POWER QUALITY GLOSSARY

 

 

 

FOR QUICKER VIEWING OF A TERM CLICK ON THE FIRST LETTER OF THE TERM.

A    B    C    D    E    F    G    H    I    J    K    L    M    N    O    P    R    S    T    U    V     W

 

AC

 

 

Alternating Current. AC current is a waveform that regularly reverses in positive and negative directions. The U.S.A. electrical power alternates 60 times per second (60 Hz). Europe and other countries have standardized on 50 Hz.

 

AMPERE (or AMP) A unit of electrical current or rate of flow of electrons. One volt across one ohm of resistance causes a current flow of one ampere. A flow of one coulomb per second equals one ampere.

 

APPARENT POWER The product of voltage and current in a circuit.

 

ARC Sparking that results when undesirable current flows between two points of differing potential. This may be due to leakage through the intermediate insulation or a leakage path due to contamination.

ARRESTER

A nonlinear device to limit the amplitude of voltage on a power line. The term implies that the device stops overvoltage problems (i.e., lightning). In actuality, voltage clamp levels, response times, and installation determine how much voltage can be removed by the operation of an arrester.

 

ATTENUATION The reduction of a signal from one point to another. For an electrical surge, attenuation refers to the reduction of an incoming surge by a limiter (attenuator). Wire resistance, arresters, and power conditioners attenuate surges to varying degrees.

 

AWG American Wire Gage. This term refers to the U.S. standard for wire size.

 

AUTOTRANSFORMER A transformer used to step voltage up or down. Because its primary and secondary windings share common turns, it does not provide isolation.

 

AUXILIARY SOURCE A power source dedicated to providing emergency power to a critical load when commercial power is interrupted.

BALANCED

LOAD An alternating current power system consisting of more than two current-carrying conductors in which these conductors all carry the same current.

 

BATTERY A collection of cells, grouped together to provide higher voltage and/or higher current than a single cell.

BLACKOUT

Total loss of commercial power.

 

BRANCH CIRCUIT A division of a load circuit with current limited by a fuse or circuit breaker.

 

BREAK-BEFORE-MAKE Operational sequence of a switch or relay where the existing connection is opened prior to making the new connection.

 

BROWNOUT An intentional reduction of voltage by a utility in response to a power demand in excess of its generation capability. Nominal reductions are 3, 5, or 8 percent.

 

BUILDING SERVICE ENTRY The point where commercial power enters the building.

 

BUSBAR A heavy, rigid conductor used for feeders.

CAPACITIVE

REACTANCE The behavior of alternating current as it interacts with capacitance encountered in a circuit.

 

CAPACITOR A device consisting of two conducting surfaces separated by an insulating material or dielectric. A capacitor stores electrical energy, blocks the flow of direct current, and permits the flow of alternating current to a degree dependent essentially on the capacitance and frequency.

 

CIRCUIT BREAKER A resettable device that responds to a preset excess of current flow by opening the circuit thereby preventing damage to circuit elements.

 

CLAMP-ON CT A current transformer which clamps around a current-carrying conductor so the conductor does not have to be opened for insertion of the transformer primary. Particularly suited for monitoring when current must be sensed at many points for relatively short periods.

 

COMMON MODE (CM) The term refers to electrical interference which is measurable as a ground referenced signal. In true common mode a signal is common to both the current carrying conductors.

 

COMMON MODE NOISE Abnormal signals that appear between a current carrying line and its associated ground.

 

CSA The abbreviation which stands for Canadian Standards Association. This is a Canadian safety assurance agency similar to Underwriter’s Laboratories.

 

CURRENT The flow of electricity in a circuit as expressed in Amperes. Current refers to the quantity or intensity of electrical flow. Voltage on the other hand refers to the pressure or force causing the electrical flow.

 

CURRENT BALANCE A term that describes the nearly equal flow of current on each leg of a three phase power system. With this flow balanced the theoretical flow of current in the neutral with respect to ground will be zero.

 

CYCLES PER SECOND This term describes the frequency of alternating current. Frequency is more properly described using the term Hertz (Hz) which is synonymous with cycles per second.

DECIBEL

The standard unit for expressing relative power levels. Decibels indicate the ratio of power output to power input: dB = 10Log10(P1/P2).

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DELTA A method of connecting a three-phase source (or load) in a closed series loop with the input (or output) connections made to each of the three junctions.

 

DELTA-DELTA The connection between a delta source and a delta load.

 

DELTA-WYE The connection between a delta source and a wye load.

 

DIRECT CURRENT (DC) Current which flows in only one direction.

 

DROPOUT A discrete voltage loss. A voltage sag (complete or partial) for a very short period of time (milliseconds) constitutes a dropout.

 

dv/dt The change in voltage per change in time. d = change v = volts t = time

EARTH

GROUND A low impedance path to earth for the purpose of discharging lightning, static, and radiated energy, and to maintain the main service entrance at earth potential.

 

EARTHING ELECTRODE A grounding electrode, water pipe, or building steel, or some combination of these, used for establishing a building’s earth ground.

 

EFFICIENCY The ratio of the output to input power times 100, expressed as a percentage. Efficiency = (Pout/Pin) x 100.

 

ELECTROMAGNETIC A magnetic field cause by an electric current. Power lines cause electromagnetic fields that can interfere with nearby data cables.

 

ELECTROMECHANICAL A mechanical device which is controlled by an electric device. Solenoids and shunt trip circuit breakers are examples of electromechanical devices.

 

ELECTROSTATIC A potential difference (electric charge) measurable between two points which is caused by the distribution of dissimilar static charge along the points. The voltage level is usually in kilovolts.

 

EMF Electromotive Force or voltage.

 

EMI Electromagnetic interference. A term that describes electrically induced noise or transients.

EQUIPMENT GROUNDING CONDUCTOR   See SAFETY GROUND

 

ESD Electrostatic Discharge (static electricity). The effects of a static discharge can range from simple skin irritation for an individual to degraded or destroyed semiconductor junctions for an electronic device.

FARAD

A farad is the measure of capacitance. One farad is equal to 1 coulomb of charge between two terminals causing one volt of potential difference.

 

FARADAY SHIELD A grounded metallic barrier which can be used for improved isolation between the windings of a transformer. In this application, the shield basically reduces the leakage capacitance between the primary and secondary.

 

FEEDERS Transmission lines supplying power to a distribution system.

 

FERRORESONANCE Resonance resulting when the iron core of an inductive component of an LC circuit is saturated, increasing the inductive reactance with respect to the capacitance reactance. This is an undesirable effect in electrical distribution systems.

 

FILTER A selective network of resistors, inductors, or capacitors which offers comparatively little opposition to certain frequencies or direct current, while blocking or attenuating other frequencies.

 

FIPS PUB 94 Federal Information Processing Standards Publication (1983, September 21) is an official publication of the National Bureau of Standards (since renamed National Institute for Standards and Technology). The document is a recommended guideline for federal agencies with respect to the electrical environment for automatic data processing (ADP) facilities. The IEEE Emerald Book 1100 is an update to FIPS PUB 94.

 

FLASHOVER Flashing due to high current flowing between two points of different potential. Usually due to insulation breakdown resulting from arcing or lightning.

 

FLUCTUATION A surge or sag in voltage amplitude, often caused by load switching or fault clearing.

 

FREQUENCY The number of complete cycles of sinusoidal variation per unit time. For AC power lines, the most widely used frequencies are 60 and 50 hertz (Hz).

FREQUENCY DEVIATION A variation from nominal frequency. See SLEW RATE

GFI

(Ground Fault Interrupter) A device whose function is to interrupt the electric circuit to the load when a fault current to ground exceeds some predetermined value that is less than that required to operate the overcurrent protective device of the supply circuit.

 

GFCI (Ground Fault Circuit Interrupter) A life safety device commonly used in outdoor, bathroom, & kitchen locations limits ground fault current to 5 milliamp.

 

GROUND A conducting connection, whether intentional or accidental, between an electrical circuit or equipment and the earth, or to some conducting body that serves in place of the earth.

 

GROUND LOOP The condition of having two or more ground references in a common system. When two or more grounds have a potential difference between them, current can flow. This flow of current is a new circuit or loop which can interfere with the normal operation of the system.

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GROUND FAULT Any undesirable current path from a point of differing potential to ground.

 

GROUNDED Connected to earth or to some conducting body that serves in place of the earth.

 

HARMONIC

 

 

A sinusoidal component of an AC voltage that is a multiple of the fundamental waveform frequency.

 

HARMONIC DISTORTION The presence of harmonics which change an AC waveform from sinusoidal to complex. This can cause overheating of circuit elements and might appear to a device as data corrupting noise.

 

HARMONIC NEUTRALIZATION A cancellation process: harmonics at the output of a circuit are inverted and fed back in their opposite phase.

 

HERTZ (Hz) Unit of frequency, one hertz (Hz) equals one cycle per second.

 

HV High voltage.

I2R

The expression of power resulting from the flow of current through a resistance: P = I2R. P = Power I = Current R = Resistance

 

IMPEDANCE The total opposition (i.e., resistance and reactance) a circuit offers to the flow of alternating current at a given frequency.

 

IMPULSE Transient voltage or current condition of positive or negative amplitude.

 

INDUCTANCE The ability of a coil to store energy and oppose changes in current flowing through it. A function of the cross sectional area, number of turns of coil, length of coil, and core material.

 

INDUCTIVE REACTANCE A term used to describe the impedance to alternating current offered by an inductive circuit.

 

INDUCTOR A conductor, usually coiled, which tends to oppose any change in the flow of current through itself.

 

INRUSH CURRENT The initial surge of current required by a load before resistance or impedance increases to its normal operating value.

 

INVERTER A device used to convert DC current to AC current. Inverters may be mechanical (motor), ferroresonant and solid state.

 

ISOLATION TRANSFORMER A transformer that contains electrostatic shields between primary and secondary windings, and no direct electrical path between primary and secondary.

JOULE

A unit of energy. One joule equals one watt/second.

KILO

(k) A metric prefix meaning 1000 or 103.

 

kVA Kilovolt amperes; apparent power.

 

kW Kilowatts; real power delivered to a load.

LC

CIRCUIT An electrical network containing both inductive and capacitive elements.

 

LIGHTNING ARRESTER A device used to pass large impulses to ground. It is vital that this device be placed upstream from the computer ground.

 

LINE A term used generally to describe a current carrying conductor.

 

LINE TO LINE A term used to describe a given condition between conductors of a multiphase feeder.

 

LINE TO NEUTRAL A term used to describe a given condition between a phase conductor and a neutral conductor.

 

LINEAR LOAD Those electrical loads in which the impedance is constant regardless of the voltage, so that if the voltage is sinusoidal the current drawn will also be sinusoidal.

 

LINE IMBALANCE Unequal loads on the phase lines of a multiphase feeder.

MAGNETIC

FIELD The lines of force that exist around an energized electrical conductor, magnet, or inductor.

 

MAIN SERVICE ENTRANCE The enclosure containing connection panels and switchgear, located at the point where the utility power lines enter a building.

METAL OXIDE VARISTOR (MOV)

A MOV is a voltage sensitive breakdown device which is commonly used to limit overvoltage conditions (electrical surges) on power and data lines. When the applied voltage exceeds the breakdown point, the resistance of the MOV decreases from a very high level (thousands of ohms) to a very low level (a few ohms). The actual resistance of the device is a function of the rate of applied voltage and current.

 

MOTOR ALTERNATOR A device that consists of an AC generator mechanically linked to an electric motor which is driven by utility power or by batteries. An alternator is an AC generator.

 

MOTOR GENERATOR A term used synonymously for motor alternator.

 

MTBF, MEAN TIME BETWEEN FAILURE A statistical estimate of the time a component, subassembly, or operating unit will operate before failure will occur.

 

MTTR, MEAN TIME TO REPAIR A statistical estimate of the repair time for a failed item.

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NEC

National Electrical Code. It is revised every three years. The current version is NFPA 70-1999.

 

NEGATIVE RESISTANCE The characteristic of a circuit in which current varies inversely with applied voltage.

NEUTRAL

Conductor used as the primary return path for current during normal operation of an electrical device. Also, the junction of the legs in a wye circuit.

 

NOISE An undesirable signal which is irregular yet oscillatory that is super imposed on the desired signal. See common mode noise and normal mode noise.

 

NONLINEAR LOAD Electrical loads in which the instantaneous current is not proportional to the instantaneous voltage, or, effectively, the load impedance varies with voltage.

 

NONSINUSOIDAL A waveform that is not capable of being expressed mathematically by using the sine function.

 

NORMAL MODE (NM) The term refers to electrical interference which is measurable between line and neutral (current carrying conductors). Normal mode interference is readily generated by the operation of lights, switches, and motors.

 

NORMAL MODE NOISE A noise signal which appears between a set of phase conductors irrespective of their associated ground conductor.

 

NOTCH Slang for a negative or subtractive impulse.

OHM

The unit of resistance. Symbol W

 

OHM’S LAW The mathematical relationship between Volts, Amperes, and Ohms: Volts = Amperes * Ohms.

OUTAGE

An outage is a long-term power interruption. From the utility perspective, an outage occurs when a component of the distribution system is not abailable to provide its normal function (i.e., the generator cannot supply power). Normally, utility companies do not include short power interruptions (grid switching) in their classification of outage history and also may only count power interruptions with a duration longer than 1 to 5 minutes.

 

OVERVOLTAGE A voltage greater than the rating of a device or component. Normally overvoltage refers to long term events (several AC cycles and longer). The term can also apply to transients and surges.

PDU

, POWER DISTRIBUTION UNIT A portable electrical distribution device that provides an easily expandable and flexible electrical environment for a computer and its associated peripherals.

 

PEAK The maximum instantaneous measurement of an electrical event.

 

PEAK LINE CURRENT Maximum instantaneous current during a cycle.

 

PHASE A term used to describe the timing between two or more events tied to the same frequency.

PHASE BALANCING The practice of placing equal electrical loads on each leg of a three phase system. See NEUTRAL

 

PHASE ROTATION The sequence in which a comparable voltage appears in all three phases: A, B, and C, of a three phase system.

 

POWER A general term which means the capacity for doing work. In the electrical environment this is usually measured in watts.

 

POWER FACTOR The ratio of real power to apparent power. Power factor will be "leading" or "lagging" depending on which way the load shifts the current’s phase with respect to the voltage’s phase. Inductive loads cause current to lag behind voltage, while capacitive loads cause current to lead voltage.

 

POWER LINE MONITOR A measuring device which reports information on the changing conditions of electrical power.

POWER OUTAGE See BLACKOUT or OUTAGE

RADIAL

ARRAY A group of earthing electrodes or conductors of equal length and ampacity, connected at a central point and extending outward at equal angles, spoke fashion, to provide a low earth impedance reference.

 

REACTANCE Opposition to the flow of alternating current. Capacitive reactance is the opposition offered by capacitors, and inductive reactance is the opposition offered by a coil or other inductance.

 

REAL POWER Watts.

 

RECLOSER The automatic closing of a circuit-interrupting device following automatic tripping.

 

RECTIFIER/CHARGER A subassembly of a UPS that performs the function of converting the incoming AC into DC for driving the Inverter and charging the batteries.

 

RESISTANCE A term describing the opposition of a circuit to alternating or direct current.

 

RESISTOR A discrete electronic component designed to produce a DC voltage drop when current passes through it.

 

RFI Radio Frequency Interference.

 

RMS Root Mean Square (RMS) is a calculation process for alternating current and voltage waveforms. The RMS calculation is intended to provide a measurement of an AC current that is equivalent to a comparable DC current.

 

RMS LINE CURRENT The square root of the average of the squares of all instantaneous current amplitudes occurring during a given cycle.

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RMS LINE VOLTAGE The square root of the average of the squares of all instantaneous voltage amplitudes occurring during a given cycle.

SAFETY GROUND

An alternate path of return current, during a fault condition, for the purpose of tripping a circuit breaker. Also, the means of establishing a load at earth level. NEC refers to it as equipment grounding conductor.

 

SAG A reduction in a voltage envelope. The duration is usually from one cycle to a few seconds. Usually, sags are caused by fault clearing or heavy load startup.

SLEW RATE

The rate of frequency change per second. A typical rate for sensitive equipment is 1Hz/second.

 

SHIELDING, ELECTROSTATIC A conductive enclosure used to protect circuits from the effects of external electrostatic fields.

 

SHUNT TRIP A type of circuit breaker that can also be activated by a circuit other than the one it is protecting.

 

SINGLE PHASE That portion of a power source which represents only a single phase of the three phases that are available.

 

SINGLE POINT GROUND The practice of tying the power neutral ground and safety ground together at the same point avoiding differential ground potential between points in a system.

 

SINUSOIDAL WAVEFORM A waveform that can be expressed mathematically by using the sine function.

 

SINE WAVE A waveform which oscillates periodically with the amplitude of points on the waveform proportional to the sine of the phase angle of the point.

 

SUBSTATION Location where high voltage transmission lines connect to switchgear and step-down transformers to produce lower voltages at lower power levels for local distribution networks.

SUPPRESSOR See ARRESTER

 

SURGE A short-term positive change in amplitude of a voltage.

TOTAL

HARMONIC DISTORTION (THD) The square root of the sum of the squares of the RMS harmonic voltages or currents divided by the RMS fundamental voltage or current. Can also be calculated in the same way for only even harmonics or odd harmonics.

 

TRANSIENT A high amplitude, short duration impulse superimposed on the normal voltage or current.

 

TRANSVERSE MODE NOISE Often used as a synonym for normal mode noise, it more clearly relates to noise that is the result of the conversion of common mode noise to normal mode noise after it passes through a transformer.

UL

The abbreviation for Underwriters Laboratories, an independent United States product safety assurance agency.

 

UNDERVOLTAGE Negative change in amplitude of a voltage.

VAC

Volts of Alternating Current

 

VAR Volt Amps Reactive

VARISTOR See MOV

 

VOLT (V) The unit of voltage or potential difference.

 

VOLT-AMPERE The unit of measurement of apparent power.

 

VOLTAGE REGULATOR A circuit that has a constant output voltage when input voltage fluctuates.

WATT

(W) The unit of power. Equal to one joule per second.

 

WYE A wye connection refers to a polyphase electrical supply where the source transformer has the conductors connected to the terminals in a physical arrangement resembling a Y. Each point of the Y represents the connection of a hot conductor. The angular displacement between each point on the Y is 120 degrees. The center point is the common return point for the neutral conductor.

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Library - Electrical Circuit Theorems

Notation
The is used for some notation and formulae. If the Greek symbols for alpha beta delta do not
appear here [ a b d ] the symbol font needs to be installed for correct display of notation and formulae.
E
G
I
R
P
voltage source
conductance
current
resistance
power
[volts, V]
[siemens, S]
[amps, A]
[ohms, W]
[watts]
V
X
Y
Z
 
voltage drop
reactance
admittance
impedance
 
[volts, V]
[ohms, W]
[siemens, S]
[ohms, W]
 

Ohm's Law

When an applied voltage E causes a current I to flow through an impedance Z, the value of the impedance Z is equal to the voltage E divided by the current I.
Impedance = Voltage / Current Z = E / I

Similarly, when a voltage E is applied across an impedance Z, the resulting current I through the impedance is equal to the voltage E divided by the impedance Z.

Current = Voltage / Impedance I = E / Z

Similarly, when a current I is passed through an impedance Z, the resulting voltage drop V across the impedance is equal to the current I multiplied by the impedance Z.

Voltage = Current * Impedance V = IZ

Alternatively, using admittance Y which is the reciprocal of impedance Z:

Voltage = Current / Admittance V = I / Y


Kirchhoff's Laws

Kirchhoff's Current Law
At any instant the sum of all the currents flowing into any circuit node is equal to the sum of all the currents flowing out of that node:
SIin = SIout

Similarly, at any instant the algebraic sum of all the currents at any circuit node is zero:
SI = 0

Kirchhoff's Voltage Law
At any instant the sum of all the voltage sources in any closed circuit is equal to the sum of all the voltage drops in that circuit:
SE = SIZ

Similarly, at any instant the algebraic sum of all the voltages around any closed circuit is zero:
SE - SIZ = 0


Thévenin's Theorem

Any linear voltage network which may be viewed from two terminals can be replaced by a voltage-source equivalent circuit comprising a single voltage source E and a single series impedance Z. The voltage E is the open-circuit voltage between the two terminals and the impedance Z is the impedance of the network viewed from the terminals with all voltage sources replaced by their internal impedances.


Norton's Theorem

Any linear current network which may be viewed from two terminals can be replaced by a current-source equivalent circuit comprising a single current source I and a single shunt admittance Y. The current I is the short-circuit current between the two terminals and the admittance Y is the admittance of the network viewed from the terminals with all current sources replaced by their internal admittances.


Thévenin and Norton Equivalence

The open circuit, short circuit and load conditions of the Thévenin model are:
Voc = E
Isc = E / Z
Vload = E - IloadZ
Iload = E / (Z + Zload)

The open circuit, short circuit and load conditions of the Norton model are:
Voc = I / Y
Isc = I
Vload = I / (Y + Yload)
Iload = I - VloadY

Thévenin model from Norton model

Voltage = Current / Admittance
Impedance = 1 / Admittance
E = I / Y
Z = Y -1

Norton model from Thévenin model

Current = Voltage / Impedance
Admittance = 1 / Impedance
I = E / Z
Y = Z -1

When performing network reduction for a Thévenin or Norton model, note that:
- nodes with zero voltage difference may be short-circuited with no effect on the network current distribution,
- branches carrying zero current may be open-circuited with no effect on the network voltage distribution.


Superposition Theorem

In a linear network with multiple voltage sources, the current in any branch is the sum of the currents which would flow in that branch due to each voltage source acting alone with all other voltage sources replaced by their internal impedances.


Reciprocity Theorem

If a voltage source E acting in one branch of a network causes a current I to flow in another branch of the network, then the same voltage source E acting in the second branch would cause an identical current I to flow in the first branch.


Compensation Theorem

If the impedance Z of a branch in a network in which a current I flows is changed by a finite amount dZ, then the change in the currents in all other branches of the network may be calculated by inserting a voltage source of -IdZ into that branch with all other voltage sources replaced by their internal impedances.


Millman's Theorem (Parallel Generator Theorem)

If any number of admittances Y1, Y2, Y3, ... meet at a common point P, and the voltages from another point N to the free ends of these admittances are E1, E2, E3, ... then the voltage between points P and N is:
VPN = (E1Y1 + E2Y2 + E3Y3 + ...) / (Y1 + Y2 + Y3 + ...)
VPN = SEY / SY

The short-circuit currents available between points P and N due to each of the voltages E1, E2, E3, ... acting through the respective admitances Y1, Y2, Y3, ... are E1Y1, E2Y2, E3Y3, ... so the voltage between points P and N may be expressed as:
VPN = SIsc / SY


Joule's Law

When a current I is passed through a resistance R, the resulting power P dissipated in the resistance is equal to the square of the current I multiplied by the resistance R:
P = I2R

By substitution using Ohm's Law for the corresponding voltage drop V (= IR) across the resistance:
P = V2 / R = VI = I2R


Maximum Power Transfer Theorem

When the impedance of a load connected to a power source is varied from open-circuit to short-circuit, the power absorbed by the load has a maximum value at a load impedance which is dependent on the impedance of the power source.

Note that power is zero for an open-circuit (zero current) and for a short-circuit (zero voltage).

Voltage Source
When a load resistance RT is connected to a voltage source ES with series resistance RS, maximum power transfer to the load occurs when RT is equal to RS.

Under maximum power transfer conditions, the load resistance RT, load voltage VT, load current IT and load power PT are:
RT = RS
VT = ES / 2
IT = VT / RT = ES / 2RS
PT = VT2 / RT = ES2 / 4RS

Current Source
When a load conductance GT is connected to a current source IS with shunt conductance GS, maximum power transfer to the load occurs when GT is equal to GS.

Under maximum power transfer conditions, the load conductance GT, load current IT, load voltage VT and load power PT are:
GT = GS
IT = IS / 2
VT = IT / GT = IS / 2GS
PT = IT2 / GT = IS2 / 4GS

Complex Impedances
When a load impedance ZT (comprising variable resistance RT and variable reactance XT) is connected to an alternating voltage source ES with series impedance ZS (comprising resistance RS and reactance XS), maximum power transfer to the load occurs when ZT is equal to ZS* (the complex conjugate of ZS) such that RT and RS are equal and XT and XS are equal in magnitude but of opposite sign (one inductive and the other capacitive).

When a load impedance ZT (comprising variable resistance RT and constant reactance XT) is connected to an alternating voltage source ES with series impedance ZS (comprising resistance RS and reactance XS), maximum power transfer to the load occurs when RT is equal to the magnitude of the impedance comprising ZS in series with XT:
RT = |ZS + XT| = (RS2 + (XS + XT)2)½
Note that if XT is zero, maximum power transfer occurs when RT is equal to the magnitude of ZS:
RT = |ZS| = (RS2 + XS2)½

When a load impedance ZT with variable magnitude and constant phase angle (constant power factor) is connected to an alternating voltage source ES with series impedance ZS, maximum power transfer to the load occurs when the magnitude of ZT is equal to the magnitude of ZS:
(RT2 + XT2)½ = |ZT| = |ZS| = (RS2 + XS2)½


Kennelly's Star-Delta Transformation

A star network of three impedances ZAN, ZBN and ZCN connected together at common node N can be transformed into a delta network of three impedances ZAB, ZBC and ZCA by the following equations:
ZAB = ZAN + ZBN + (ZANZBN / ZCN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZCN
ZBC = ZBN + ZCN + (ZBNZCN / ZAN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZAN
ZCA = ZCN + ZAN + (ZCNZAN / ZBN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZBN

Similarly, using admittances:
YAB = YANYBN / (YAN + YBN + YCN)
YBC = YBNYCN / (YAN + YBN + YCN)
YCA = YCNYAN / (YAN + YBN + YCN)

In general terms:
Zdelta = (sum of Zstar pair products) / (opposite Zstar)
Ydelta = (adjacent Ystar pair product) / (sum of Ystar)


Kennelly's Delta-Star Transformation

A delta network of three impedances ZAB, ZBC and ZCA can be transformed into a star network of three impedances ZAN, ZBN and ZCN connected together at common node N by the following equations:
ZAN = ZCAZAB / (ZAB + ZBC + ZCA)
ZBN = ZABZBC / (ZAB + ZBC + ZCA)
ZCN = ZBCZCA / (ZAB + ZBC + ZCA)

Similarly, using admittances:
YAN = YCA + YAB + (YCAYAB / YBC) = (YABYBC + YBCYCA + YCAYAB) / YBC
YBN = YAB + YBC + (YABYBC / YCA) = (YABYBC + YBCYCA + YCAYAB) / YCA
YCN = YBC + YCA + (YBCYCA / YAB) = (YABYBC + YBCYCA + YCAYAB) / YAB

In general terms:
Zstar = (adjacent Zdelta pair product) / (sum of Zdelta)
Ystar = (sum of Ydelta pair products) / (opposite Ydelta)

Resistance

The resistance R of a circuit is equal to the applied direct voltage E divided by the resulting steady current I:
R = E / I


Resistances in Series

When resistances R1, R2, R3, ... are connected in series, the total resistance RS is:
RS = R1 + R2 + R3 + ...


Voltage Division by Series Resistances

When a total voltage ES is applied across series connected resistances R1 and R2, the current IS which flows through the series circuit is:
IS = ES / RS = ES / (R1 + R2)

The voltages V1 and V2 which appear across the respective resistances R1 and R2 are:
V1 = ISR1 = ESR1 / RS = ESR1 / (R1 + R2)
V2 = ISR2 = ESR2 / RS = ESR2 / (R1 + R2)

In general terms, for resistances R1, R2, R3, ... connected in series:
IS = ES / RS = ES / (R1 + R2 + R3 + ...)
Vn = ISRn = ESRn / RS = ESRn / (R1 + R2 + R3 + ...)
Note that the highest voltage drop appears across the highest resistance.


Resistances in Parallel

When resistances R1, R2, R3, ... are connected in parallel, the total resistance RP is:
1 / RP = 1 / R1 + 1 / R2 + 1 / R3 + ...

Alternatively, when conductances G1, G2, G3, ... are connected in parallel, the total conductance GP is:
GP = G1 + G2 + G3 + ...
where Gn = 1 / Rn

For two resistances R1 and R2 connected in parallel, the total resistance RP is:
RP = R1R2 / (R1 + R2)
RP = product / sum

The resistance R2 to be connected in parallel with resistance R1 to give a total resistance RP is:
R2 = R1RP / (R1 - RP)
R2 = product / difference


Current Division by Parallel Resistances

When a total current IP is passed through parallel connected resistances R1 and R2, the voltage VP which appears across the parallel circuit is:
VP = IPRP = IPR1R2 / (R1 + R2)

The currents I1 and I2 which pass through the respective resistances R1 and R2 are:
I1 = VP / R1 = IPRP / R1 = IPR2 / (R1 + R2)
I2 = VP / R2 = IPRP / R2 = IPR1 / (R1 + R2)

In general terms, for resistances R1, R2, R3, ... (with conductances G1, G2, G3, ...) connected in parallel:
VP = IPRP = IP / GP = IP / (G1 + G2 + G3 + ...)
In = VP / Rn = VPGn = IPGn / GP = IPGn / (G1 + G2 + G3 + ...)
where Gn = 1 / Rn
Note that the highest current passes through the highest conductance (with the lowest resistance).


Capacitance

When a voltage is applied to a circuit containing capacitance, current flows to accumulate charge in the capacitance:
Q = ̣idt = CV

Alternatively, by differentiation with respect to time:
dq/dt = i = C dv/dt
Note that the rate of change of voltage has a polarity which opposes the flow of current.

The capacitance C of a circuit is equal to the charge divided by the voltage:
C = Q / V = ̣idt / V

Alternatively, the capacitance C of a circuit is equal to the charging current divided by the rate of change of voltage:
C = i / dv/dt = dq/dt / dv/dt = dq/dv


Capacitances in Series

When capacitances C1, C2, C3, ... are connected in series, the total capacitance CS is:
1 / CS = 1 / C1 + 1 / C2 + 1 / C3 + ...

For two capacitances C1 and C2 connected in series, the total capacitance CS is:
CS = C1C2 / (C1 + C2)
CS = product / sum


Voltage Division by Series Capacitances

When a total voltage ES is applied to series connected capacitances C1 and C2, the charge QS which accumulates in the series circuit is:
QS = ̣iSdt = ESCS = ESC1C2 / (C1 + C2)

The voltages V1 and V2 which appear across the respective capacitances C1 and C2 are:
V1 = ̣iSdt / C1 = ESCS / C1 = ESC2 / (C1 + C2)
V2 = ̣iSdt / C2 = ESCS / C2 = ESC1 / (C1 + C2)

In general terms, for capacitances C1, C2, C3, ... connected in series:
QS = ̣iSdt = ESCS = ES / (1 / CS) = ES / (1 / C1 + 1 / C2 + 1 / C3 + ...)
Vn = ̣iSdt / Cn = ESCS / Cn = ES / Cn(1 / CS) = ES / Cn(1 / C1 + 1 / C2 + 1 / C3 + ...)
Note that the highest voltage appears across the lowest capacitance.


Capacitances in Parallel

When capacitances C1, C2, C3, ... are connected in parallel, the total capacitance CP is:
CP = C1 + C2 + C3 + ...


Charge Division by Parallel Capacitances

When a voltage EP is applied to parallel connected capacitances C1 and C2, the charge QP which accumulates in the parallel circuit is:
QP = ̣iPdt = EPCP = EP(C1 + C2)

The charges Q1 and Q2 which accumulate in the respective capacitances C1 and C2 are:
Q1 = ̣i1dt = EPC1 = QPC1 / CP = QPC1 / (C1 + C2)
Q2 = ̣i2dt = EPC2 = QPC2 / CP = QPC2 / (C1 + C2)

In general terms, for capacitances C1, C2, C3, ... connected in parallel:
QP = ̣iPdt = EPCP = EP(C1 + C2 + C3 + ...)
Qn = ̣indt = EPCn = QPCn / CP = QPCn / (C1 + C2 + C3 + ...)
Note that the highest charge accumulates in the highest capacitance.


Inductance

When the current changes in a circuit containing inductance, the magnetic linkage changes and induces a voltage in the inductance:
dy/dt = e = L di/dt
Note that the induced voltage has a polarity which opposes the rate of change of current.

Alternatively, by integration with respect to time:
Y = ̣edt = LI

The inductance L of a circuit is equal to the induced voltage divided by the rate of change of current:
L = e / di/dt = dy/dt / di/dt = dy/di

Alternatively, the inductance L of a circuit is equal to the magnetic linkage divided by the current:
L = Y / I

Note that the magnetic linkage Y is equal to the product of the number of turns N and the magnetic flux F:
Y = NF = LI


Mutual Inductance

The mutual inductance M of two coupled inductances L1 and L2 is equal to the mutually induced voltage in one inductance divided by the rate of change of current in the other inductance:
M = E2m / (di1/dt)
M = E1m / (di2/dt)

If the self induced voltages of the inductances L1 and L2 are respectively E1s and E2s for the same rates of change of the current that produced the mutually induced voltages E1m and E2m, then:
M = (E2m / E1s)L1
M = (E1m / E2s)L2
Combining these two equations:
M = (E1mE2m / E1sE2s)½ (L1L2)½ = kM(L1L2)½
where kM is the mutual coupling coefficient of the two inductances L1 and L2.

If the coupling between the two inductances L1 and L2 is perfect, then the mutual inductance M is:
M = (L1L2)½


Inductances in Series

When uncoupled inductances L1, L2, L3, ... are connected in series, the total inductance LS is:
LS = L1 + L2 + L3 + ...

When two coupled inductances L1 and L2 with mutual inductance M are connected in series, the total inductance LS is:
LS = L1 + L2 ± 2M
The plus or minus sign indicates that the coupling is either additive or subtractive, depending on the connection polarity.


Inductances in Parallel

When uncoupled inductances L1, L2, L3, ... are connected in parallel, the total inductance LP is:
1 / LP = 1 / L1 + 1 / L2 + 1 / L3 + ...


Time Constants

Capacitance and resistance
The time constant of a capacitance C and a resistance R is equal to CR, and represents the time to change the voltage on the capacitance from zero to E at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance).

Similarly, the time constant CR represents the time to change the charge on the capacitance from zero to CE at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance).

If a voltage E is applied to a series circuit comprising a discharged capacitance C and a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vC across the capacitance and the charge qC on the capacitance are:
i = (E / R)e - t / CR
vR = iR = Ee - t / CR
vC = E - vR = E(1 - e - t / CR)
qC = CvC = CE(1 - e - t / CR)

If a capacitance C charged to voltage V is discharged through a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vC across the capacitance and the charge qC on the capacitance are:
i = (V / R)e - t / CR
vR = iR = Ve - t / CR
vC = vR = Ve - t / CR
qC = CvC = CVe - t / CR

Inductance and resistance
The time constant of an inductance L and a resistance R is equal to L / R, and represents the time to change the current in the inductance from zero to E / R at a constant rate of change of current E / L (which produces an induced voltage E across the inductance).

If a voltage E is applied to a series circuit comprising an inductance L and a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vL across the inductance and the magnetic linkage yL in the inductance are:
i = (E / R)(1 - e - tR / L)
vR = iR = E(1 - e - tR / L)
vL = E - vR = Ee - tR / L
yL = Li = (LE / R)(1 - e - tR / L)

If an inductance L carrying a current I is discharged through a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vL across the inductance and the magnetic linkage yL in the inductance are:
i = Ie - tR / L
vR = iR = IRe - tR / L
vL = vR = IRe - tR / L
yL = Li = LIe - tR / L

Rise Time and Fall Time
The rise time (or fall time) of a change is defined as the transition time between the 10% and 90% levels of the total change, so for an exponential rise (or fall) of time constant T, the rise time (or fall time) t10-90 is:
t10-90 = (ln0.9 - ln0.1)T » 2.2T

The half time of a change is defined as the transition time between the initial and 50% levels of the total change, so for an exponential change of time constant T, the half time t50 is :
t50 = (ln1.0 - ln0.5)T » 0.69T

Note that for an exponential change of time constant T:
- over time interval T, a rise changes by a factor 1 - e -1 (» 0.63) of the remaining change,
- over time interval T, a fall changes by a factor e -1 (» 0.37) of the remaining change,
- after time interval 3T, less than 5% of the total change remains,
- after time interval 5T, less than 1% of the total change remains.


Power

The power P dissipated by a resistance R carrying a current I with a voltage drop V is:
P = V2 / R = VI = I2R

Similarly, the power P dissipated by a conductance G carrying a current I with a voltage drop V is:
P = V2G = VI = I2 / G

The power P transferred by a capacitance C holding a changing voltage V with charge Q is:
P = VI = CV(dv/dt) = Q(dv/dt) = Q(dq/dt) / C

The power P transferred by an inductance L carrying a changing current I with magnetic linkage Y is:
P = VI = LI(di/dt) = Y(di/dt) = Y(dy/dt) / L


Energy

The energy W consumed over time t due to power P dissipated in a resistance R carrying a current I with a voltage drop V is:
W = Pt = V2t / R = VIt = I2tR

Similarly, the energy W consumed over time t due to power P dissipated in a conductance G carrying a current I with a voltage drop V is:
W = Pt = V2tG = VIt = I2t / G

The energy W stored in a capacitance C holding voltage V with charge Q is:
W = CV2 / 2 = QV / 2 = Q2 / 2C

The energy W stored in an inductance L carrying current I with magnetic linkage Y is:
W = LI2 / 2 = YI / 2 = Y2 / 2L


Batteries

If a battery of open-circuit voltage EB has a loaded voltage VL when supplying load current IL, the battery internal resistance RB is:
RB = (EB - VL) / IL

The load voltage VL and load current IL for a load resistance RL are:
VL = ILRL = EB - ILRB = EBRL / (RB + RL)
IL = VL / RL = (EB - VL) / RB = EB / (RB + RL)

The battery short-circuit current Isc is:
Isc = EB / RB = EBIL / (EB - VL)


Voltmeter Multiplier

The resistance RS to be connected in series with a voltmeter of full scale voltage VV and full scale current drain IV to increase the full scale voltage to V is:
RS = (V - VV) / IV

The power P dissipated by the resistance RS with voltage drop (V - VV) carrying current IV is:
P = (V - VV)2 / RS = (V - VV)IV = IV2RS


Ammeter Shunt

The resistance RP to be connected in parallel with an ammeter of full scale current IA and full scale voltage drop VA to increase the full scale current to I is:
RP = VA / (I - IA)

The power P dissipated by the resistance RP with voltage drop VA carrying current (I - IA) is:
P = VA2 / RP = VA(I - IA) = (I - IA)2RP


Wheatstone Bridge

The Wheatstone Bridge consists of two resistive potential dividers connected to a common voltage source. If one potential divider has resistances R1 and R2 in series and the other potential divider has resistances R3 and R4 in series, with R1 and R3 connected to one side of the voltage source and R2 and R4 connected to the other side of the voltage source, then at the balance point where the two resistively divided voltages are equal:
R1 / R2 = R3 / R4

 

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