# Electrical and Electronics Engineering Formulas and Equations

**List of All Electrical and Electronics ****Engineering**** Formulas**

Note: Click on the desired toggle box below to see related electrical and electronics engineering formulas and equation with details.

### 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**

- V
_{L}= √3 V_{PH}or V_{L}= √3 E_{PH}[Star Connection] - V
_{L}= V_{PH}[Delta Connection]

**Electrical Resistance Formulas**

**Electrical Resistance & Impedance Formulas in DC Circuits**

- R = V/I
- R = P/I
^{2} - R = V
^{2}/P

**Electrical Resistance & Impedance Formulas in AC Circuits**

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

- Z = √(R
^{2}+ X_{L}^{2})… In case of Inductive Load - Z = √(R
^{2}+ X_{C}^{2})… In case of Capacitive Load - Z = √(R
^{2}+ (X_{L}– X_{C})^{2}… In case of both inductive and capacitive Loads.

**Good to know:**

Where;

X_{L }= Inductive reactance

X_{L }= 2πfL…Where L = Inductance in Henry

And;

X_{C} = Capacitive reactance

X_{C} = 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**:

Where

- V
_{n}= Voltage across Resistor R_{n} - V
_{s}= Supplied voltage or total voltage across resistance network - R
_{n}= Resistance of resistor, where n = 1,2,3..

**Voltage Divider Rule For AC Circuit**:

Where

- V
_{n}= Voltage across Impedance Z_{n} - V
_{s}= Supplied voltage or total voltage across impedance network - Z
_{n}= 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

- I
_{n}= current through Resistor R_{n} - I
_{s}= Supplied current or total current through the resistance network - R
_{n}= Resistance of resistor, where n = 1,2,3..

**Current Divider Rule For AC Circuit**:

Where

- I
_{n}= Current through Impedance Z_{n} - I
_{s}= Supplied current or total current through impedance network - Z
_{n}= 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

### 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**.

**Φ _{E} = EA cosϴ**

Where

- Φ
_{E }= 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 = Φ _{E} /A**

Where:

- D = Density
- Φ
_{E}= 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.

**Φ _{b} = BA cosϴ**

Where

- Φ
_{b}= 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.

- Related Post: Basic Magnetic Terms definition with Formulas

### 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:**

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

R_{Eq} = R1 + R2 + R3 +…

**Resistance In Parallel:**

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

Where

R_{Eq} 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 in Series & Parallel Equations

**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

C_{Eq} = C1 + C2 + C3 +…

Where

C_{Eq} 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:**

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

L_{Eq} = L1 + L2 + L3 +…

**Inductance In Parallel:**

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

Where

L_{Eq} 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:

- k is the dielectric constant
- ε
_{d }is the permittivity of the dielectric - ε
_{0 }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

- X
_{C }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 = X _{C}/ESR**

Where

**X _{C }**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/X _{C}**

Where

- DF is the dissipation factor
- δ is the angle between capacitive reactance victor & negative axis.
- X
_{C }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 CV ^{2}**

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:

**P**_{av}** = CV ^{2} / 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

- V
_{C}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

**Formula & Equations For Inductor:**

**Inductance of Inductor:**

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

**Voltage Across 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.
- i
_{to}= 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 _{L }= ωL = 2πfL**

Where

- X
_{L }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 = X _{L}/ESR**

Where

- X
_{L }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/X _{L}**

Where

- DF is the dissipation factor
- δ is the angle between capacitive reactance victor & negative axis.
- X
_{C }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 Li ^{2}**

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:

**P**_{av}** = Li ^{2} / 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:

**During Discharging:**

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

Where

- I
_{C}is the current of the inductor - I
_{0}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:**

**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 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 = I
^{2}R - P = V
^{2}/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| = √(P^{2}+Q^{2})

*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.
**X**,_{L}**X**and_{C}**R**known as**Inductive reactance**,**capacitive 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.

**Cosϴ = P / V I****Cosϴ = kW / kVA****Cosϴ =****True Power/ Apparent Power**

**Real Power Of Single Phase & 3-Phase Current**

Where

- V
_{rms }& I_{rms}is the root mean square value of voltage & current respectively. - V
_{L-N }& I_{L-N }is the line-to-neutral voltage & current respectively. _{ }V_{L-L}& I_{L-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 Motors Formulas

**Shunt Motor:**

**Voltage Equation Of Shunt Motor:**

**V = E _{b }+ I_{a} R_{a}**

Where

- V is the terminal voltage
- E
_{b }is the induced back e.m.f - I
_{a }is the armature current - R
_{a }is the armature resistance

**The Shunt Field Current:**

**I _{sh }= V / R_{sh}**

Where

- I
_{sh }is the shunt field current - R
_{sh }is the shunt field resistance

**Induced Back EMF:**

The armature induced voltage E_{b }is proportional to the speed & it is given by:

**E _{b }= k_{f}Φω**

Where

- K
_{f }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.

**E _{b} = 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

- N
_{nl }= No load speed of the motor - N
_{fl }= Full load speed of the motor

**Input & Output Power:**

**P _{in }= VI_{a}**

**P _{out }= 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_{a }+ I_{a} R_{a} + I_{a}R_{se}

V = E_{a }+ I_{a}(R_{a}+R_{se})

Where

- E
_{a }is the armature induced voltage - I
_{a }is the armature current - R
_{a }is the armature resistance - R
_{se}is the series field resistance

**Armature Induced Voltage & Torque:**

The armature induced voltage E_{a }is proportional to the speed & armature current whereas the torque T_{a} of series motor is directly proportional to the square of armature current & it is given by:

**E _{a }= k_{f}ΦωI_{a}**

**T _{a} = k_{f }Φ I_{a}^{2}**

Where

- K
_{f }is a constant based on machine construction - Φ is the magnetic flux
- ω is the angular speed

**Speed Of Series Motor:**

**Input & Output Power**

The input power of a series motor is given by:

**P _{in }= VI_{a}**

The output power is given by

**P _{out }= ωT**

- Related Post: Voltage And Power Equations of a DC Motor

**Efficiency Of DC Motor:**

**Electrical Efficiency:**

η_{e }= Converted power in armature / Input electrical Power

**Mechanical Efficiency:**

η_{m }= Converted power in armature / output mechanical power

**Overall Efficiency:**

η = Output mechanical Power / Input electrical Power

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

Where

**P**is the useful output power_{out}**P**_{a}_{ }is the armature copper loss**P**is the field copper loss_{f}**P**is the constant losses that contains_{k}**core losses**&**mechanical losses**

### DC Generator Formulas

**Shunt Generator:**

**Terminal Voltage:**

**V = E _{a }– I_{a} R_{a}**

Where

- E
_{a }is the armature induced voltage - I
_{a }is the armature current - R
_{a }is the armature resistance

**Terminal Current:**

**I _{a} = I_{f }+ I_{L}**

where I_{f }Is the field current & I_{L }is the load current

**The Field Current:**

**I _{f }= V / R_{sh}**

Where

- I
_{f }is the field current - R
_{sh }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:

**E _{g }= 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 = k _{f}ΦI_{a}**

Where

- K
_{f }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:

**P _{g }= ωT = E_{a}I_{a}**

**P _{L} = VI_{L}**

Where I_{L }is the load current

**Series Generator:**

**Terminal Voltage:**

**V = E _{a }– (I_{a} R_{a }+ I_{a} R_{se})**

**V = E _{a }– I_{a}(R_{a }+ R_{se})**

Where

- E
_{a }is the armature induced voltage - I
_{a }is the armature current - R
_{a }is the armature resistance - R
_{se }is the series field resistance

The series field current is equal to the armature current;

**I _{a }= I_{se}**

**Armature Induced Voltage & Torque:**

The armature induced voltage E_{a }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:

**E _{a }= k_{f}ΦωI_{a}**

**T = k _{f }Φ I_{a}^{2}**

Where

- K
_{f }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:

**P _{g }= ωT = E_{a}I_{a}**

**P _{L} = VI_{L}**

Where I_{L }is the load current

**Input Power**:

**P _{in }= ωT**

Where

- ω is the angular speed of armature
- T is the torque applied

**Converted Power:**

**P _{con }= P_{in }– Stray losses – mechanical losses – core losses**

**P _{con }= E_{a}I_{a}**

Where

**E**is the induced voltage_{a}**I**is the armature current_{a}

**Output Power**

**P _{out }= P_{con }– Electrical losses (I^{2}R)**

**P _{out }= VI_{L}**

Where

- V is the terminal voltage
- I
_{L }is the load current

##### Efficiency Of DC Generator:

**Mechanical Efficiency:**

**Electrical Efficiency:**

**Overall Efficiency:**

Where

**P**is the useful output power_{out}**P**_{a}_{ }is the armature copper loss**P**is the field copper loss_{f}**P**is the constant losses that contains_{k}**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 (I^{2}R) 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 = P _{a} = I_{a}^{2 }R_{a}**

Where

- I
_{a }is the armature current - R
_{a}is the armature resistance

**Field Loss:**

**Field cu Losses = P _{f} = I_{f}^{2}R_{f}**

Where

- I
_{f }is the field current - R
_{f }is the field resistance

**For Shunt Field:**

Where

- I
_{sh }is the shunt field current - R
_{sh}is the shunt field resistance

**For Series Field:**

Where

- I
_{se }is the series field current - R
_{se}is the series field resistance

**Iron/Core Losses**

**Hysteresis Loss:**

Where

- η = hysteresis or Steinmetz’s constant
- B
_{max }= 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

- K
_{e }is the electrical constant of the core material - B
_{max }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:

Where

- N
_{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 = E _{b} + I_{a}(R_{a} + jX_{s})**

Where

- V = voltage applied
- E
_{b }= Back emf - I
_{a}= Armature current - R
_{a }= Armature resistance - X
_{s }= synchronous reactance

**Resultant Voltage:**

The difference between the voltage applied V & back emf is known as resultant voltage E_{R}

**E _{R} = V – E_{b}**

**E _{R} = I_{a}(R_{a} + jX_{s})**

**Internal Angle:**

It is the angle by which the armature current I_{a} lags behind the resultant voltage in armature E_{R}, and it is given by;

**Back EMF Generated:**

**E _{b }= K_{a}φ_{a}N_{s}**

Where

- K
_{a }= constant of the armature winding - φ
_{a }= magnetic Flux per pole of the rotor - N
_{s }= synchronous speed of the rotor

**Different Excitations:**

**E**_{b }= V*Normal Excitation*Lagging Power Factor**E**_{b}< V*Under-Excitation*Lagging Power Factor**E**_{b}> V*Over- Excitation*Leading Power Factor

**Input Power:**

The input power of synchronous motor is given by:

Where

Φ is the angle between V & I_{a}

**Mechanical Power In Rotor:**

Where

**α**is the load angle between E_{b }& V- Φ is the angle between V & I
_{a} **T**is gross torque produced_{g }**N**is the synchronous speed_{s }

### Synchronous Generator Formulas

**Synchronous Generator:**

- Related Post: EMF Equation of an Alternator and AC Generator

**Output Electrical Frequency:**

Where

- f
_{e }= Electrical frequency - N
_{r}= speed of rotor in RPM - P = Number of poles

**Voltage Generated:**

**E _{a }= Kφ_{a}N_{s}**

Where

- K = constant representing the construction of machine
- φ
_{a }= magnetic Flux per pole of the rotor - N
_{s }= synchronous speed of the rotor

**Total Phase Voltage:**

**V _{φ} = E_{a} – jX_{s}I_{a} – R_{a}I_{a}**

Where

- X
_{s}= Synchronous reactance of machine - I
_{a}= Armature current - R
_{a}= Armature resistance

**Three Phase Terminal Voltage:**

##### Power Of Synchronous Generator:

Where

- T
_{app}= Torque applied - T
_{ind}= Torque induced in rotor - ω
_{r}= mechanical speed of rotor

**Voltage Regulation:**

Where

- V
_{nl }= Voltage at no load - V
_{fl}= Voltage at full load

**Efficiency:**

**η = (P _{out} / P_{in}) * 100%**

**P _{in} = P_{out} + P_{Cu} + P_{iron} + P_{mech }+ P_{stray}**

### Induction Motors Formulas

#### Induction Motor:

**Induced EMF:**

**e _{ind }= vBl**

where

- e
_{ind }= 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:**

Terms used in Motor Torque Equations and formulas.

**N**= Synchronous speed_{s}**s**= slip of the motor**s**= breakdown or pull-out slip_{b }**E**= stator voltage or input voltage_{1}**E**= Rotor EMF per phase at a standstill_{2}**R**= Rotor Resistance Per Phase_{2}**X**= Rotor Reactance Per Phase_{2}**V**= supply voltage**K**= rotor/stator turn ration per Phase

**Starting Torque**

**Maximum Starting Torque Condition**

R_{2 }= X_{2}

**Starting Torque Relation With Supply Voltage**

T_{st } α V^{2}

**Torque In Running Condition**

**Gross Torque**

**Maximum Running Torque Condition**

R_{2 }= sX_{2}

**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;

**N**(Speed in RPM)_{slip }= N_{s }– N**ω**(Angular speed in Rad/sec)_{slip }= ω_{s }– ω

Where

- N
_{slip }= slip speed - N
_{s }= 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)N**(Speed in RPM)_{s }**ω = (1-s) ω**(Angular speed in Rad/sec)_{ s}

**Electrical Frequency On The Rotor: **

Where

- f
_{r }= Rotor Frequency - f = Line Frequency
- P = Number of Poles

**Power Of Induction Motor:**

Related terms used in Motor Power Formulas and Equation.

- P
_{1}= Stator input Power - P
_{2}= Rotor Input power - P
_{m}= Rotor Gross Output Power - P
_{out}= Output Power - T
_{g}= gross torque - T
_{sh}= shaft torque

**Rotor Input Power:**

P_{2} = T_{g}ω_{s}

**Rotor Gross Output Power:**

P_{m} = T_{g}ω

**Output Power:**

P_{out} = T_{sh}ω

P1 = P2 + stator Losses = P_{m }+ Rotor Copper Losses = P_{out} + Windage & friction Losses

_{}

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

Where

P_{cr} = I^{2}R = 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 Formulas

### Stepper Motors Formulas

### Stepper Motor:

**Step Angle:**

Where

- β = step angle, the angle of rotation of the shaft with each pulse.
- N
_{s}= number of stator poles or teeth - N
_{r}= 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 Formulas & Equations

### Transformer:

**EMF Induced In Primary & Secondary Windings**:

Where

- E
_{1 }= EMF induced in primary winding - E
_{2}= EMF induced in Secondary winding - N
_{1}= Number of Turns in Primary winding - N
_{2}= Number of Turns in Secondary winding - f = Line frequency
- φ
_{m }= Maximum Flux in Core - B
_{m }= Maximum flux density - A = Area of Core

Related Post: EMF Equation Of a Transformer

**Voltage Transformation Ratio:**

Where

- K = voltage transformation ratio of transformer
- V
_{1}I_{1}= Primary voltage & current Respectively - V
_{2}I_{2}= Secondary voltage & current Respectively

**Equivalent Resistance Of Transformer Windings**:** **

Where

**R**= Resistance of Primary winding in Secondary_{1}^{’}**R**= Resistance of Secondary winding in primary_{2}^{’}**R**= Equivalent resistance of transformer from primary side_{01 }**R**= Equivalent resistance of transformer from Secondary side_{02 }**R**= Primary winding Resistance_{1}**R**= Secondary Winding Resistance_{2}

**Leakage Reactance:**

Where

- X
_{1}= Primary leakage Reactance - X
_{2}= Secondary leakage Reactance - e
_{L1}= Self-Induced EMF in primary - e
_{L2}= Self-Induced EMF in Secondary

**Equivalent Reactance Of Transformer Windings**:** **

Where

**X**= Reactance of Primary winding in Secondary_{1}^{’}**X**= Reactance of Secondary winding in primary_{2}^{’}**X**= Equivalent reactance of transformer from primary side_{01 }**X**= Equivalent reactance of transformer from Secondary side_{02 }

**Total Impedance Of Transformer Winding:**

Where

**Z**= Impedance of primary winding_{1}**Z**= Impedance of Secondary winding_{2}**Z**= Equivalent Impedance of transformer from primary side_{01 }**Z**= Equivalent Impedance of transformer from Secondary side_{02 }

#### Input & Output Voltage Equations** **

**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

- W
_{h}= Hysteresis loss - W
_{e}= Eddy current loss - η = Steinmetz Hysteresis coefficient
- K
_{e}= Eddy current constant - B
_{max }= 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

_{0}V_{2 }= No load Secondary voltage- V
_{2}= Full load Secondary voltage - V
_{1}= No load Primary voltage - V
_{2}^{’ }= V_{2}/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 **V _{1}** to

**V**, where the voltage regulation is given by:

_{1}^{’}**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**

**W _{cu} = W_{i}**

Where

- W
_{i}= W_{h }+ W_{e} - W
_{cu }= I_{1}^{2}R_{01}= I_{2}^{2}R_{02}

**Load Current For Maximum Efficiency:**

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

### 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;

Where

- X
_{L}= Inductive reactance - X
_{C}= Capacitive reactance

**Power Factor:**

The power factor for this circuit is

Cos ϴ = Z/R

**Resonance Frequency:**

When inductive reactance X_{L} & capacitive reactance X_{c} 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.

**Bandwidth:**

B.W = f_{r} / Q

**Resonant Circuit Current:**

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

At resonance, the X_{L }= X_{c }, so Z = R

I_{T} = 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

I_{mag} = Q I_{T}

**Characteristic Equation:**

**Neper Frequency For Parallel RLC Circuit:**

**Resonant Radian Frequency For Parallal RLC Circuit:**

**Voltage Response:**

**Over-Damped Response**

When

ω_{0}^{2 }< α^{2}

The roots s_{1} & s_{2} are real & distinct

**Under-Damped Response**

When

ω_{0}^{2 }> α^{2}

The roots s_{1} & s_{2} are complex & conjugate of each other

**Critically Damped Response**

When

ω_{0}^{2 }= α^{2}

The roots s_{1} & s_{2} 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 X_{L} = Capacitive reactance X_{c} is known as resonance frequency.

Where

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

**Quality Factor:**

**Bandwidth:**

B.W = (f_{r} / Q)

- B.W = (R / L) in rad/s
- B.W = (R / 2πL) in hz

**Lower Cutoff Frequency & Upper Cutoff Frequency:**

f_{h} = f_{r} + ½ B.W

f_{l} = f_{r} – ½ B.W

##### Characteristic** Equation:**

**Neper Frequency For series RLC Circuit:**

**Resonant Radian Frequency For series RLC Circuit:**

**Voltage Response:**

**Over-Damped Response**

When

ω_{0}^{2 }< α^{2}

The roots s_{1} & s_{2} are real & distinct

**Under-Damped Response**

When

ω_{0}^{2 }> α^{2}

The roots s_{1} & s_{2} are complex & conjugate of each other

**Critically Damped Response**

When

ω_{0}^{2 }= α^{2}

The roots s_{1} & s_{2} are real & equal

### Diode Formula & Equations

**Diode:**

**Schockley Diode Equation:**

Where

- I
_{D }= current through the diode - V
_{D }= diode voltage - I
_{s}= 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
_{T }= 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 10^{23 }J/K

**Diode Rectifier:**

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

**Output DC Power:**

**P _{dc }= V_{dc} I_{dc}**

Where

- V
_{dc }is the average output voltage - I
_{dc }is the average output current

**Output AC Power:**

**P _{ac }= V_{rms} I_{rms}**

Where

- V
_{rms }Is the rms of output voltage - I
_{rms }is the rms of output current

**Rectifier Efficiency:**

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

Where

- P
_{dc }is the output DC power - P
_{ac }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 (BJT) Formulas

### BiPolar Junction Transistor:

**Current Gains in BJT:**

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

Where

- I
_{E }is the emitter current - I
_{C }is the collector current - I
_{B }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 A_{V }is the ratio of output voltage to input voltage:

Where

- α = I
_{C}/ I_{E} - R
_{L}is the load resistance - R
_{in }is the input resistance

**Common Emitter Configuration:**

**Forward Current Gain:**

It is the ratio of output current i.e. the collector current I_{C }to the input current i.e. the base current I_{B.}

**β _{F} = h_{FE} = I_{C}/I_{B}**

Where

- Β
_{F}is the forward current gain - I
_{C}is the collector current - I
_{B }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.

**I**_{E }= I_{C}+ I_{B}**I**_{E }= I_{C}/**α****I**_{E }= I_{B }(1+ β)

**Collector Current:**

The collector current for BJT is given by:

**I**_{C }= β_{F}I_{B }+ I_{CEO }≈ β_{F}I_{B}**I**_{C }=**α I**_{E}**I**_{C }= I_{E}– I_{B}

Where

I_{CEO } 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:**

**V _{CE }= V_{CB }+ V_{BE}**

Where

- V
_{CE }is the collector-to-Emitter voltage - V
_{CB }is the collector-to-base voltage - V
_{BE }is the base-to-emitter voltage

**Common Collector Configuration:**

**Current Gain:**

The current gain A_{i} of common collector BJT is given by the ratio of output current I_{E }to input Current I_{B}:

**A**_{i }= I_{E}/ I_{B}**A**_{i }= (I_{C }+ I_{B}) / I_{B}**A**_{i}= (I_{C }/ I_{B}) + 1**A**_{i }= β + 1

### Operational Amplifier (OP-AMP) Formulas

**Operational Amplifiers:**

**Inverting Amplifier:**

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

- R
_{f}= Feedback resistor - R
_{in }= Input Resistor - V
_{in }= Input voltage - V
_{out}= Output voltage - A
_{v}= 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 R_{1} = R_{2 }= R_{3} = R_{n }= R

**Summed Output:**

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

If R_{f} = R_{1} = R_{2 }= R_{3} = R_{n }= R;

V_{out} = – (V_{1 }+ V_{2} + V_{3} +… + V_{n})

**Non-Inverting Amplifier:**

Terms used for Non-Inverting Amplifier formulas and equations.

- R
_{f}= Feedback resistor - R = Ground Resistor
- V
_{in }= Input voltage - V
_{out}= Output voltage - A
_{v}= Voltage Gain

Related Post: Negative Feedback Amplifier Systems

**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. R_{f} = 0, the non-inverting amplifier will become voltage follower/buffer.

**Differential Amplifier:**

Terms used for Differential Amplifier formulas.

- R
_{f}= Feedback resistor - R
_{a }= Inverting Input Resistor - R
_{b }= Non Inverting Input Resistor - R
_{g}= Non Inverting ground Resistor - V
_{a }= Inverting Input voltage - V
_{b }= Non Inverting Input voltage - V
_{out}= Output voltage - A
_{v}= Voltage Gain

**General Output:**

the output voltage of the above given circuits is;

**Scaled Differential Output:**

If the resistor R_{f} = R_{g }& R_{a }= R_{b }, 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. R_{a} = R_{b }= R_{f }= R_{g }= R, the amplifier will provide output that is the difference of input voltages;

V_{out }= V_{b }– V_{a}

**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 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;

**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 / ω
_{c }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.

**Transfer Function:**

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

**Time Constant:**

It will also remain same.

- τ = 1 / ω
_{c }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 f_{c} is given by;

**Cutoff Frequency:**

There are two cutoff frequency in band pass filters i.e. Lower cutoff ω_{c1 }& upper cutoff ω_{c2 }, any frequency before ω_{c1 }and after ω_{c2 }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 = ω_{c2} – ω_{c1}

- β = R/L For Series RLC
- β = 1/RC For Parallel RLC

**Quality Factor:**

**Transfer Function:**

**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:**

**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 = R_{2} / R_{1}

**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’ = k
_{m }R - L’ = k
_{m}L - C’ = C / k
_{m}

**Frequency Scaling**

When you only want to scale the frequency of the filter

- R’ = R
- L’ = L / k
_{f} - C’ = C / k
_{f}

**Simultaneous Scaling**

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

- R’ = k
_{m }R - L’ = (k
_{m}/k_{f}) L - C’ = (1/k
_{m}k_{f}) C - R’ = scaled resistance
- L’ = scaled inductance
- C’ = scaled capacitance
- k
_{m}= Magnitude scaling factor - k
_{f}= frequency scaling factor

Related Post: Electrical / Electronics Engineering Formulas

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