and I_C, and V_CB = V_CN - V_BN is the potential across the wattmeter W₂. The total power is the algebraic sum:

P_T = P₁ + P₂

Analysis for a Balanced System

In a balanced three-phase supply and a balanced three-phase load the line voltages have equal magnitudes and are phase-displaced by 120°. The phase difference between a line voltage (for example V_AB) and the corresponding phase voltage (V_AN) is 30°. If the load is inductive so that the line current lags its phase voltage by angle φ, then the phase difference between I_A and V_AB is (30° + φ).

• For a balanced system, V_AB = V_CB = V_L, the line voltage magnitude.

Thus, the wattmeter readings are

P₁ = V_L I_L cos(φ + 30°)

P₂ = V_L I_L cos(φ - 30°)

The sum is

P₁ + P₂ = V_L I_L [cos(φ + 30°) + cos(φ - 30°)]

Using the identity cos(a + b) + cos(a - b) = 2 cos a cos b, we get

P₁ + P₂ = 2 V_L I_L cos φ cos 30° = √3 V_L I_L cos φ

Therefore the algebraic sum of the two wattmeter readings gives the total active power for a three-phase balanced load:

P_T = √3 V_L I_L cos φ

• This is the total real power consumed by the load.
• One wattmeter reading (commonly W₁) can become negative if the power factor angle φ is greater than 60°; the algebraic sum still gives the correct total power.

Determination of Power Factor from Two Wattmeter Readings

Let W₁ and W₂ be the two wattmeter readings for a balanced three-phase system. Then the total real power is

P = W₁ + W₂

The apparent power for a balanced three-phase system is

S = √3 V_L I_L

Using the wattmeter expressions, one can obtain the power factor angle φ from the difference and sum of readings. The useful relations are

W₁ - W₂ = √3 V_L I_L sin φ

W₁ + W₂ = √3 V_L I_L cos φ = P

From these,

tan φ = (W₁ - W₂) / (W₁ + W₂) · (1/√3)

and the power factor is

cos φ = (W₁ + W₂) / S

Measurement of Power Factor

In alternating-current systems three power quantities are important: active power (P), reactive power (Q), and apparent power (S). These apply both to sinusoidal and non-sinusoidal steady-state conditions, but the definitions must be used carefully when waveforms are non-sinusoidal.

1. Classical Definition (Sinusoidal)

For purely sinusoidal voltage and current the power factor (PF) is defined as the cosine of the phase angle between the fundamental components of voltage and current:

PF = cos φ = P / S

Measuring the Phase Angle (Zero-Cross Detection)

• Power factor measurement often reduces to measuring the phase difference between voltage and current.
• Zero-cross detection is a practical method: it detects instants when voltage and current waveforms cross zero and measures the time difference between their corresponding zero crossings.
• From the measured time difference Δt and the known system frequency f, the phase angle is

• Implementation requires accurate detection of zero crossings. A comparator circuit is commonly used to convert sinusoidal waveforms into digital pulses at the zero crossings.
• The LM358 dual operational amplifier is often used as a comparator in low-cost implementations; one amplifier can condition the voltage waveform into a square pulse at its zero crossings, another can do the same for current (after appropriate scaling/isolation).
• Precision of the measured phase angle depends on sampling/processing resolution and timing jitter of the zero-cross circuit.

Power Factor for Non-Sinusoidal Currents

When voltage is sinusoidal but current is distorted (contains harmonics), the classical PF = cos φ no longer fully describes the relation between real and apparent power. In such cases the following definitions are used:

• The Total Harmonic Distortion (THD) of current quantifies the contribution of harmonic components to the RMS current.
• The distortion factor and displacement factor combine to give the true power factor. The relationship between THD and distortion factor follows from harmonic composition:

• If V is purely sinusoidal and the phase of the fundamental current relative to the voltage is φ1, the general formula for power factor becomes

Here the power factor has two contributions: the displacement factor cos φ1 (due to phase shift of the fundamental) and the distortion factor (due to harmonics). A common result is

• PF = (I1 / Irms) × cos φ1 where I1 is the RMS value of the fundamental component of current and Irms is the total RMS current.

Measurement of Energy

Energy, heat, work and power are related physical concepts. Energy is the capacity to do work. When a force moves an object a distance, work is done and energy is transferred; in electrical systems energy is transferred by current and voltage.

• The SI unit of energy, heat and work is the joule (J). Other commonly used units in power systems are the kilowatt-hour (kWh) and the British thermal unit (Btu).
• Power is the rate at which energy is transferred. One watt equals one joule per second.
• For example, a 100 W lamp uses 100 joules per second of electrical energy. If it runs for 1 hour it consumes 0.1 kWh of energy.

Classification of Energy (Watt-Hour) Meters

• Type of display: analog or digital meters.
• Metering point: grid, substation, distribution or consumer end.
• End application: domestic, commercial, industrial.
• Technical classification: single-phase, three-phase, HT (high tension) / LT (low tension), accuracy class, etc.

Electro-Mechanical Induction Type Energy Meter

• This is the traditional rotating-disc watt-hour meter used for decades for billing consumer energy.
• Its main parts are a light aluminium disc mounted on a spindle, two laminated magnetic circuits (series and shunt magnets), a shading or braking magnet, and a mechanical register with gear train.
• The series magnet produces flux proportional to the line current, while the shunt (or pressure) magnet produces flux proportional to the line voltage. The interaction produces a driving torque proportional to instantaneous power and hence disc speed proportional to power.
• A braking magnet provides a retarding torque proportional to the disc speed so that steady rotation speed becomes proportional to power. The mechanical counter integrates disc revolutions to record energy (watt-hours).

Electronic Energy Meters

• Electronic meters use voltage and current sensors, analogue-to-digital converters and digital processing (microcontrollers or DSP) to compute instantaneous power and integrate it to energy.
• They are more accurate, have better precision, consume less power, and start measuring instantaneously when connected to a load.
• Analog electronic meters convert power to a proportional frequency or pulse rate; a counter or frequency-to-digital circuit integrates pulses to yield energy.
• Digital electronic meters sample voltages and currents, perform discrete calculations of instantaneous power, and compute real, reactive and apparent energy. They also support calibration, diagnostics and communications.

Smart Energy Meters

Smart meters are advanced electronic meters with communication and control capabilities.

• They enable two-way communication: meters can transmit consumption data, power quality parameters and alarms to the utility, and receive commands such as connect/disconnect, tariff updates or firmware upgrades from the utility.
• Smart meters support features such as time-of-use (TOU) billing, remote disconnect/reconnect, load control, tamper detection and better metrology accuracy.

Basic Formula to Calculate Energy Consumed by a Load

The electrical energy consumed by an appliance over a period can be calculated easily in kilowatt-hours (kWh) using the appliance power rating and operating duration:

Energy (kWh) = [number of hours per day] × [number of days] × ([appliance power in watts] / 1,000)

Divide watts by 1,000 to convert to kilowatts. The result is the energy used in kWh.

Summary

This chapter covered measurement principles for power, power factor and energy in DC and AC systems. For sinusoidal systems the instantaneous and average power expressions lead to the important relation P = Vrms Irms cos φ. For three-phase systems the two-wattmeter method gives total power and permits power factor determination. For non-sinusoidal conditions, harmonic distortion reduces the usable portion of current and modifies the power factor; THD and the distortion factor must be used together with the displacement factor to obtain true PF. Energy is the time integral of power and is measured with induction meters, electronic meters, and smart meters according to application and accuracy needs.