Electrical and Electronics Engineering Formulas – 5000+ Formulas

5000+ Electrical and Electronics Engineering Formulas You Must Know

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;

XL = Inductive reactance

XL = 2πfL…Where L = Inductance in Henry

And;

XC = Capacitive reactance

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

Electrical Power Formulas

Electrical Power formulas in DC Circuits

• P = V x I
• P = I2R
• P = V2/R

Electrical Power formulas in Single Phase AC Circuits

• P = V x I Cosθ
• P = I2 x R Cosθ
• P = (V2/R) Cosθ

Electrical Power Formulas in Three Phase AC Circuits

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:

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:

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:

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:

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.

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.

You may find more about KVL & KCL Here.

Coulomb’s Law

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

Where

• 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:

Force per unit charge is known as electric field intensity.

E = F/Q

Electric Flux:

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

ΦE = EA cosϴ

Where

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

It’s a vector quantity.

Electric Flux Density:

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

D = ΦE /A

It is a scalar quantity.

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:

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

B = Φ/A

It is a scalar quantity.

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:

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:

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

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.

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

From Wye (Y) to Delta (Δ) Interconnection

Capacitance:

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

Capacitance In Series:

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

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:

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

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:

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

Where

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

Equations For Capacitors:

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:

Where

• k is the dielectric constant
• εd is 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:

Where

• XC is 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

XC is 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.
• XC is 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:

During Discharging:

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

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:

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:

Inductance Of Inductor:

The inductance of the inductor from the basic formula of 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:

XL = ωL = 2πfL

Where

• XL is 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

• XL is 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.
• XC is 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 upto 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:

During Discharging:

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

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 τ “Tau” Formulas:

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:

For 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:

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

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

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.

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

• 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|.

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.

Where

|S| = √(P2+Q2)

Real Power Of Single Phase & 3-Phase Current

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:

Where

ϴ is the phase angle

DC Machines:

DC MOTOR:

Shunt Motor:
Voltage Equation Of Shunt Motor:

V = Eb + Ia Ra

Where

• V is the terminal voltage
• Eb is the induced back e.m.f
• Ia is the armature current
• R­­­a is 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 Eb is proportional to the speed & it is given by:

Eb = kfΦω

Where

• Kf is 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:

And

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.

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 = Ea + Ia Ra + IaRse

V = Ea + Ia(Ra+Rse)

Where

• a is the armature induced voltage
• Ia is the armature current
• R­­­a is the armature resistance
• Rse is the series field resistance
Armature Induced Voltage & Torque:

The armature induced voltage Ea is 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:

Ea = kfΦωIa

Ta = kf Φ Ia2

Where

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

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:

ηe =  Converted power in armature / Input electrical Power

Mechanical Efficiency:

ηm = Converted power in armature / output mechanical power

Overall Efficiency:

η = Output mechanical Power / Input electrical Power

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

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

Generator:

Shunt Generator:
Terminal Voltage:

V = Ea – Ia Ra

Where

• a is the armature induced voltage
• Ia is the armature current
• R­­­a is the armature resistance

Ia = If ­+ IL

­where If Is the field current & IL is the load current

The Field Current:

f ­= V / Rsh

Where

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

The EMF generated per conductor in a DC generator is:

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;

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

Eg = 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

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

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

Power Generated & Load Power

The power generated by a shunt generator is given by:

Pg = ωT = EaIa

PL = VIL

Where IL is the load current

Series Generator:
Terminal Voltage:

V = Ea – (Ia Ra + Ia Rse)

V = Ea – Ia(Ra + Rse)

Where

• a is the armature induced voltage
• Ia is the armature current
• R­­­a is the armature resistance
• Rse ­is the series field resistance

The series field current is equal to the armature current;

Ia = Ise

Armature Induced Voltage & Torque:

The armature induced voltage Ea is 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:

Ea = kfΦωIa

T = kf Φ Ia2

Where

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

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

Power Generated & Load Power

The power generated by a series generator is given by:

Pg = ωT = EaIa

PL = VIL

Where IL is 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
• IL is the load current
Efficiency Of DC Generator:

Overall Efficiency:

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

Cupper loss = Core & mechanical loss

Cupper 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 a Machine:

Copper Losses:
Armature Loss:

Armature Cu Losses = Pa = Ia2 Ra

Where

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

Field cu Losses = Pf = If2Rf

Where

• If is the field current
• Rf is the field resistance

For Shunt Field:

Where

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

For Series Field:

Where

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

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

Where

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

Where

• e 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

AC Machines:

Synchronous Machine:

Speed Of Synchronous Machine:

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

Where

• s 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
• Eb = Back emf
• Ia = Armature current
• Ra = Armature resistance
• Xs = 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;

Back EMF Generated:

Eb = KaφaNs

Where

• Ka = constant of the armature winding
• φa = magnetic Flux per pole of the rotor
• Ns = synchronous speed of the rotor
Different Excitations:

Eb = 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:

Where

Φ is the angle between V & Ia

Mechanical Power In Rotor:

Where

• α is the load angle between E­b & V
• Φ is the angle between V & Ia
• Tg is gross torque produced
• Ns is the synchronous speed

Synchronous Generator:

Output Electrical Frequency:

Where

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

Ea = KφaNs

Where

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

Vφ = Ea – jXsIa – RaIa

Where

• Xs = Synchronous reactance of machine
• Ia = Armature current
• Ra = Armature resistance

Power Of Synchronous Generator:

Where

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

Where

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

η = (Pout / Pin) * 100%

Pin = Pout + PCu + Piron + P­mech ­+ Pstray

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:

Torque Induced:
• Ns = Synchronous speed
• s = slip of the motor
• sb = 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

• Maximum Starting Torque Condition

R2 = X2

• Starting Torque Relation With Supply Voltage

Tst  α  V2

• Torque In Running Condition

• Gross Torque

• Maximum Running Torque Condition

R2 = sX2

• Maximum Running Torque

• Breakdown Slip

• Torque Relation With Max Torque

Slip Speed & Slip of Induction Motor:

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

Nslip = N­s – N­                         (Speed in RPM)

ωslip = ωs – 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:

Where

S is the slip of induction motor

Rotor Speed:

The rotor speed of induction motor is given by

N = (1-s)Ns                (Speed in RPM)

ω = (1-s) ω s             (Angular speed in Rad/sec)

Electrical Frequency On The Rotor:

Where

fr  = Rotor Frequency

f  = Line Frequency

P = Number of Poles

Power Of Induction Motor:

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 = Pm + Rotor Copper Losses = Pout + Windage & friction Losses

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

­Where

Pcr = I2R = rotor Copper loss

Synchronous Watt:

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

Efficiency Of Induction Motor:

• Rotor Efficiency:

• Overall Efficiency

Linear Induction Motor:

Synchronous Speed:

Where

vs  = linear synchronous speed

w = width of one pole-pitch

f = line frequency

Slip:

Where

vs = linear synchronous speed

v = Actual speed

Thrust Or Force:

Where

P2 = Rotor input Power

Stepper Motor:

Step Angle:

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;

Higher the resolution, higher the accuracy of stepper motor.

Motor Speed:

Where

n = motor speed in revolution per second

f = stepping pulse frequency

Transformer:

EMF Induced In Primary & Secondary Windings:

Where

E1 = 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

φm = Maximum Flux in Core

Bm = Maximum flux density

A = Area of Core

Voltage Transformation Ratio:

Where

K = voltage transformation ratio of transformer

V1I1 = Primary voltage & current Respectively

V2I2 = Secondary voltage & current Respectively

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:

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:

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:

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

Losses In Transformer:

Core / Iron Losses

The losses that occur inside the core;

• Hysteresis Loss

Due to magnetization and demagnetization of the core

• Eddy Current Loss

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

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

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

0V2 = 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 Down

Regulation “Down” is commonly referred as regulation

• 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

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 Reactance 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.

Total losses = Cu loss + Iron Loss

Efficiency At Any Load:

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

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.

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 = W+ We

Wcu = I12 R01 = I22 R02

Load Current For Maximum Efficiency:

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

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;

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.

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.

B.W = fr / Q

Resonant Circuit Current:

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

At resonance, the XL = Xc , 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

Voltage Response:
• Over-Damped Response

When

ω02 < α2

The roots s1 & s2 are real & distinct

• Under-Damped Response

When

ω02 > α2

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

• Critically Damped Response

When

ω02 = α2

The roots s1 & s2 are real & equal

Series RLC Circuit:

Impedance:

The total impedance of the series RLC circuit is;

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.

where

• L = Inductance of the inductor
• C = Capacitance of the capacitor

Bandwidth:

B.W = (fr / Q)

B.W = (R / L)              in rad/s

B.W = (R / 2πL)         in hz

fh = fr + ½ B.W

fl = fr – ½ B.W

Voltage Response:
• Over-Damped Response

When

ω02 < α2

The roots s1 & s2 are real & distinct

• Under-Damped Response

When

ω02 > α2

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

• Critically Damped Response

When

ω02 = α2

The roots s1 & s2 are real & equal

Diode:

Schockley Diode Equation:

Where

• ID = current through the diode
• VD = 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.
• VT = thermal voltage which is

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:

­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:

Form Factor:

The ratio of RMS voltage to the average dc voltage,

Ripple Factor:

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

BiPolar Junction Transistor:

Current Gains in BJT:

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

Where

• IE is the emitter current
• IC is the collector current
• Iis the base current

Common Base Configuration:

Common Base Voltage Gain

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

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 I­C to 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
• IB is the base current
Emitter Current:

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

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

The collector current for BJT is given by:

• C = βFIB + ICEO ≈ βFIB
• C = α IE
• IC = 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 IE to input Current IB:

• Ai = IE / IB
• Ai = (IC + IB) / IB
• Ai = (IC / IB) + 1
• Ai = β + 1

Operational Amplifiers:

Inverting Amplifier:

• 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;

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:

Output Voltage:

The general output of this given circuit above is;

Inverted Amplified Sum of Input Voltage:

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

If R1 = R2 = R3 = Rn = R

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 = R2 = R3 = Rn = R;

Vout = – (V1 + V2 + V3 +… + Vn)

Non-Inverting Amplifier:

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

The total gain of non-inverting amplifier is;

Output Voltage:

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

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.

Differential Amplifier:

• Rf = Feedback resistor
• Ra = Inverting Input Resistor
• Rb = 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;

Scaled Differential Output:

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

Unity Gain Difference:

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

Vout = Vb – Va

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;

Triangular wave input => Rectangular wave output

Sine wave input => Cosine wave output

Integrator Amplifier

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

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

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;

Transfer Function:

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

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.

Cutoff Frequency:

Same as the Low pass filter.

Transfer Function:

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

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;

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.

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

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.

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.

Cutoff Frequency:

The cutoff frequency for both high pass & low pass 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;

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’ = km R
• 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’ = km R
• 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

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One Comment

1. Mulu hailu says:

This is good for electrical knowledge.

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