Unit 2: Fundamentals of A.C. circuits - Subjective Questions

Alternating Current (AC): An alternating current is a current whose magnitude changes continuously with time and whose direction reverses periodically. A direct current (DC) flows only in one direction with a constant or non-reversing magnitude.

Alternating Voltage: An alternating voltage is a voltage whose magnitude varies with time and whose polarity reverses periodically.

Mathematical Representation:

where:

• = maximum (peak) values
• = angular frequency in rad/s
• = frequency in Hz

Differences from DC:

• Direction: AC reverses direction periodically; DC flows in one direction only.
• Magnitude: AC magnitude varies sinusoidally; DC magnitude is constant.
• Representation: AC is represented by phasors; DC by a single value.
• Transmission: AC can easily be stepped up/down using transformers; DC cannot (without electronics).

The sinusoidal waveform completes one full cycle (positive half + negative half) in a time period .

In AC circuit analysis, different notations are used to distinguish between instantaneous, maximum, and effective (RMS) values of current and voltage.

Lowercase letters (, ):

• Represent instantaneous values of current and voltage.
• These are functions of time: and .
• Example: gives the value of current at any instant .

Uppercase letters (, ):

• Represent constant (RMS or effective) values of current and voltage.
• = RMS value of current, = RMS value of voltage.
• These are the values indicated by AC meters.

Summary Table:

Key point: The relationship between RMS and peak values is:

Proper use of notations avoids confusion in circuit analysis.

(a) Amplitude:

• The amplitude of an AC signal is its maximum value (peak value) attained during a cycle.
• For , the amplitude is .
• It is always a positive quantity measured from the zero (mean) axis to the peak.

(b) Phase:

• Phase is the fraction of a cycle that has elapsed with reference to a chosen origin (starting point).
• It is measured as an angle in radians or degrees.
• For , the term is the phase angle indicating the position of the waveform relative to the reference.

(c) Phase Difference:

• When two AC quantities of the same frequency do not reach their maximum or zero values at the same instant, they are said to have a phase difference.
• Example:
  
  
• Here the current lags the voltage by angle .

Terminology:

• Leading: A quantity reaching its peak earlier.
• Lagging: A quantity reaching its peak later.
• In phase: Phase difference = .

Phase difference is crucial in determining power factor and circuit behaviour.

Definition: The RMS (Root Mean Square) value of an AC current is that steady DC current which, when flowing through a given resistance for a given time, produces the same amount of heat as produced by the AC through the same resistance in the same time.

Derivation:

Let the alternating current be:

The RMS value is obtained by:

1. Squaring the instantaneous values
2. Taking the mean over one cycle
3. Taking the square root

Substituting :

Using the identity :

Thus, the RMS value of a sinusoidal current is 0.707 times its peak value.

Definition: The average value of an alternating quantity is the arithmetic mean of all instantaneous values over a given period.

Average over a full cycle:
For a symmetrical sinusoidal wave, the positive half is equal and opposite to the negative half. Hence their algebraic sum = 0, giving:

Therefore, the average is calculated over half a cycle.

Derivation (Half Cycle):

Let .

Thus the average value over half a cycle is 0.637 times the peak value.

Form Factor:

• It is the ratio of the RMS value to the average value of an alternating quantity.
  

Peak Factor (Crest Factor):

• It is the ratio of the maximum (peak) value to the RMS value.
  

For a Sinusoidal Waveform:

We know:

• RMS value
• Average value

Form Factor:

Peak Factor:

Significance:

• The form factor indicates the shape of the waveform.
• The peak factor is important for designing insulation (it reflects the stress on dielectric materials).

Impedance ():

• Impedance is the total opposition offered by an AC circuit to the flow of alternating current. It combines resistance and reactance.
• In complex form:
  
  where:
• = resistance (real part)
• = reactance (imaginary part)

Magnitude and Angle:

Reactance ():

• Inductive reactance: (positive, +j)
• Capacitive reactance: (negative, -j)

So:

Admittance ():

• It is the reciprocal of impedance (ease with which current flows).
  
  where:
• = conductance
• = susceptance

Significance:

• Complex representation simplifies AC circuit analysis by treating phasors as vectors.
• A positive angle indicates an inductive (lagging) circuit, and a negative angle indicates a capacitive (leading) circuit.

Series RL Circuit:
A resistor and inductor are connected in series across an AC supply .

Voltage Equations:

• Voltage across resistor: (in phase with current)
• Voltage across inductor: (leads current by )

Phasor Addition:
Since and are apart:

Impedance:

In complex form:

Current:

Phase Angle:

The current lags the voltage by angle .

Phasor Diagram:

• Take current as reference (horizontal).
• is along .
• leads by (vertical).
• The resultant makes angle with .

Power Factor:

The RL circuit has a lagging power factor.

Series RC Circuit:
A resistor and capacitor are connected in series across an AC supply .

Voltage Equations:

• Voltage across resistor: (in phase with current)
• Voltage across capacitor: (lags current by )

Phasor Addition:

Impedance:

In complex form:

Current:

Phase Angle:

The current leads the voltage by angle .

Phasor Diagram:

• Take current as reference.
• is along .
• lags by (downward).
• The resultant lies below the reference axis.

Power Factor:

The RC circuit has a leading power factor.

Series RLC Circuit:
Resistor , inductor , and capacitor are connected in series across an AC supply .

Voltage Across Elements:

• (in phase with )
• (leads by )
• (lags by )

Net Reactance:
Since and are opposite in phase:

Impedance:

In complex form:

Current:

Phase Angle:

Three Possible Conditions:

• Case 1: → Circuit is inductive. Current lags voltage; power factor is lagging.
• Case 2: → Circuit is capacitive. Current leads voltage; power factor is leading.
• Case 3: → Circuit is purely resistive (resonance). (minimum), current is maximum, power factor = unity.

Series Resonance:
Resonance in a series RLC circuit occurs when the inductive reactance equals the capacitive reactance, so that they cancel each other and the circuit behaves purely resistive.

Condition for Resonance:

Derivation of Resonant Frequency:

Since :

Characteristics at Resonance:

• Impedance is minimum and equal to : .
• Current is maximum: .
• Power factor is unity ().
• Voltage across and are equal and opposite (cancel out).
• The circuit is called an acceptor circuit (accepts maximum current at resonance).

Quality Factor (Q):

A higher Q indicates sharper resonance and better selectivity.

Applications: Tuning circuits in radios, filters, and signal selection.

Quality Factor (Q):

• The quality factor is a measure of the sharpness of resonance of a resonant circuit.

• It is defined as the ratio of the reactive power to the resistive power at resonance, or equivalently the ratio of resonant frequency to bandwidth.
  

• It also represents voltage magnification across L or C at resonance:
  

Bandwidth (BW):

• The bandwidth is the range of frequencies between the two half-power points (where current falls to of maximum).
  
  where and are the lower and upper cut-off frequencies.

Relationship Between Q and Bandwidth:

Therefore:

Interpretation:

• A high Q means narrow bandwidth → sharp, selective resonance.
• A low Q means wide bandwidth → broad, less selective resonance.

This trade-off is important in the design of tuning and filter circuits.

Power Factor (p.f.):

• Power factor is the cosine of the phase angle between voltage and current in an AC circuit.
  

• It is a dimensionless quantity ranging between 0 and 1.

• Lagging p.f. → inductive circuit; Leading p.f. → capacitive circuit.

Types of Power:

1. Active (Real) Power ():

• The power actually consumed/dissipated in the circuit (does useful work).
  

2. Reactive Power ():

• The power that oscillates between source and reactive elements (not consumed).
  

3. Apparent Power ():

• The product of RMS voltage and RMS current.
  

Power Triangle:

Significance of Power Factor:

• A low power factor means higher current for the same power, increasing losses and requiring larger conductors.
• A high (near unity) power factor improves efficiency and reduces transmission losses.
• Industries use capacitor banks for power factor improvement.

Series RL Circuit Power Analysis:

Let the applied voltage and resulting current be:


(current lags voltage by )

Instantaneous Power:

Using the identity :

This shows the instantaneous power has:

• A constant component: (gives real power)
• A fluctuating component: (averages to zero over a cycle)

Average (Active) Power:
The average of the fluctuating term over a full cycle is zero. Hence:

Expressing in RMS values (, ):

where is the power factor of the RL circuit (lagging). Power is consumed only in the resistance; the inductor consumes no real power.

Comparison of Series RL and RC Circuits:

Key Points:

• In RL, voltage leads current → inductive behaviour.
• In RC, voltage lags current → capacitive behaviour.
• Both consume real power only in ; reactive elements store and return energy.

Generation of Three-Phase Voltages:

• A three-phase supply is generated using three identical coils placed apart on the stator of an alternator.
• As the magnetic field (rotor) rotates, EMFs are induced in each coil that are equal in magnitude and frequency but displaced by in phase.

EMF Equations:

Phase Sequence: R → Y → B (standard sequence).

Important Property: The sum of three instantaneous voltages at any instant is zero:

Advantages of Three-Phase over Single-Phase:

• More power output: For the same frame size, a 3-phase machine delivers more power.
• Constant power: The total instantaneous power is constant (not pulsating), giving smoother operation.
• Self-starting motors: Three-phase induction motors are self-starting; single-phase motors need auxiliary starting arrangements.
• Economical transmission: Requires less conductor material for transmitting the same power.
• Better efficiency: Three-phase machines are more efficient and have better power factor.
• Flexibility: Both single-phase and three-phase loads can be supplied.

Interconnection of Three Phases:
Instead of using six separate wires for three phases, the three windings are interconnected to reduce the number of conductors. There are two standard methods: Star and Delta.

1. Star (Y) Connection:

• The similar ends (either all starting ends or all finishing ends) of the three windings are joined together at a common point called the neutral point (N).
• The other three ends are brought out as line terminals (R, Y, B).
• Provides a 4-wire system (three lines + neutral) or 3-wire system.
• Suitable for supplying both 3-phase and 1-phase loads.

Features of Star:

2. Delta (Mesh) Connection:

• The windings are connected end-to-end in series forming a closed loop (mesh).
• The finishing end of one winding is connected to the starting end of the next.
• Line terminals are taken from the three junction points.
• Provides a 3-wire system (no neutral).

Features of Delta:

Comparison:

Star (Y) Connection Analysis:

In a star connection, the three phase windings are joined at a common neutral point. Line terminals are taken from the free ends.

Current Relationship:

• Each line conductor carries the current of the respective phase winding (connected in series with the line).
  

Voltage Relationship:

• The line voltage is the phasor difference between two phase voltages.

For line voltage between lines R and Y:

Since phase voltages are apart, the magnitude of the phasor difference is:

Summary for Star Connection:

Power in Star Connection:

Thus in a star-connected system, line voltage is times the phase voltage, while line current equals phase current.

Delta (Mesh) Connection Analysis:

In a delta connection, the three windings are connected in a closed loop, and line terminals are taken from the junctions.

Voltage Relationship:

• Each phase winding is connected directly across two line terminals.
  

Current Relationship:

• The line current is the phasor difference of two phase currents meeting at a junction.

For line current at a junction:

Since phase currents are apart, the magnitude of the phasor difference is:

Summary for Delta Connection:

Power in Delta Connection:

Thus in a delta-connected system, line voltage equals phase voltage, while line current is times the phase current. Interestingly, the total power expression is the same for both star and delta connections.

Comparison of Star and Delta Connections:

Balanced Load Condition:

• A three-phase load is said to be balanced when the impedances of all three phases are equal in magnitude and phase angle:
  

Significance of Balanced Load:

• The three phase currents are equal in magnitude and displaced by .
• In a star system, the neutral current is zero:
  
• The total instantaneous power becomes constant (non-pulsating), ensuring smooth operation of machines.
• Analysis is simplified since only one phase needs to be solved and results applied to the others.

Unbalanced Load: If impedances differ, neutral current flows (in star) and analysis becomes more complex.