Power Factor & Measuring Instruments Notes

Last reviewed 16 Sept 2026 · 8 min read

Power factor

Power factor = cos ϕ = \frac{Active power (kW)}{Apparent power (kVA)}

φ = phase angle between voltage and current.

Causes of low (lagging) power factor

Induction motors — especially at light load (magnetising current is large relative to load current).
Transformers at light load (magnetising current).
Arc lamps, discharge lamps, arc furnaces, welding transformers.
Industrial heating furnaces with reactors.

Disadvantages of low power factor

For a given power $P = 3 V I cos ϕ$ , lower pf means higher current:

Larger kVA rating of generators, transformers and switchgear.
Larger conductors needed.
Higher copper losses ( $I^{2} R$ ) → lower efficiency.
Poor voltage regulation (larger voltage drops).
Reduced handling capacity of the system.
Consumers may face penalties under tariffs (and reward for good pf, depending on regulations).

Methods of power factor improvement

Method	Principle	Features
Static capacitors	Draw leading reactive current to offset lagging current	Low losses, no moving parts, easy installation; fixed or automatic power factor correction (APFC) panels; can be damaged by over-voltage/harmonics
Synchronous condenser	Over-excited synchronous motor at no load supplies reactive power	Smooth, continuous control; used for large installations/grid; costly, has losses and maintenance
Phase advancers	AC exciter in rotor circuit of induction motors	For large induction motors

FormulaCapacitor rating

Q_{C} = P (tan ϕ_{1} - tan ϕ_{2})

$P$ = active power; $ϕ_{1}$ = original angle; $ϕ_{2}$ = improved angle.

Capacitance per phase:

Star connection: $C = \frac{Q _{C} /3}{2 π f V _{p h}^{2}}$
Delta connection: $C = \frac{Q _{C} /3}{2 π f V _{L}^{2}}$ (delta requires smaller capacitance but higher voltage rating)

Economics: raising pf from low values to about 0.9–0.95 is usually economical; approaching unity requires disproportionately large capacitors.

Measuring instruments

Classification

Basis	Types
Absolute vs secondary	Absolute — give values in terms of instrument constants and deflection (e.g. tangent galvanometer); secondary — calibrated against absolute/standard instruments (most practical instruments)
Function	Indicating (ammeter, voltmeter, wattmeter — instantaneous value on a scale), recording (continuous record — chart recorders), integrating (total over time — energy meters, ampere-hour meters)
Principle	Magnetic effect (PMMC, moving iron), electrodynamic, induction, heating (hot-wire, thermocouple), electrostatic, electronic/digital

Essential torques in indicating instruments

Deflecting torque — moves the pointer in proportion to the measured quantity.
Controlling torque — opposes deflection so that the pointer comes to rest at the correct position and returns to zero — spring control (most common) or gravity control.
Damping torque — prevents oscillations so the pointer settles quickly — air friction, fluid friction, or eddy current damping (best, used in PMMC).

Types of instruments

Instrument	Principle	Use	Scale	Features
PMMC (permanent magnet moving coil)	Current-carrying coil in the field of a permanent magnet	DC only (reads average)	Uniform (linear)	High accuracy, low power consumption, sensitive; eddy current damping; cannot measure AC directly (used with rectifiers in multimeters)
Moving iron (attraction and repulsion types)	Soft iron piece attracted/repelled by coil's magnetic field	AC and DC (reads RMS)	Non-uniform (cramped at start)	Robust, cheap; errors due to hysteresis, frequency, stray fields
Electrodynamometer	Fixed and moving coils — torque ∝ product of currents	AC and DC; mainly wattmeters; transfer instruments	Non-uniform (as ammeter/voltmeter), uniform as wattmeter	Accurate, used for calibration
Induction type	Eddy currents induced by alternating fluxes	AC only — traditional energy meters	—	Robust; being replaced by electronic meters
Hot-wire / thermocouple	Heating effect	AC/DC, high frequency	Non-uniform	—
Electrostatic	Force between charged plates	High voltages	—	Negligible current drawn
Rectifier type	PMMC with rectifier	AC (calibrated in RMS for sine waves)	—	Multimeters
Digital instruments	ADC and display	AC/DC	Numeric	High accuracy, no parallax, high input impedance

Energy meters

Induction-type (electromechanical) energy meter — aluminium disc rotates at speed proportional to power; revolutions counted; meter constant in rev/kWh.
Electronic (static) energy meters — measure voltage and current samples digitally; meter constant in impulses/kWh (LED blinks); features: tamper detection, maximum demand, time-of-day tariffs.
Smart meters — two-way communication, remote reading, prepaid options.

Extension of range

FormulaShunts and multipliers

Ammeter shunt (low resistance in parallel with meter):

R_{s h} = \frac{I _{m} R _{m}}{I - I _{m}}

Voltmeter multiplier (high resistance in series):

R_{s} = \frac{V}{I _{m}} - R_{m}

$I_{m}$ = full-scale meter current; $R_{m}$ = meter resistance; $I$ , $V$ = new ranges.

Ammeters have low resistance and are connected in series; voltmeters have high resistance and are connected in parallel.
Voltmeter sensitivity (Ω/V) $= 1/ I_{m}$ .

Instrument transformers

Current transformer (CT) — steps down large currents (e.g. 1000/5 A) for ammeters, energy meters and relays; secondary must never be open-circuited while primary carries current (dangerous high voltage and core saturation).
Potential (voltage) transformer (PT/VT) — steps down high voltages (e.g. 11 kV/110 V).
Provide isolation from high voltage and standardised instrument ranges.

Other instruments

Instrument	Purpose
Wattmeter (electrodynamometer / digital)	Power — current coil in series, pressure (voltage) coil in parallel
Megger (insulation resistance tester)	Insulation resistance (megaohms) of cables, windings, installations — applies high DC test voltage (e.g. 500 V, 1000 V)
Earth tester	Earth electrode resistance — fall-of-potential method with auxiliary electrodes
Multimeter (analogue/digital)	Voltage, current, resistance, continuity, diode tests
Clamp meter (tong tester)	Current without breaking the circuit (CT principle / Hall effect)
Tachometer	Speed
Power factor meter, frequency meter, synchroscope	pf, frequency, synchronisation
Cathode ray oscilloscope (CRO) / digital storage oscilloscope	Waveforms, frequency, phase, voltage vs time
Wheatstone bridge / Kelvin double bridge	Medium / very low resistance measurement
Phase sequence indicator	Phase order

Errors in measurement

Gross errors (human mistakes — reading, recording).
Systematic errors — instrumental (calibration, friction, loading effect), environmental (temperature, stray fields), observational (parallax).
Random errors — unpredictable; reduced by averaging.
Accuracy (closeness to true value) vs precision (repeatability); resolution, sensitivity, accuracy class (e.g. class 0.5, 1.0).
Loading effect — a low-resistance voltmeter across a high-resistance circuit reads low.

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