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Basic / Important Electrical FormulasBasic ConceptsElectric Circuit Analysis

Electrical and Electronics Engineering Formulas and Equations

List of All Electrical and Electronics Engineering Formulas

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5000+ Electrical and Electronics Engineering Formulas

Basic Electrical Engineering Formulas

Basic Electrical Engineering Formulas

Electrical Current Formulas

Electrical Current Formulas in DC Circuit

  • I=V/R
  • I = P/V
  • I = √P/R

Where

  • I = Current in Amperes (A)
  • V = Voltage in Volts (V)
  • P = Power in Watts (W)
  • R = Resistance in Ohm (Ω)

Electrical Current Formulas in Single Phase AC Circuit

  • I = P / (V x Cosθ)
  • I=(V/Z) …Where Z = impedance = Resistance of AC Circuits

Electrical Current Formulas in Three Phase AC Circuit

I = P / √3 x V x Cosθ

Voltage or Electrical Potential Formulas

Electrical Potential or Voltage Formula in DC Circuits

  • V = I x R
  • V = P / I
  • V = √ (P x R)

Voltage or Electrical Potential Formulas in Single Phase AC Circuits

  • V = P/(I x Cosθ)
  • V = I / Z… Where Z = impedance = Resistance of AC Circuits

Voltage Formulas in Three Phase AC Circuits

  • VL = √3 VPH or VL = √3 EPH     [Star Connection]
  • VL = VPH     [Delta Connection]

Electrical Resistance Formulas

Electrical Resistance & Impedance Formulas in DC Circuits

  • R = V/I
  • R = P/I2
  • R = V2/P

Electrical Resistance & Impedance Formulas in AC Circuits

In AC Circuits (Capacitive or inductive Load), Resistance = Impedance i.e., R = Z

  • Z = √(R2 + XL2)… In case of Inductive Load
  • Z = √(R2 + XC2)… In case of Capacitive Load
  • Z = √(R2 + (XL– XC)2… In case of both inductive and capacitive Loads.

Good to know:

Where;

X= Inductive reactance

X= 2πfL…Where L = Inductance in Henry

And;

XC = Capacitive reactance

XC = 1/2πfC… Where C = Capacitance in Farads.

Ohm’s Law

Ohm’s law shows the relationship between current “I” & the voltage “V” where the resistance “R” is a constant in an electrical circuit.

For DC:

  • I = V/R            For current calculation
  • V = IR             For voltage calculation
  • R = V/I            For resistance calculation

For AC:

  • I = V/Z             For current calculation
  • V = IZ              For voltage calculation
  • Z = V/I             For impedance calculation

Voltage Divider Rule

It is only applicable when there is more than one resistance or impedance in series. In the parallel combination of resistors, the voltage remains the same.

Voltage Divider Rule For DC Circuit:

Voltage Divider Rule For DC Circuit

Where

  • Vn = Voltage across Resistor Rn
  • Vs = Supplied voltage or total voltage across resistance network
  • Rn = Resistance of resistor, where n = 1,2,3..

Voltage Divider Rule For AC Circuit:

Voltage divider for AC Circuit

Where

  • Vn = Voltage across Impedance Zn
  • Vs = Supplied voltage or total voltage across impedance network
  • Zn = Impedance, where n = 1,2,3..

Current Divider Rule

It is only applicable when the resistance network is connected in a parallel combination. In series combination, the current remains the same through the resistance network.

Current Divider Rule For DC Circuit:

Current Divider For DC Circuit

Where

  • In = current through Resistor Rn
  • Is = Supplied current or total current through the resistance network
  • Rn = Resistance of resistor, where n = 1,2,3..

Current Divider Rule For AC Circuit:

Current Divider For AC Circuit

Where

  • In = Current through Impedance Zn
  • Is = Supplied current or total current through impedance network
  • Zn = Impedance, where n = 1,2,3..

Kirchhoff's Laws

Kirchhoff’s Current Law

Summation of all currents entering a node or junction is 0.

Kirchhoff’s Current Law

Current entering the node is denoted with positive sign.

Current leaving the node is written with a negative sign.

Kirchhoff’s Voltage Law

Summation of all potential differences in a circuit loop is 0.

Kirchhoff’s Voltage Law

You may find more about KVL & KCL Here.

Coulomb's Law

It provides the force of attraction or repulsion between two charges.

Coulomb’s Law

WhereCoulomb’s constant

  • ε0 = permittivity in space
  • εr = relative permittivity of material
  • q1,q2 = 1st & 2nd amount of charge respectively in coulombs
  • r = Distance between the charges in meters

Electric Field Intensity Formula

Force per unit charge is known as electric field intensity.

E = F/Q

Where:

  • E = Electric Field Intensity
  • F = Force
  • Q = Electric Charge

Electric Flux Formula

Electric flux is the electric field lines passing through an area A.

ΦE = EA cosϴ

Where

  • Φ= Electric flux
  • E = Electric field
  • ϴ = Angle between E & A

It’s a vector quantity.

Electric Flux Density Formula

Electric Flux Density:

The electric flux per unit area is called the electric flux density.

D = ΦE /A

Where:

  • D = Density
  • ΦE = Electric flux
  • A = Area

It is a scalar quantity.

Magnetic Flux Formula

Magnetic Flux:

The number of magnetic lines passing through area A is known as Magnetic flux.

Φb = BA cosϴ

Where

  • Φb = magnetic flux
  • B = Magnetic field
  • ϴ = angle between B & A

It is a vector quantity.

Magnetic Flux Density Formula

The magnetic flux per unit area is called magnetic flux density.

B = Φ/A

Where:

  • B = Magnetic Flux Density
  • Φ = Magnetic Flux
  • A = Area

It is a scalar quantity.

Resistance in Series & Parallel Equations

Electrical Elements in Series & Parallel Combination:

Resistance:

The total equivalent resistance of resistors connected in series or parallel configuration is given the following formulas:

Resistance In Series:

Series resistance

When two or more than two resistors are connected in series as shown in figure their equivalent resistance is calculated by

REq = R1 + R2 + R3 +…

Resistance In Parallel:

Resistance in parallel

when the resistors are in parallel configuration the equivalent resistance becomes:

Resistance In Parallel

Where

REq is the equivalent resistance of all resistors (R1, R2, R3…)

Delta Δ to Wye Y (Pi to Tee) Conversion:

The delta (Δ) interconnection is also referred to as Pi interconnection & the wye (Y) interconnection is also referred to as Tee (T) interconnection.

delta to wye conversion

From Delta (Δ) to Wye (Y) Interconnection:

Delta (Δ) to Wye

From Wye (Y) to Delta (Δ) Interconnection

Wye (Y) to Delta

Capacitance in Series & Parallel Equations

Capacitance:

Total capacitance of the capacitor connected in parallel & series configuration are given below:

Capacitance In Series:

Capacitance in series

When the capacitors are connected in series configuration the equivalent capacitance becomes:

Capacitance In Series

Capacitance In Parallel:

Capacitance in parallel

The capacitance sums up together when they are connected together in a parallel configuration

CEq = C1 + C2 + C3 +…

Where

CEq is the equivalent Capacitance of all capacitors (C1, C2, C3…)

Inductance in Series & Parallel Equations

Inductance:

The calculation of total Inductance of inductors inside a circuit resembles resistors.

Inductance In Series:

Inductance in series

When the inductors are in series as shown in the figure, their inductance adds up together.

LEq = L1 + L2 + L3 +…

Inductance In Parallel:

Inductance in parallel

In parallel combination, the equivalent Inductance of the inductors is given by

Inductance in parallel

Where

LEq is the equivalent Inductance of all inductors (L1, L2, L3…)

Formulas for Capacitors

Formula & Equations For Capacitor:

The capacitance is the amount of charge stored in a capacitor per volt of potential between its plates.

The Capacitance Of The capacitor:

Capacitance can be calculated when charge Q & voltage V of the capacitor are known:

C = Q/V

Charge Stored in a Capacitor:

If capacitance C & voltage V is known then the charge Q can be calculated by:

Q = C V

Voltage Of The Capacitor:

And you can calculate the voltage of the capacitor if the other two quantities (Q & C) are known:

V = Q/C

Where

  • Q is the charge stored between the plates in Coulombs
  • C is the capacitance in farads
  • V is the potential difference between the plates in Volts

Capacitance Formula

The capacitance between two conducting plates with a dielectric between then can be calculated by:Capacitance

Wheredielectric constant

  • k is the dielectric constant
  • εis the permittivity of the dielectric
  • ε0 ­­is the permittivity of space which is equal to 8.854 x 10-12 F/m
  • A is the area of the plates
  • d is the separation between the plates

Reactance of the Capacitor:

Reactance is the opposition of capacitor to Alternating current AC which depends on its frequency and is measured in Ohm like resistance. Capacitive reactance is calculated using:

capacitive reactance

Where

  • Xis the capacitive reactance
  • F is the applied frequency
  • C is the capacitance

Quality Factor Of Capacitor:

Q factor or Quality factor is the efficiency of the capacitor in terms of energy losses & it is given by:

QF = XC/ESR

Where

Xis the capacitive reactance

ESR is the equivalent series resistance of the capacitor.

Dissipation Factor Of Capacitor:

D factor or dissipation factor is the inverse of the Quality factor, it shows the power dissipation inside the capacitor & is given by:

DF = tan δ = ESR/XC

Where

  • DF is the dissipation factor
  • δ is the angle between capacitive reactance victor & negative axis.
  • Xis the capacitive reactance
  • ESR is the equivalent series resistance of the circuit.

Energy Stored In Capacitor:

The Energy E stored in a capacitor is given by:

E = ½ CV2

Where

  • E is the energy in joules
  • C is the capacitance in farads
  • V is the voltage in volts

Average Power Of Capacitor

The Average power of the capacitor is given by:

Pav = CV2 / 2t

where

t is the time in seconds.

Capacitor Voltage During Charge / Discharge:

When a capacitor is being charged through a resistor R, it takes upto 5 time constant or 5T to reach upto its full charge. The voltage at any specific time can by found using these charging and discharging formulas below:

During Charging:

The voltage of capacitor at any time during charging is given by:

Capacitor charging

During Discharging:

The voltage of capacitor at any time during discharging is given by:

Capacitor discharging

Where

  • VC is the voltage across the capacitor
  • Vs is the voltage supplied
  • t  is the time passed after supplying voltage.
  • RC = τ is the time constant of the RC charging circuit

Ohm’s Law For Capacitor:

Q = CV

By differentiating the equation, we get:Ohm’s Law for capacitor

where

  • i is the instantaneous current through the capacitor
  • C is the capacitance of the capacitor
  • Dv/dt is the instantaneous rate of change of voltage applied.

Formula for Inductor

Formula & Equations For Inductor:

Inductance of Inductor:

The inductance of the inductor from the basic formula of inductor:

inductance

Voltage Across Inductor:

Voltage Across Inductor

Current Of The Inductor:

Current Of The Inductor

Where

  • V is the voltage across inductor
  • L is the inductance of the inductor in Henry
  • Di/dt is the instantaneous rate of current change through the inductor.
  • ito = current at time t=0.

Reactance Of The Inductor:

Inductive reactance is the opposition of inductor to alternating current AC, which depends on its frequency f and is measured in Ohm just like resistance. Inductive reactance is calculated using:

X= ωL = 2πfL

Where

  • Xis the Inductive reactance
  • F is the applied frequency
  • L is the Inductance in Henry

Quality Factor Of Inductor:

The efficiency of the inductor is known as quality factor & its measured by:

QF = XL/ESR

Where

  • Xis the Inductive reactance
  • ESR is the equivalent series resistance of the circuit.

Dissipation Factor Of Inductor:

It is the inverse of the quality factor and it shows the power dissipation inside the inductor & its given by:

DF = tan δ = ESR/XL

Where

  • DF is the dissipation factor
  • δ is the angle between capacitive reactance victor & negative axis.
  • Xis the capacitive reactance
  • ESR is the equivalent series resistance of the circuit.

Energy Stored In Inductor:

The energy E stored in inductor is given by:

E = ½ Li2

Where

  • E is the energy in joules
  • L is the inductance in Henry
  • i is the current in Amps

Average Power Of Inductor

The average power for the inductor is given by:

Pav = Li2 / 2t

Where

t = is the time in seconds.

Inductor Current During Charge / Discharge:

Just like capacitor, the inductor takes up to 5 time constant to fully charge or discharge, during this time the current can be calculated by:

During Charging:

Instantaneous current of the inductor during charging is given by:

Inductor Charging

During Discharging:

The current during the discharging at any time t is given by:

Inductor discharging

Where

  • IC is the current of the inductor
  • I0 is the current at time t=0
  • t  is the time passed after supplying current.
  • τ = L/R is the time constant of the RL circuit

Time Constant Tau Formulas

Time Constant τ “Tau” Formulas for RC, RL, RLC Circuits

Time constant τ is a constant parameter of any capacitive or inductive circuit. It differs from circuit to circuit &also used in different equations. The time constant for some of these circuits are given below:

For RC Circuit:

In this circuit, resistor having resistance R is connected in series with the capacitor having capacitance C, whose time constant τ is given by:

τ = RC

Where

  • R is the resistance in series
  • C is the capacitance of the capacitor

For RL Circuit:

Inductor of inductance L connected in series with resistance R, whose time constant τ is given by:

τ = R/L

Where

  • R is the resistance in series
  • L is the Inductance of the Inductor

For RLC Circuit:

In RLC circuit, we have both RL & RC time constant combined, which makes a problem calculating the time constant. So we calculate what we call the Q-Factor (quality factor).

For Series RLC Circuit:

Q factor Series RLC Circuit

For Parallel RLC Circuit:

Q factor parallel RLC Circuit

Where

  • R is the resistance in series
  • L is the Inductance of the Inductor
  • C is the capacitance of the capacitor

Resistance & Conductance Formulas

Resistance

Resistance is the opposition to the flow of electrical current denoted bu R and measured in ohms. For any metal conductor R is given by:

R = ρl/A

Where

  • ρ (Greek word Rho)is specific electrical resistance of the conductor
  • l is the length of the conductor
  • A is the cross-sectional area of the conductor

Conductance

The conductance is the inverse of resistance. It is the allowance of the electrical current through a conductor, denoted by G & measured in Siemens.

G = σA/l

Where

σ (Greek word sigma) is the electrical conductivity

Impedance & Admittance Formulas

Impedance:

The opposition of a circuit to the current when voltage is applied is impedance, denoted by Z & it is measured in Ohms.

Z= R + jX

Where

  • R is the real part, resistance of the circuit
  • X is the imaginary part, reactance of the circuit.

Admittance:

The inverse of Impedance is Admittance denoted by Y & it is measured in Siemens:

Y = 1/Z

Y = G + JB

Where

  • G is the real part known as Conductance of the circuit
  • B is the imaginary part known as Susceptance

Power Formulas

Power

DC Power:

  • P = IV
  • P = I2R
  • P = V2/R

AC Power:

Complex Power & Apparent Power:

When there is an inductor or capacitor in a circuit, the power becomes complex power “S”, meaning it has two parts i.e. real & imaginary part. The magnitude of Complex power is called Apparent power |S|.

Complex power & apparent power

Where

  • P is the real power
  • Q is the reactive power
Active or Real Power & Reactive Power:

The real part is Complex power “S” is known as active or real power “P” & the imaginary part is known as reactive power “Q”.

  • S = P + jQ
  • P = V I cosϴ
  • Q = V I sinϴ

Where

ϴ is the phase angle between voltage & current.

Power Factor:

Power factor “PF” is the ratio of real power “P” to apparent power “|S|”. Mathematically, Power factor is the cosine of angle ϴ between real power & apparent power.

Power Factor

Where

|S| = √(P2+Q2)

Other formulas used for Power Factor are as follow:

Cosϴ = R/Z

Where:

  • Cosϴ = Power Factor
  • R = Resistnace
  • Z = Impedence (Resistance in AC circuits i.e. XLXC and R known as Inductive reactancecapacitive reactance and resistance respectively).

Cosϴ = kW / kVA

Where

  • Cosϴ = Power Factor
  • kW = Real Power in Watts
  • kVA = Apparent Power in Volt-Amperes or Watts

Additional formulas used for power factor.

Real Power Of Single Phase & 3-Phase Current

Real Power Of Single Phase & 3-Phase

Where

  • Vrms  & Irms is the root mean square value of voltage & current respectively.
  • VL-N  & IL-N  is the line-to-neutral voltage & current respectively.
  •  VL-L & IL-L  is the line-to-line voltage & current respectively.
  • Cosϴ is the power factor PF.
Reactive Power Of Single & 3-Phase Current:

Reactive power of single & 3-phase

Where

ϴ = is the phase angle

DC Motors Formulas

Shunt Motor:
Voltage Equation Of Shunt Motor:

V = E+ Ia Ra

Where

  • V is the terminal voltage
  • Eis the induced back e.m.f
  • Iis the armature current
  • R­­­is the armature resistance
The Shunt Field Current:

sh ­= V / Rsh

Where

  • Ish is the shunt field current
  • Rsh ­is the shunt field resistance
Induced Back EMF:

The armature induced voltage Eis proportional to the speed & it is given by:

E= kfΦω

Where

  • Kis a constant based on machine construction
  • Φ is the magnetic flux
  • ω is the angular speed
Maximum Power Condition:

The output mechanical power is of shunt dc motor is maximum when the back e.m.f. produced is equal to the half of its terminal voltage i.e.

Eb = V/2

Torque & Speed:

Torque & Speed of DC shunt Motor

And

Machine constant

Where

  • N = speed of the motor in RPM
  • P = No of poles
  • Z = number of armature conductors
  • A = number of armature parallel path
Speed Regulation:

It is a term expressed in percentage that shows the change of motor speed when the load is changed.

Speed Regulation of DC Shunt Motor

Where

  • nl = No load speed of the motor
  • Nfl = Full load speed of the motor
Input & Output Power:

Pin = VIa

Pout = T ω

Where

  • V = terminal voltage
  • I­a = armature current
  • T = torque of the motor
  • ω = speed of the motor
Series Motor:
Voltage Equation Of Series Motor:

V = E+ Ia Ra + IaRse

V = E+ Ia(Ra+Rse)

Where

  • is the armature induced voltage
  • Iis the armature current
  • R­­­is the armature resistance
  • Rse is the series field resistance
Armature Induced Voltage & Torque:

The armature induced voltage Eis proportional to the speed & armature current whereas the torque Ta of series motor is directly proportional to the square of armature current & it is given by:

E= kfΦωIa

Ta = kΦ Ia2

Where

  • Kis a constant based on machine construction
  • Φ is the magnetic flux
  • ω is the angular speed
Speed Of Series Motor:

Speed of DC series motor

Input & Output Power

The input power of a series motor is given by:

Pin = VIa

The output power is given by

Pout = ωT

Efficiency Of DC Motor:
Electrical Efficiency:

η=  Converted power in armature / Input electrical Power

Electrical efficiency of DC motor

Mechanical Efficiency:

η= Converted power in armature / output mechanical power

Mechanical Efficiency of DC Motor

Overall Efficiency:

η = Output mechanical Power / Input electrical Power

η = (Input Power – Total losses) / Input Power

Overall Efficiency of DC Motor

Where

  • Pout is the useful output power
  • Pa ­­is the armature copper loss
  • Pf is the field copper loss
  • Pk is the constant losses that contains core losses & mechanical losses

DC Generator Formulas

Shunt Generator:
Terminal Voltage:

V = E– Ia Ra

Where

  • is the armature induced voltage
  • Iis the armature current
  • R­­­is the armature resistance

Terminal Current:

Ia = If ­+ IL

where IIs the field current & Iis the load current

The Field Current:

­= V / Rsh

Where

  • Iis the field current
  • Rsh ­is the shunt field resistance
EMF Equation For DC Generator:

The EMF generated per conductor in a DC generator is:

EMF equation for DC generator

Where

  • Z = number of conductors
  • P = number of Poles
  • N = Speed of rotor in RPM
  • A = number of parallel paths

The EMF generated per path for a wave winding & lap-winding;

EMF generated per path for a wave winding & lap-winding

So the generalized equation for generated EMF of DC generator is:

E= kΦω

Where

  • K = ZP/2πA = constant of the machine
  • ω = 2πN/60 = angular speed in rads per second
Torque:

the torque of generator is directly proportional to the armature current & it is given by:

T = kfΦIa

Where

  • Kis a constant based on machine construction
  • Φ is the magnetic flux
  • ω is the angular speed

angular speed of generator

Where N is the speed in Rotation Per Minute (RPM)

Power Generated & Load Power

The power generated by a shunt generator is given by:

P= ωT = EaIa

PL = VIL

Where Iis the load current

Series Generator:
Terminal Voltage:

V = E– (Ia R+ Ia Rse)

V = E– Ia(R+ Rse)

Where

  • is the armature induced voltage
  • Iis the armature current
  • R­­­is the armature resistance
  • Rse ­is the series field resistance

The series field current is equal to the armature current;

I= Ise

Armature Induced Voltage & Torque:

The armature induced voltage Eis proportional to the speed & armature current whereas the torque T of series generator is directly proportional to the square of armature current & it is given by:

E= kfΦωIa

T = kΦ Ia2

Where

  • Kis a constant based on machine construction
  • Φ is the magnetic flux
  • ω is the angular speed

angular speed of series generator

Where N is the speed in Rotation Per Minute (RPM)

Power Generated & Load Power

The power generated by a series generator is given by:

P= ωT = EaIa

PL = VIL

Where Iis the load current

Input Power:

Pin = ωT

Where

  • ω is the angular speed of armature
  • T is the torque applied
Converted Power:

Pcon = Pin – Stray losses – mechanical losses – core losses

Pcon = EaIa

Where

  • Ea is the induced voltage
  • Ia is the armature current
Output Power

Pout = Pcon – Electrical losses (I2R)

Pout = VIL

Where

  • V is the terminal voltage
  • Iis the load current
Efficiency Of DC Generator:
Mechanical Efficiency:

Mechanical efficiency of DC generator

Electrical Efficiency:

Electrical Efficiency of DC generator

Overall Efficiency:

Overall Efficiency of dc generator

Where

  • Pout is the useful output power
  • Pa ­­is the armature copper loss
  • Pf is the field copper loss
  • Pk is the constant losses that contains core losses & mechanical losses
Maximum Efficiency:

The efficiency of the dc generator is Maximum, when;

Variable power loss = Constant power loss

Copper loss = Core & mechanical loss

Copper loss (I2R) such as armature & field copper loss are variable loss because they depend on current. While core loss such as hysteresis & eddy current loss, mechanical loss such as friction losses are all constant losses.

Losses in Machines Formulas

Copper Losses:
Armature Loss:

Armature Cu Losses = Pa = IaRa

Where

  • Iis the armature current
  • Ra is the armature resistance
Field Loss:

Field cu Losses = Pf = If2Rf

Where

  • Iis the field current
  • Ris the field resistance

For Shunt Field:

Shunt field copper loss

Where

  • sh is the shunt field current
  • sh is the shunt field resistance

For Series Field:

Series field copper loss

Where

  • se is the series field current
  • se is the series field resistance
Iron/Core Losses
Hysteresis Loss:

Hysteresis loss in DC machine

Where

  • η = hysteresis or Steinmetz’s constant
  • Bmax = maximum value of the magnetic flux density
  • f = frequency of magnetization
  • V= volume of the core

Also

frequency of magnetization

Where

  • P is the number of poles
  • N is the speed in RPM
Eddy Current Loss:

Eddy Current loss in DC machine

Where

  • is the electrical constant of the core material
  • Bmax is the maximum flux density
  • f is the frequency of magnetization
  • t is the thickness of lamination
  • V is the volume of the core

Synchronous Motor Formulas

AC Machines:

Synchronous Machine:

Speed Of Synchronous Machine:

Synchronous machine are designed to be operated at synchronous speed, which is given by:

synchronous speed

Where

  • is the synchronous speed
  • f is the line voltage frequency
  • P is the number of poles in machine

Synchronous Motor:

Voltage Equation Of Synchronous Motor:

V = Eb + Ia(Ra + jXs)

Where

  • V = voltage applied
  • E= Back emf
  • Ia = Armature current
  • R= Armature resistance
  • X= synchronous reactance
Resultant Voltage:

The difference between the voltage applied V & back emf is known as resultant voltage ER

ER = V – Eb

ER = Ia(Ra + jXs)

Internal Angle:

It is the angle by which the armature current Ia lags behind the resultant voltage in armature ER, and it is given by;

Internal Angle in synchronous motor

Back EMF Generated:

E= KaφaNs

Where

  • K= constant of the armature winding
  • φ= magnetic Flux per pole of the rotor
  • N= synchronous speed of the rotor
Different Excitations:
  • E= V             Normal Excitation                            Lagging Power Factor
  • Eb < V             Under-Excitation                              Lagging Power Factor
  • Eb > V             Over- Excitation                                Leading Power Factor
Input Power:

The input power of synchronous motor is given by:

Input Power to synchronous motor

Where

Φ is the angle between V & Ia

Mechanical Power In Rotor:

Mechanical power in rotor of synchronous motor

Where

  • α is the load angle between E­& V
  • Φ is the angle between V & Ia
  • Tis gross torque produced
  • Nis the synchronous speed

Synchronous Generator Formulas

Synchronous Generator:

Output Electrical Frequency:

Output Electrical frequency of synchronous generator

Where

  • f = Electrical frequency
  • Nr = speed of rotor in RPM
  • P = Number of poles
Voltage Generated:

E= KφaNs

Where

  • K = constant representing the construction of machine
  • φ= magnetic Flux per pole of the rotor
  • N= synchronous speed of the rotor
Total Phase Voltage:

Vφ = Ea – jXsIa – RaIa

Where

  • Xs = Synchronous reactance of machine
  • Ia = Armature current
  • Ra = Armature resistance
Three Phase Terminal Voltage:

Three Phase Terminal voltage of synchronous generator

Power Of Synchronous Generator:

Power of synchronous generator

Where

  • Tapp = Torque applied
  • Tind = Torque induced in rotor
  • ωr = mechanical speed of rotor
Voltage Regulation:

Voltage Regulation of synchronous generator

Where

  • Vnl ­­= Voltage at no load
  • Vfl = Voltage at full load
Efficiency:

η = (Pout / Pin) * 100%

Pin = Pout + PCu + Piron + P­mech ­+ Pstray

Induction Motors Formulas

Induction Motor:

Induced EMF:

eind = vBl

where

  • eind = induced EMF
  • v = velocity of the rotor
  • B = magnetic flux density
  • l = length of conductors inside magnetic field
Rotor Current:

The rotor current is given by:

Rotor current of induction motor

Torque Induced:

Terms used in Motor Torque Equations and formulas.

  • Ns = Synchronous speed
  • s = slip of the motor
  • s= breakdown or pull-out slip
  • E1 = stator voltage or input voltage
  • E2 = Rotor EMF per phase at a standstill
  • R2 = Rotor Resistance Per Phase
  • X2 = Rotor Reactance Per Phase
  • V = supply voltage
  • K = rotor/stator turn ration per Phase

Starting Torque

Starting torque of induction motor

  • Maximum Starting Torque Condition

R= X2

  • Starting Torque Relation With Supply Voltage

Tst  α  V2

  • Torque In Running Condition

Torque in running condition of induction motorGross torque of induction motor

  • Gross Torque

Gross torque of induction motor

  • Maximum Running Torque Condition

R= sX2

  • Maximum Running Torque

Maximum running torque of induction motor

  • Breakdown Slip

Breakdown slip of induction motor

  • Torque Relation With Max Torque

Torque relation with max torque of induction motor

Slip Speed & Slip of Induction Motor:

Slip speed is the difference between synchronous speed and rotor speed;

  • Nslip = N­– N­                         (Speed in RPM)
  • ωslip = ω– ω­                        (Angular speed in Rad/sec)

Where

  • N­­slip = slip speed
  • Ns ­= synchronous speed = 120f/P
  • N­­ = rotor speed of motor

The slip of induction motor is a relative term expressed in percentage. It is given by:

Slip of Induction motor

Where

S is the slip of induction motor

Rotor Speed:

The rotor speed of induction motor is given by

  • N = (1-s)N               (Speed in RPM)
  • ω = (1-s) ω s             (Angular speed in Rad/sec)
Electrical Frequency On The Rotor: 

Electrical frequency on the rotor of induction motor

Where

  • f = Rotor Frequency
  • f  = Line Frequency
  • P = Number of Poles
Power Of Induction Motor:

Related terms used in Motor Power Formulas and Equation.

  • P1 = Stator input Power
  • P2 = Rotor Input power
  • Pm = Rotor Gross Output Power
  • Pout = Output Power
  • Tg = gross torque
  • Tsh = shaft torque

Rotor Input Power:

P2 = Tgωs

  • Rotor Gross Output Power:

Pm = Tgω

  • Output Power:

Pout = Tshω

P1 = P2 + stator Losses = P+ Rotor Copper Losses = Pout + Windage & friction Losses

Rotor Input Power: Output Mechanical Power: Rotor Cu loss ratio:

Power loss relation in induction motor

Where

Pcr = I2R = rotor Copper loss

Synchronous Watt:

The torque at which the machine at synchronous speed will generate one watt;

Synchronous watt

Efficiency Of Induction Motor:
  • Rotor Efficiency:

Rotor Efficiency of induction motor

  • Overall Efficiency

Overall Efficiency of induction motor

Linear Induction Motor Formulas

Linear Induction Motor:

Synchronous Speed:

Synchronous speed of linear induction motor

Where

  • v = linear synchronous speed
  • w = width of one pole-pitch
  • f = line frequency
Slip:

Slip of linear induction motor

Where

  • v= linear synchronous speed
  • v = Actual speed
Thrust Or Force:

Thrust Or Force of linear induction motor

Where

P2 = Rotor input Power

Rotor Cu Loss:

Rotor Cu loss of linear induction motor

Gross Mechanical Power:Gross Mechanical Power of linear induction motor

Stepper Motors Formulas

Stepper Motor:

Step Angle:

Step Angle of stepper motor

Where

  • β = step angle, the angle of rotation of the shaft with each pulse.
  • Ns = number of stator poles or teeth
  • Nr = number of rotor poles or teeth

Resolution Of Stepper Motor:

The number of steps required to complete one revolution, its given by;

Resolution of stepper motor

Higher the resolution, higher the accuracy of stepper motor.

Motor Speed:

Stepper Motor Speed

Where

  • n = motor speed in revolution per second
  • f = stepping pulse frequency

Transformer Formulas & Equations

Transformer:

EMF Induced In Primary & Secondary Windings:

EMF induced in Primary & Secondary Windings of transformer

Where

  • E= EMF induced in primary winding
  • E2 = EMF induced in Secondary winding
  • N1 = Number of Turns in Primary winding
  • N2 = Number of Turns in Secondary winding
  • f  = Line frequency
  • φ= Maximum Flux in Core
  • B= Maximum flux density
  • A = Area of Core

Related Post: EMF Equation Of a Transformer

Voltage Transformation Ratio:

Voltage Transformation ratio of transformer

Where

  • K = voltage transformation ratio of transformer
  • V1I1 = Primary voltage & current Respectively
  • V2I2 = Secondary voltage & current Respectively

Equivalent Resistance Of Transformer Windings:       

Equivalent resistance of transformer windings

Where

  • R1 = Resistance of Primary winding in Secondary
  • R2 = Resistance of Secondary winding in primary
  • R01 = Equivalent resistance of transformer from primary side
  • R02 = Equivalent resistance of transformer from Secondary side
  • R1 = Primary winding Resistance
  • R2 = Secondary Winding Resistance

Leakage Reactance:

Leakage Reactance of transformer

Where

  • X1 = Primary leakage Reactance
  • X2 = Secondary leakage Reactance
  • eL1 = Self-Induced EMF in primary
  • eL2 = Self-Induced EMF in Secondary

Equivalent Reactance Of Transformer Windings:                    

Equivalent reactance of transformer windings

Where

  • X1 = Reactance of Primary winding in Secondary
  • X2 = Reactance of Secondary winding in primary
  • X01 = Equivalent reactance of transformer from primary side
  • X02 = Equivalent reactance of transformer from Secondary side

Total Impedance Of Transformer Winding:

Total Impedance of Transformer winding

Where

  • Z1 = Impedance of primary winding
  • Z2 = Impedance of Secondary winding
  • Z01 = Equivalent Impedance of transformer from primary side
  • Z02 = Equivalent Impedance of transformer from Secondary side

Input & Output Voltage Equations 

Input & Output Voltage equations of transformer

Losses In Transformer:

Core / Iron Losses

The losses that occur inside the core;

  • Hysteresis Loss

Due to magnetization and demagnetization of the core

Hysteresis loss in transformer

  • Eddy Current Loss

Due to the induced EMF produced inside the core causes the flow of eddy current.

Eddy Current Loss in transformer

Where

  • Wh = Hysteresis loss
  • We = Eddy current loss
  • η = Steinmetz Hysteresis coefficient
  • Ke = Eddy current constant
  • Bmax  = Maximum magnetic flux
  • f = frequency of flux
  • V = Volume of the core
  • t = thickness of the lamination
Copper Loss:

The loss due to the resistance of the winding

Copper Loss in transformer

Voltage Regulation Of Transformer:

When the input voltage to the transformer primary is kept constant and a load is connected to the secondary terminal, the secondary voltage decreases due to internal impedance.

The comparison of no load secondary voltage to the full load secondary voltage is called voltage regulation

Voltage Regulation of transformer

  • 0V= No load Secondary voltage
  • V2 = Full load Secondary voltage
  • V1 = No load Primary voltage
  • V2’ = V2/K = Full load Secondary voltage from primary side
  • Regulation Up

Regulation Up

  • Regulation Down

Regulation Down

Regulation “Down” is commonly referred as regulation

  • Regulation in Primary Voltage Terms:

Regulation in primary voltage terms

  • Regulation When Secondary Voltage Supposed to be Constant

After connecting load, the primary voltage needs to be increased from V1 to V1, where the voltage regulation is given by:

Regulation when secondary voltage supposed to be constant

Percentage Resistance, Reactance & Impedance:

These quantities are measured at full load current with the voltage drop, & expressed as the percentage of normal voltage.

  • Percentage Resistance at Full Load:

Percentage resistance at full load

  • Percentage Reactance at Full Load:

  • Percentage Impedance at Full Load:

Percentage Impedance at Full Load

Transformer Efficiency:

The efficiency of the transformer is given by the output power divide by the input power. Some of the input power is wasted in internal losses of the transformer.

Transformer Efficiency

Total losses = Cu loss + Iron Loss

Efficiency At Any Load:

The efficiency of the transformer at an actual load can be given by;

transformer Efficiency at any load

Where

x = Ratio of Actual load to full load kVA

All Day Efficiency:

The ratio of energy delivered in Kilo Watt-Hour (kWh) to the energy input in kWh of the transformer for 24 hours is called all day efficiency.

All Day Efficiency of transformer

Condition For Maximum Efficiency:

The copper lost must be equal to the iron loss, which the combination of hysteresis loss & eddy current loss.

Cu Loss = Iron Loss

Wcu = Wi

Where

  • Wi = Wh­ + We
  • Wcu = I12 R01 = I22 R02
Load Current For Maximum Efficiency:

The load current required for the maximum efficiency of the transformer is;

Load Current for Maximum Efficiency

RLC Circuits - Series & Parallel Equations

RLC Circuit:

When the resistor, inductor & capacitor are connected together in parallel or series combination, it operates as an oscillator circuit whose equations are given below:

Parallel RLC Circuit

When they are connected in parallel combination

Impedance:

Total impedance of the circuit is;

Total impedance of parallel RLC circuit

Where

  • XL = Inductive reactance
  • XC = Capacitive reactance
Power Factor:

The power factor for this circuit is

Cos ϴ = Z/R

Resonance Frequency:

When inductive reactance XL & capacitive reactance X­c of the circuit is equal.

Resonance Frequency of parallel RLC circuit

Where

  • L = Inductance of inductor
  • C = Capacitance of capacitor
Quality Factor:

It is the ratio of stored energy to the energy dissipated in the circuit.

Quality Factor of parallel RLC circuit

Bandwidth:

B.W = fr / Q

Resonant Circuit Current:

The total current through the circuit when the circuit is at resonance.

At resonance, the X= X, so Z = R

IT = V/R

Current Magnification

Parallel resonance RLC circuit is also known current magnification circuit. Because, current flowing through the circuit is Q times the input current

Imag = Q IT

Characteristic Equation:

Characteristic equation of parallel RLC circuit

Neper Frequency For Parallel RLC Circuit:

Neper frequency for Parallel RLC circuit

Resonant Radian Frequency For Parallal RLC Circuit:

Resonant radian frequency for parallal RLC circuit

Voltage Response:
  • Over-Damped Response

When

ω0< α2

The roots s1 & s2 are real & distinct

  • Under-Damped Response

When

ω0> α2

The roots s1 & s2 are complex & conjugate of each other

  • Critically Damped Response

When

ω0= α2

The roots s1 & s2 are real & equal

Series RLC Circuit:

 Impedance:

The total impedance of the series RLC circuit is;

total impedance of the series RLC circuit

Power Factor:

The power factor of Series RLC circuit;

Cos ϴ = R/Z

Resonance Frequency:

The frequency at which the inductive reactance XL = Capacitive reactance X­­c is known as resonance frequency.

Resonance Frequency of series RLC circuit

Where

  • L = Inductance of the inductor
  • C = Capacitance of the capacitor
Quality Factor:

Quality Factor of series RLC circuit

Bandwidth:

B.W = (fr / Q)

  • B.W = (R / L)              in rad/s
  • B.W = (R / 2πL)         in hz
Lower Cutoff Frequency & Upper Cutoff Frequency:

fh = fr + ½ B.W

fl = fr – ½ B.W

Characteristic Equation:

Characteristic equation of series RLC circuit

Neper Frequency For series RLC Circuit:

Neper Frequency For series RLC Circuit

Resonant Radian Frequency For series RLC Circuit:

Resonant Radian Frequency For series RLC Circuit

Voltage Response:
  • Over-Damped Response

When

ω0< α2

The roots s1 & s2 are real & distinct

  • Under-Damped Response

When

ω0> α2

The roots s1 & s2 are complex & conjugate of each other

  • Critically Damped Response

When

ω0= α2

The roots s1 & s2 are real & equal

Diode Formula & Equations

Diode:

Schockley Diode Equation:

Schockley diode equation

Where

  • I= current through the diode
  • V= diode voltage
  • Is = leakage or reverse saturation current
  • n = emission coefficient or ideality factor, for germanium n=1, for silicon it ranges in 1.1-1.8.
  • V= thermal voltage which is

thermal voltage

Where

  • q = charge of electron = 1.6022 x 10-19 coulomb
  • T = absolute temperature in Kelvin (K = 273 + °C)
  • k = Boltzmann’s constant = 1.3806 x 1023 J/K

Diode Rectifier:

A rectifier’s output contains DC as well as AC components, So;

Output DC Power:

Pdc = Vdc Idc­

Where

  • Vdc is the average output voltage
  • Idc­ is the average output current
Output AC Power:

Pac = Vrms Irms

Where

  • Vrms Is the rms of output voltage
  • Irms is the rms of output current
Rectifier Efficiency:

The efficiency of the rectifier denote by η is given by:

Rectifier Efficiency

Where

  • Pdc is the output DC power
  • Pac is the output AC power
Output AC Voltage:

The rms of AC component of the output voltage is:

Output AC voltage Rectifier

Form Factor:

The ratio of RMS voltage to the average dc voltage,

Form Factor

Ripple Factor:

It’s the ratio between the AC & DC component of the rectifier. It shows the purity of the DC output.

Ripple Factor

Bipolar Junction Transistor (BJT) Formulas

BiPolar Junction Transistor:

Current Gains in BJT:

There are two types of current gain in BJT i.e. α & β.

Current gains in BJT

Where

  • Iis the emitter current
  • Iis the collector current
  • IB­ is the base current

Common Base Configuration:

Common Base Voltage Gain

In common base configuration, BJT is used as voltage gain amplifier, where the gain Ais the ratio of output voltage to input voltage:

Common Base Voltage Gain

Where

  • α = IC / IE
  • RL is the load resistance
  • Rin ­is the input resistance

Common Emitter Configuration:

Forward Current Gain:

It is the ratio of output current i.e. the collector current I­to the input current i.e. the base current IB.

βF = hFE = IC/IB

Where

  • ΒF is the forward current gain
  • IC is the collector current
  • Iis the base current
Emitter Current:

The emitter current is the combination of collector & base current. It can be calculated using any of these equations.

  • I= IC + IB
  • I= IC / α
  • I= I(1+ β)
Collector Current:

The collector current for BJT is given by:

  • = βFI+ ICEO ≈ βFIB
  • α IE
  • I= IE – IB

Where

ICEO  is the collector to emitter leakage current (Open base).

Alpha α to Beta β Conversion Formula:

The gain alpha & beta are inter-convertible, & they can be converted using,

  • α = β / (β + 1)
  • β = α / (1- α)
Collector-to-Emitter Voltage:

VCE = VCB + VBE

Where

  • VCE is the collector-to-Emitter voltage
  • VCB is the collector-to-base voltage
  • VBE is the base-to-emitter voltage

Common Collector Configuration:

Current Gain:

The current gain Ai of common collector BJT is given by the ratio of output current Ito input Current IB:

  • A= IE / IB
  • A= (I+ IB) / IB
  • Ai = (I/ IB) + 1
  • A= β + 1

Operational Amplifier (OP-AMP) Formulas

Operational Amplifiers:

Inverting Amplifier:

Inverting Amplifier

The following terms are used in the formulas and equations for Operational Amplifies.

  • Rf = Feedback resistor
  • Rin = Input Resistor
  • Vin ­­­= Input voltage
  • Vout = Output voltage
  • Av = Voltage Gain
Voltage Gain:

The close loop gain of an inverting amplifier is given by;

gain of an inverting amplifier

Output Voltage:

The output voltage is out of phase with the input voltage that is why it is known as the inverting amplifier.

Summing Amplifier:

Summing Amplifier

Output Voltage:

The general output of this given circuit above is;

Summing Amplifier

Inverted Amplified Sum of Input Voltage:

if the input resistors are same, the output is a scaled inverted sum of input voltages,

If R1 = R= R3 = R= R

Inverted Amplified sum of input voltage

Summed Output:

When all the resistors in the above given circuit are same, the output is an inverted sum of input voltages.

If Rf = R1 = R= R3 = R= R;

Vout = – (V+ V2 + V3 +… + Vn)

Non-Inverting Amplifier:

Non-Inverting Amplifier

Terms used for Non-Inverting Amplifier formulas and equations.

  • Rf = Feedback resistor
  • R = Ground Resistor
  • Vin ­­­= Input voltage
  • Vout = Output voltage
  • Av = Voltage Gain

Related Post: Negative Feedback Amplifier Systems

Gain Of Amplifier:

The total gain of non-inverting amplifier is;

Non-Inverting Amplifier

Output Voltage:

The output voltage of non-inverting amplifier is in-phase with its input voltage & it’s given by;

output voltage of non-inverting amplifier

Unity Gain Amplifier / Buffer / Voltage Follower:

If the feedback resistor in removed i.e. Rf = 0, the non-inverting amplifier will become voltage follower/buffer.

Unity gain amplifier

Differential Amplifier:

Differential Amplifier

Terms used for Differential Amplifier formulas.

  • Rf = Feedback resistor
  • R= Inverting Input Resistor
  • R= Non Inverting Input Resistor
  • Rg = Non Inverting ground Resistor
  • Va ­­­= Inverting Input voltage
  • Vb ­­­= Non Inverting Input voltage
  • Vout = Output voltage
  • Av = Voltage Gain
General Output:

the output voltage of the above given circuits is;

Differential Amplifier

Scaled Differential Output:

If the resistor Rf = Rg  & R= R, then the output will be scaled difference of the input voltage;

Scaled Differential Output

Unity Gain Difference:

If all the resistors used in the circuit are same i.e. Ra = R= R= R= R, the amplifier will provide output that is the difference of input voltages;

Vout = V– Va

Differentiator Amplifier

Differentiator Amplifier

This type of Operational Amplifier provides the output voltage which is directly proportional to the changes in the input voltage. The output voltage is given by;

Differentiator Amplifier

Triangular wave input => Rectangular wave output

Sine wave input => Cosine wave output

Integrator Amplifier

Integrator Amplifier

This amplifier provides an output voltage which is the integral of the input voltages.

Integrator Amplifier

Frequency Filters Active / Passive Filters Formulas

Frequency Filters:

Passive Filters

The type of frequency selecting circuits that are made of only passive components such as resistor, capacitor & inductor.

Low Pass Filter:

It passes low input frequency without any attenuation & blocks high frequency after a fix point known as cutoff frequency.

The output is taken across C & R in RC & RL circuit respectively.

Related Posts:

Cutoff Frequency:

The frequency where the output signal becomes the 70.7% of the input signal is called cutoff, corner or breakpoint frequency, & it is given by;

Cutoff Frequency of Low Pass Filter

Transfer Function:

The transfer function for both series RC & RL circuit is same;

transfer function low pass filter

Time Constant:

Time constant plays an important role in defining the cutoff frequency of the ciruit.

  • τ = 1 / ωc                           ­For Both circuit
  • τ = L / R                      For RL circuit
  • τ = RC                        For RC circuit
High Pass Filter:

This type of filter allows high frequency component from its input signal. The circuit used for HPF is same as LPF but the output is taken across R & L in RC & RL circuit respectively.

Related Posts:

Cutoff Frequency:

Same as the Low pass filter.

Cutoff Frequency of High Pass Filter

Transfer Function:

Only transfer function is changed due to changing the output element.

transfer function high pass filter

Time Constant:

It will also remain same.

  • τ = 1 / ωc                           ­For Both circuit
  • τ = L / R                      For RL circuit
  • τ = RC                        For RC circuit
Band-Pass Filter:

it allows a fixed range of frequency & blocks every other frequency component before or after that allowable region.

Center Frequency:

the center of the allowable band of frequency fc is given by;

Center Frequency of band Pass Filter

Cutoff Frequency:

There are two cutoff frequency in band pass filters i.e. Lower cutoff ωc1 & upper cutoff ωc2 , any frequency before ωc1 and after ωc2 is being blocked by the filter.

Cutoff Frequency of band Pass Filter

Bandwidth:

The total range of the allowable frequency is known as bandwidth, from lower cutoff to upper cutoff frequency.

β = ωc2 – ωc1

  • β = R/L                       For Series RLC
  • β = 1/RC                    For Parallel RLC
Quality Factor:

Quality Factor of band pass filter

Transfer Function:

transfer function band pass filter

Band Reject Filter:

Band reject filter has the same circuit to a band pass filter, except the output is taken across both inductor L & Capacitor C. Thus only the transfer function changes.

Transfer Function:

transfer function band reject filter

Active Filters:

They allow specific frequencies with a gain which can be modified using the resistor network.

First Order Low Pass & High Pass Filter:

The first order filter contains only one reactive component.

Active Low Pass & High Pass Filter

Cutoff Frequency:

The cutoff frequency for both high pass & low pass active filter;

cutoff frequency active filter

Gain:

Total output voltage gain for this filter is given by;

K = R2 / R­1

Transfer Function:

The transfer function for both low pass & high pass active filter with the gain K is given by;

transfer function for both low pass & high pass active filter

Scaling:

Scaling allow us to use more realistic values of resistors, inductors & capacitors while keeping the quality of the filter. It can be used in passive as well as active filters. There are two types of scaling i.e. magnitude scaling & frequency scaling.

Magnitude Scaling

if you only want to scale the magnitude of the filter.

  • R’ = kR
  • L’ = km L
  • C’ = C / km

Frequency Scaling

When you only want to scale the frequency of the filter

  • R’ = R
  • L’ = L / kf
  • C’ = C / kf

Simultaneous Scaling

When you want to scale the both frequency & magnitude of the filter;

  • R’ = kR
  • L’ = (km/kf) L
  • C’ = (1/kmkf) C
  • R’ = scaled resistance
  • L’ = scaled inductance
  • C’ = scaled capacitance
  • km = Magnitude scaling factor
  • kf = frequency scaling factor

Electrical and Electronics Engineering Formulas & Equations

 

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One Comment

  1. Mulu hailu says:

    This is good for electrical knowledge.

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