Single phase and Three phase formula's used in electrical engineering

Electrical Formulas

Electrical engineering required certain formula which must be known for an electrical engineer and even a non engineer person so to know certain parameters of appliances around everyone.

Lets discuss 1^st about basic parameters used in electrical systems:-

I = Amperes

E = Volts

kW = Kilowatts

kVA = Kilo volt-Amperes

HP = Horsepower

% eff. = Percent Efficiency

pf = Power Factor

For Single-Phase load here are the formula’s as below:-

TO FIND:-

§  Amperes when kVA is known –>   I = kVA x 1000 / E

§  Amperes when horsepower is known –>  ( HP x 746) / ( E  x  % eff.  x pf )

§  Amperes when kilowatts are known –>  ( kW x 1000 ) / ( E x pf )

§  Kilowatts  –>  ( I x E x pf ) /1000

§  Kilovolt-Amperes  –>  ( I x E ) / 1000

§  Horsepower  –>  ( I x E x % eff. x pf  ) / 746

§  Watts   –>  E x I x pf

§  Energy Efficiency  –>  Load Horsepower x 746 / Load Input kVA x 1000

§  Power Factor  @ cos θ –>  Power Consumed /Apparent Power ( W / VA ) @ ( kW / kVA)

Two-Phase

TO FIND :-

§  Amperes when kVA is known –>   I = ( kVA x 1000 )  / ( E x 2 )

§  Amperes when horsepower is known –>  ( HP x 746) / ( E  x  2  x % eff.  x pf )

§  Amperes when kilowatts are known –>  ( kW x 1000 ) / ( E x 2 x pf )

§  Kilowatts  –>  ( I x E x 2 x pf ) /1000

§  Kilovolt-Amperes  –>  ( I x E x 2 ) / 1000

§  Horsepower  –>  ( I x E x 2 x % eff. x pf  ) / 746

§  Watts   –>  E x I x 2 x pf

§  Energy Efficiency –>  Load Horsepower x 746 / Load Input kVA x 1000

§  Power Factor @ cos θ –>  Power Consumed /Apparent Power ( W / VA ) @ ( kW / kVA)

For findingThree-Phase parameters formula’s are as below:-

TO FIND :-

§  Amperes when kVA is known –>   I = ( kVA x 1000 )  / ( E x 1.73 )

§  Amperes when horsepower is known –>  ( HP x 746) / ( E  x  1.73  x % eff.  x pf )

§  Amperes when kilowatts are known –>  ( kW x 1000 ) / ( E x 1.73 x pf )

§  Kilowatts  –>  ( I x E x 1.73 x pf ) /1000

§  Kilovolt-Amperes  –>  ( I x E x 1.73 ) / 1000

§  Horsepower  –>  ( I x E x 1.73 x % eff. x pf  ) / 746

§  Watts   –>  E x I x 1.73 x pf

§  Energy Efficiency  –>  Load Horsepower x 746 / Load Input kVA x 1000

§  Power Factor  @ cos θ –>  Power Consumed /Apparent Power ( W / VA ) @ ( kW / kVA)

Others Formula

§  kW = hp x .746

§  Torque in lb-ft = hp x 5250 / rpm

§  Motor synchronous speed in rpm = 120 x Hz / number of poles

§  Three-phase full-load amp= hp x .746 / 1.73 x kV x effi ciency x power factor

§  Rated motor kVA = hp (.746) / efficiency x power factor

§  kW loss = hp (.746) (1.0 – effi ciency) / efficiency

§  kVA in-rush = percent in-rush x rated kVA

§  Approximate voltage drop (%) = motor kVA in-rush x transformer impedance / transformer kVA

§  Stored kinetic energy in kW-sec = 2.31 x (total Wk2) x rpm2 x 107

§  Inertia constant (H) in seconds = stored kinetic energy in kW-seconds / hp (.746)

§  Conversion factors: CV = (metric hp) = 735.5 watts = 75 kg-m/sec Wk2 (lb-ft) = 5.93 x GD2 (kg-m2)

§  Ventilating-air requirements: 100-125 cfm of 400C air at 1/2-in. water pressure for each kW of loss

§  Degrees C = (Degrees F-32) x 5/9

§  Degrees F = [(Degrees C) x 9/5 ] + 32

Comparison between Three Phase Overhead and Underground cables

🔌 Power Transmission: Overhead vs Underground Cables

⚡ Key Electrical Aspect

·         Overhead Lines (OHTL): Inductance is predominant.

·         Underground Cables (UGC): Capacitance is predominant.

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✅ Advantages of Overhead Transmission Lines

1.      Lower Conductor Cost:

o    Overhead conductors run cooler in open air → smaller cross-section needed compared to underground.

o    Results in cheaper conductor material.

2.      Cheaper Insulation:

o    OHTL uses bare conductors, spaced by air (a natural insulator).

o    UGC requires costly insulation: paper tapes, metal sheath, oil/gas filling, storage vessels.

3.      Lower Installation Cost:

o    Poles/towers are cheaper to install than trenching and laying underground cables.

4.      Easy Capacity Addition:

o    OHTL can be upgraded (adding more circuits or reconductoring) faster and at lower cost.

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✅ Advantages of Underground Transmission Cables

1.      Public Safety:

o    No risk of electrocution from exposed conductors.

o    Less chance of accidental contact.

2.      No Interference & Better Aesthetics:

o    No electromagnetic interference to nearby communication lines.

o    Cityscape remains uncluttered → better appearance in urban areas.

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📊 Comparative Snapshot

Feature	Overhead Lines (OHTL)	Underground Cables (UGC)
Electrical Property	Inductance predominant	Capacitance predominant
Conductor Cost	Lower	Higher
Insulation Requirement	Minimal (air insulation)	Complex (paper, oil, sheath, vessels)
Installation Cost	Lower	Very high
Capacity Upgradation	Easy	Difficult
Public Safety	Exposed conductors, safety concern	Safer (no exposure)
Interference	May interfere with nearby comm. lines	No interference
Aesthetics	Visible structures, cluttered look	Hidden, better appearance

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Aspect	Overhead (OHTL)	Underground (UGC)
Initial Cost	Baseline—e.g., ~$285k/mile (69 kV)	Approximately 3× to 14×, typically ~5× or 4.5× higher
Reliability	Vulnerable to weather, vegetation, etc.	Highly reliable—e.g., 95% SAIDI improvement, 4% outage vs 24%
Maintenance & Repairs	Quick repairs, lower cost	Slower, more expensive maintenance
Lifetime Costs	Lower overall	Higher—12–28% more in distribution; 2–4× for transmission
Urban Use Case	Less preferred—visible infrastructure	Preferred—saves ₹500 cr/year in Hyderabad example
Real-World Costs	~$40 m/15 km UK, or AU$700M+	~$330 m/15 km UK; AU$2–4.5 b; DKK 30M/km in Denmark

 

Skin Effect Three phase lines; Factors effecting skin effect; Why skin effect not occur on DC?

Skin Effect in Transmission Lines: Meaning, Causes & Factors

What is Skin Effect?

Skin effect is a phenomenon that occurs in transmission lines carrying alternating current (AC), where the current is not uniformly distributed across the entire cross-section of the conductor. Instead, the current density is higher near the surface (or skin) of the conductor and much lower at the core.

In contrast, direct current (DC) flows uniformly throughout the cross-section of the conductor, hence skin effect does not occur in DC systems.

This uneven distribution of AC leads to an increase in effective resistance of the conductor compared to DC.

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Why Does Skin Effect Occur in AC?

The root cause of skin effect lies in the electromagnetic flux linkages created by alternating current:

• The inner filaments of the conductor produce flux that links with both inner and outer filaments.

• The outer filaments, however, only produce flux that links with themselves.

• This means flux linkage is higher for inner filaments, resulting in higher inductive reactance at the core compared to the surface.

• Since current always prefers a path of lower opposition (impedance), it shifts towards the surface of the conductor.

Thus, most of the AC flows along the outer periphery, giving rise to the skin effect.

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Mathematical Understanding (Simplified)

Imagine splitting a solid conductor into n annular filaments:

• If the total current is I, then each filament carries a small current i, where:

  $$I = n \cdot i$$

• The inner filaments experience flux from the whole cross-section → higher reactance.

• The outer filaments experience less flux linkage → lower reactance.

This imbalance in reactance distribution pushes current outward to the conductor surface.

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Factors Affecting Skin Effect in Transmission Lines

The intensity of skin effect in AC conductors depends on several parameters:

1. Frequency of Operation

• Skin effect increases with frequency.

• At 50 Hz (standard power frequency), it is noticeable but not dominant.

• At higher frequencies (radio, microwave), the effect is very significant.

2. Diameter of Conductor

• Larger diameters cause more pronounced skin effect.

• This is why hollow conductors are often used at high voltage AC lines.

3. Shape of Conductor

• Conductors with irregular or stranded shapes (like stranded conductors) reduce the skin effect compared to solid conductors.

4. Material of Conductor

• Conductors with higher permeability (like iron) exhibit more skin effect.

• Materials like copper and aluminum (low permeability) are preferred for transmission lines.

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Practical Implications of Skin Effect

• Increases effective resistance of conductors in AC systems.

• Increases power losses in transmission lines.

• Engineers often use stranded conductors (like ACSR – Aluminum Conductor Steel Reinforced) to minimize this issue.

• At very high frequencies, hollow tubes are preferred since current only flows on the surface anyway.

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FAQs on Skin Effect

Q1: Why does skin effect not occur in DC?

In DC, current is steady and does not produce alternating flux. Thus, the current distribution remains uniform across the conductor cross-section.

Q2: Does skin effect depend on voltage?

No, skin effect depends on frequency and conductor properties, not on the voltage level directly.

Q3: How can skin effect be minimized?

• Use stranded conductors (like ACSR).

• Use hollow conductors for high-frequency applications.

• Operate at lower frequencies where possible.

Q4: Why is skin effect important in power transmission?

Skin effect increases the effective resistance of conductors, which means higher I²R losses and reduced efficiency of power transmission.

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Final Thoughts

The skin effect in transmission lines is a critical phenomenon in electrical engineering, especially for AC power transmission. Understanding its causes, effects, and mitigation methods helps in designing efficient transmission systems and minimizing power losses.

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Corona Loss in Transmission Lines; Hissing sound in transmission lines

⚡ Corona Effect in Transmission Lines

🔹 Introduction

In high-voltage transmission lines, the surrounding air acts as a dielectric medium. When the electric field intensity around conductors exceeds a certain critical value, it ionizes the surrounding air molecules. This ionization leads to partial discharge of electricity, accompanied by bluish glow, hissing noise, and ozone production.

This phenomenon is known as the Corona Effect.

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🔹 Mechanism of Corona Formation

1. Free electrons are always present in air (from cosmic rays, UV radiation, radioactivity).

2. As voltage increases → electric field gradient at conductor surface increases.

3. Electrons accelerate → collide with neutral air molecules → release more electrons.

4. This leads to an electron avalanche → ionization of surrounding air.

5. Discharge appears as faint luminous glow + hissing sound.

👉 If conductor spacing is too small (spacing-to-radius ratio < 15), flashover occurs before corona starts.

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🔹 Visual & Audible Symptoms

• Visual glow: Bluish, violet, or faint luminous glow around conductors at night.

• Audible noise: Crackling or hissing, louder in rainy/stormy weather.

• Ozone smell: Due to O₂ dissociation forming O₃.

• Energy loss: Appears as Corona Loss.

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🔹 Corona Loss (Power Dissipation)

The corona loss (empirical Peek’s formula) is:

$$P_c = 241 \times 10^{-5} (f+25)\sqrt{\frac{r}{d}} \cdot \frac{(V_p - V_0)^2}{d} \; \text{(kW/km/phase)}$$

Where:

• $f$ = frequency (Hz)

• $r$ = conductor radius (cm)

• $d$ = spacing between conductors (cm)

• $V_p$ = phase-to-neutral RMS voltage (kV)

• $V_0$ = critical disruptive voltage (kV)

• $d$ = air density correction factor

$$V_0 = g \, r \, d \, \ln\left(\frac{D}{r}\right)$$

with $g ≈ 30 \, kV/cm$ (at NTP).

👉 Key takeaways:

• Corona loss ∝ $(V_p - V_0)^2$ → increases sharply above disruptive voltage.

• Larger conductor diameter reduces corona.

• Higher spacing also reduces corona effect (up to a limit).

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🔹 Factors Affecting Corona

1. Conductor Parameters

• Diameter: Larger → lower electric field → reduced corona.

• Surface condition: Polished/smooth → higher disruptive voltage; rough/dirty → more corona.

• Bundled conductors: Used in EHV lines (400 kV, 765 kV) to reduce corona.

2. Line Voltage

• Below 30 kV → negligible corona.

• 110 kV – 220 kV → corona starts appearing.

• 400 kV → severe corona → special design needed.

3. Spacing Between Conductors

• Small spacing → flashover risk.

• Very large spacing → weak field, corona less likely.

• Optimized spacing balances mechanical & electrical design.

4. Atmospheric Conditions

• Humidity & Rain: Increases corona (more free ions).

• Pressure: At higher altitudes (lower air density), corona appears at lower voltages.

• Temperature: Higher temperature reduces air density → easier ionization.

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🔹 Effects of Corona on Transmission Lines

✅ Advantages

• Acts as a safety valve: reduces steepness of lightning surges.

• Provides some protective cushioning for insulation.

❌ Disadvantages

• Power Losses: Continuous energy dissipation as corona loss.

• Noise Pollution: Hissing/crackling in EHV lines.

• Interference: Causes radio/television disturbances.

• Material Degradation: Ozone produced corrodes nearby insulators & conductors.

• Voltage Regulation Issues: Loss increases with voltage, affecting efficiency.

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🔹 Practical Engineering Solutions to Reduce Corona

1. Increase conductor diameter

   • Use hollow conductors or ACSR (Aluminium Conductor Steel Reinforced) for strength & larger radius.

2. Bundled Conductors

   • Common in 400 kV, 765 kV lines: splitting one phase into 2–4 sub-conductors → reduces field intensity.

3. Smooth conductor surface

   • Prevents dirt, dust, and corrosion which increase corona.

4. Optimized spacing

   • Proper phase-to-phase spacing minimizes losses without causing flashover.

5. Weather considerations

   • Lines in humid/polluted regions designed with extra margins for corona inception voltage.

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🔹 Real-World Example

• 400 kV Single Circuit Line (India):

• Uses quad-bundled conductors to minimize corona.

• If single conductor were used, corona loss could reach 50–100 kW/km/phase in wet conditions.

• With bundled conductors, loss reduces to <10 kW/km/phase.

Key Insights into Corona Phenomenon

1. Why the Violet Glow?

• The glow arises from ionized nitrogen molecules emitting photons as they return to a lower energy state—primarily in the blue-violet and ultraviolet spectra WikipediaEEP - Electrical Engineering Portal.

• It often becomes visible at higher voltages, especially under nighttime or low-light conditions wazipoint.com.

2. What Causes Corona?

• When the field gradient at the conductor surface exceeds ~30 kV/cm (at sea level), air ionization begins, leading to corona discharge Wikipedia+1.

• Sharp or irregular conductor surfaces intensify field stress, making corona more likely. This is why smooth, polished conductors or corona rings are often used to mitigate the effect Wikipedia.

3. Associated Effects

• Audible noise: The ionization process generates small pressure waves, heard as hissing or crackling, especially in wet weather EEP - Electrical Engineering Portal.

• Ozone and NOₓ formation: Ionization leads to dissociation of oxygen and nitrogen molecules, producing ozone and nitrogen oxides, which can be corrosive EEP - Electrical Engineering Portal.

• Electromagnetic interference (EMI): The corona-generated current pulses inject noise into nearby communication systems EEP - Electrical Engineering Portal.

4. Engineering Countermeasures

• Increase conductor diameter: Reduces electric field intensity.

• Use bundled conductors: Spread the field over a larger area.

• Apply corona rings or grading rings: These rings help smoothen the electric field gradient, lowering the chance of ionization Wikipedia.

• Maintain conductor surface: Clean and smooth surfaces resist corona initiation.

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🔹 Conclusion

The corona effect is an inevitable phenomenon in EHV (Extra High Voltage) transmission lines, but its impact on efficiency, noise, and interference must be carefully minimized through design. Engineering solutions such as increasing conductor size, using bundled conductors, maintaining proper spacing, and ensuring smooth surfaces are essential for reliable and economic power transmission.

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Induction Motors Torque Equation; Torque equation

Torque Equation of Induction Motor: Explained with Factors Affecting Speed-Torque Characteristics

Induction motors are widely used in industries due to their robustness, low cost, and simple construction. One of the most important aspects of analyzing an induction motor is its torque equation, which helps in understanding how the motor develops torque under different operating conditions.

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Torque Equation of an Induction Motor

The general torque equation of an induction motor is:

$$T = \frac{3}{\omega_s} \cdot \frac{V^2 \cdot \frac{R_2'}{s}}{(R_1 + \frac{R_2'}{s})^2 + (X_1 + X_2')^2}$$

Where:

• T = Torque developed

• V = Applied stator voltage per phase

• R1 = Stator resistance

• R2’ = Rotor resistance (referred to stator side)

• X1 = Stator reactance

• X2’ = Rotor reactance (referred to stator side)

• s = Slip

• ωs = Synchronous angular speed

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Key Relation: Torque is Proportional to Voltage Squared

From the above equation, we see:

$$T \propto V^2$$

This means the torque developed is directly proportional to the square of the applied voltage. Even a small reduction in supply voltage leads to a significant drop in torque, which is crucial for applications requiring high starting torque.

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Factors Affecting Speed-Torque Characteristics of Induction Motor

The speed-torque characteristics of an induction motor are influenced by several parameters:

(a) Applied Voltage

• Torque is proportional to $V^2$.

• Reduction in voltage decreases both starting torque and running torque.

• Motors may fail to start under reduced voltage conditions in heavy-load applications.

(b) Supply Frequency

• Increase in supply frequency reduces starting torque.

• Higher frequency increases synchronous speed ($N_s = 120f/P$), shifting the torque-speed curve.

• Motors designed for fixed frequency (50 Hz/60 Hz) must not be operated at significantly higher frequencies.

(c) Rotor Resistance ($R_2’$)

• The maximum torque (Tmax) is independent of rotor resistance.

• Increasing rotor resistance:

  • Improves starting torque.

  • Shifts the slip at which Tmax occurs to a higher value (lower motor speed).

• Useful in wound rotor induction motors for controlled starting.

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Practical Implications

• Industrial Applications: Motors with high starting torque (e.g., cranes, hoists) may use external rotor resistance for better performance.

• Energy Efficiency: Voltage drops in long transmission lines can drastically affect induction motor torque.

• Variable Frequency Drives (VFDs): Allow frequency and voltage control to optimize speed-torque characteristics.

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✅ Summary:

• Torque in induction motors is directly proportional to the square of voltage.

• Supply voltage, frequency, and rotor resistance significantly affect the speed-torque curve.

• Understanding these factors helps in motor selection, efficiency improvement, and troubleshooting in industrial applications.

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Slip Ring induction motors starting; Slip ring induction motor starter

Starting Of Slip Ring Induction Motors

Slip ring Induction motors had external resistance connected in line. These motors are usually started with full line voltage applied across its terminals. During starting of slip ring induction motors the value of starting current is adjusted or kept minimum, by increasing the resistance of the rotor circuit.  The external resistance is connected in star and kept at maximum during starting so to minimize starting current. By increasing the rotor resistance it will not only reduces the rotor current but the stator current too.

This means that whenever a resistance is added in rotor circuit that will leads to reduced starting current. Thus because of this, the starting torque is increased due to the improvement in power factor.

Usually resistance is added during starting and slowly made out of circuit when motor attains the speed and this resistance is disconnected by using a contactor in line.  Slip ring can be taken in line by using manually also. The 3-phase supply to the stator has a switching contactor along with over-load and no or low-voltage protective devices. There might be also an interlock provided to ensure the proper sequential operation of the control gear and starting devices.

Slip ring motors circuit diagram is as shown above.

Torque curve of Slip ring motors is shown above

As per torque formula

Torque is directly proportional to resistance.

So as the resistance is high in slip ring motors torque is also high during starting.

As these motors have considerably high starting torque with low starting current, these motors can be started on load. The external resistance is used only for the starting purpose, after which the motor gradually picks up the speed, the resistance gradually cut-off. These rings are isolated after the motor reaches its rated speed. The carbon brushes are lifted and the rings are short circuited thus making them very similar to squirrel cage motors.

Applications of Slip Ring Induction Motors

These motors are used where the load is intermittent and comes on very sharply for brief periods, such as a punching machine. A heavy flywheel is fitted in the drive, preferably between the work and any speed-reduction gears. The flywheel shares the load with the motor, thus enabling a motor of lower rating to be employed. For load sharing to take place automatically, the motor speed should drop considerably as the load increases and this is ensured by using a motor having a high full-load slip, say for example 10%.

Characteristics Of Slip Ring Induction Motor

As other induction motors consists of Stator and Rotor circuits slip ring motors also have same Stator circuit there is only difference in rotor circuit. Rotor circuit consists of external resistance in the circuit. The stator circuit is slip ring motors is rated as same in the squirrel cage motor, but the rotor is rated in frame voltage or short circuit current. The frame voltage is the open circuit voltage when the rotor is not rotating and gives the measure of turns ratio between the stator and rotor. The short circuit current is the current flowing when the motor is operating at full speed, with the slip rings shorted and the full load applied to the motor shaft.

Advantages of Slip-Ring Motors:

1. These are used where there are high Inertia loads  as these motors have excellent starting torque.

2. These motors have low starting current then other induction motors.

3.  It is easy to control the speed of the motor from 50% to 100% of the full speed,

Disadvantages of Slip Ring Motors

1. Higher brush and slip ring maintenance required,

2. As the brush wears out, it may lead to intermittent contact, and thus heavy sparking.

3. Also speed control of motor comes along with increased losses as heat comes across resistance.