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

Alternating current (AC) is an electric current whose magnitude varies with time and whose direction reverses periodically. Alternating voltage is a voltage whose magnitude and polarity vary periodically.

A sinusoidal voltage and current can be represented as:

The important characteristics are:

• Instantaneous value: Value of voltage or current at a particular instant, represented by or .
• Maximum or peak value: Largest value attained by the signal, represented by or .
• Time period: Time required to complete one cycle, denoted by seconds.
• Frequency: Number of cycles completed per second:
  
• Angular frequency:
  
• Phase angle: Angular position of the waveform relative to a reference, represented by .

A sinusoidal AC signal repeats after every time period and has equal positive and negative half-cycles.

The notations used in AC circuit analysis distinguish instantaneous quantities from RMS values or phasors:

• or : Instantaneous value of current, which varies with time.
• or : Instantaneous value of voltage, which varies with time.
• : Usually represents the RMS current or the current phasor, depending on context.
• : Usually represents the RMS voltage or the voltage phasor.
• and : Peak values of current and voltage.

For example, if

then:

• Peak voltage is .
• RMS voltage is
  
• The corresponding voltage phasor is
  

Thus, lowercase letters generally indicate time-domain instantaneous quantities, while uppercase letters generally indicate RMS values or phasor quantities.

Amplitude is the maximum value measured from the zero reference of an alternating waveform. For

the amplitude is .

Phase specifies the angular position of a waveform relative to a chosen reference. In the above expression, is the initial phase angle.

Phase difference is the angular displacement between two sinusoidal signals of the same frequency. If

and

then the phase difference is

• If , voltage leads current by .
• If , voltage lags current.
• If , the signals are in phase.
• A phase difference of means that the signals are in phase opposition.

The time displacement corresponding to a phase difference is

The root mean square (RMS) value of an alternating current is the value of direct current that produces the same heating effect in a resistor over the same period.

For a periodic current,

Given

substitute it into the RMS expression:

Using

we obtain

The average of over one complete cycle is zero. Therefore,

Similarly, for a sinusoidal voltage,

The average value of a periodic signal is the arithmetic mean of its instantaneous values over a specified interval.

For

the average value over the positive half-cycle is

Therefore,

Similarly,

Over a complete cycle, the positive and negative half-cycles are equal in magnitude and opposite in sign. Hence,

For this reason, the average value of a symmetrical AC waveform is normally quoted over a half-cycle or for its rectified waveform.

In sinusoidal steady-state analysis, circuit quantities are represented using complex numbers. The impedance is generally written as

where is resistance, is reactance, and .

The impedances of basic elements are:

• Resistor:
  
  Voltage and current are in phase.

• Inductor:
  
  where
  
  Current lags voltage by .

• Capacitor:
  
  where
  
  Current leads voltage by .

An impedance may be converted from rectangular to polar form:

where

The phasor form of Ohm's law is

Consider a resistance and inductance connected in series to an RMS supply voltage of angular frequency .

The individual impedances are

Therefore, the total impedance is

Its magnitude is

and its phase angle is

Thus,

Using phasor Ohm's law,

If the supply voltage is the reference, , then

Hence, the RMS current magnitude is

The current lags the supply voltage by . The voltage drops are , in phase with current, and , leading current by . Therefore,

In an RL circuit, current lags voltage by an angle . The power factor is

Since the current lags the voltage, it is called a lagging power factor.

The different forms of power are:

• Active or real power:
  
  Unit: watt or .

• Reactive power:
  
  Unit: volt-ampere reactive or .

• Apparent power:
  
  Unit: volt-ampere or .

These quantities form the power triangle:

The complex power is

where is the complex conjugate of the current phasor. For an inductive circuit, is positive.

For a resistance and capacitance connected in series, the capacitive reactance is

The total impedance is

Its magnitude is

and its impedance angle is

Therefore,

If voltage is used as the reference, then

Hence,

The current leads the supply voltage by . The resistor voltage is in phase with current, whereas the capacitor voltage lags current by . The supply voltage magnitude is

In a series RC circuit, current leads the applied voltage by an angle . Therefore, the circuit has a leading power factor.

The power factor is

The powers are calculated as follows:

• Active power:
  
• Capacitive reactive power:
  
• Apparent power:
  

The negative sign of indicates that reactive power is supplied back by the capacitor rather than absorbed as in an inductor.

The complex power is

The magnitudes satisfy

An ideal capacitor consumes no average power because its voltage and current are out of phase. Only the resistor consumes active power.

For a series RLC circuit, the individual impedances are

where

The total impedance is

Its magnitude is

and its phase angle is

Therefore, the RMS current is

The nature of the circuit depends on the net reactance:

• If , the circuit is inductive and current lags voltage.
• If , the circuit is capacitive and current leads voltage.
• If , the circuit is purely resistive and current is in phase with voltage.

The supply voltage satisfies the phasor relation

Series resonance occurs in a series RLC circuit when the inductive reactance equals the capacitive reactance:

Therefore,

Rearranging,

and hence

The resonant frequency is

At resonance:

• Net reactance is zero: .
• Impedance is minimum and purely resistive:
  
• Current is maximum:
  
• Phase angle is zero.
• Power factor is unity.
• Active power is maximum:
  
• Inductor and capacitor voltages are equal in magnitude and opposite in phase:
  

The quality factor of a series resonant circuit is

A high quality factor indicates sharp resonance and a narrow bandwidth.

For a series RLC circuit,

and

The power factor is

with

The power components are:

• Active power:
  
• Reactive power:
  
• Apparent power:
  
• Complex power:
  

The operating cases are:

• If , then and the power factor is lagging.
• If , then and the power factor is leading.
• If , then , , and the power factor is unity.

The power triangle obeys

The comparison is as follows:

• Pure resistive circuit:

  • Impedance: .
  • Voltage and current are in phase.
  • Phase angle: .
  • Power factor: unity.
  • Average power: .
  • Resistance is independent of frequency for an ideal resistor.

• Pure inductive circuit:

  • Impedance: .
  • Inductive reactance: .
  • Current lags voltage by .
  • Power factor: zero lagging.
  • Average power over a complete cycle is zero.
  • Reactance increases linearly with frequency.

• Pure capacitive circuit:

  • Impedance: .
  • Capacitive reactance: .
  • Current leads voltage by .
  • Power factor: zero leading.
  • Average power over a complete cycle is zero.
  • Reactance decreases as frequency increases.

Thus, only a pure resistor continuously converts electrical energy into heat, whereas ideal inductors and capacitors alternately store and return energy.

A balanced three-phase system consists of three sinusoidal voltages having:

• Equal RMS magnitudes.
• The same frequency.
• A phase displacement of electrical between successive phases.

A balanced set may be expressed as

Phase sequence is the order in which the phase voltages attain their positive maximum values. Common sequences are:

• Positive sequence: R-Y-B or 1-2-3.
• Negative sequence: R-B-Y or 1-3-2.

Reversing any two line conductors reverses the phase sequence. This reverses the direction of rotation of a three-phase motor.

The phases can be interconnected in two principal ways:

• Star or wye connection: One end of each phase is joined to form a common neutral point. The remaining ends form the three line terminals.
• Delta or mesh connection: The end of each phase is connected to the beginning of the next phase, forming a closed loop.

A star system may use three or four wires, whereas a delta system normally uses three wires.

In a balanced star connection, one terminal of each phase is joined at the neutral point. The phase voltages are separated by .

Let

and

The line voltage between R and Y is

Using rectangular components,

This gives

Therefore,

Each line conductor is directly in series with one phase impedance. Hence,

Thus, for a balanced star network:

The line voltage leads the corresponding phase voltage by . The total three-phase active power is

In a delta connection, the end of each phase is connected to the beginning of the next phase, producing a closed triangular loop. The three junctions are connected to the line conductors.

Since each phase is directly connected between two line conductors,

At a line terminal, line current is the phasor difference of two phase currents. For example,

For a balanced load, the phase currents have equal magnitude and are separated by . Their phasor difference gives

relative to the corresponding phase current. Therefore,

Thus, for a balanced delta network:

The line current is displaced by from the corresponding phase current. The exact lead or lag description depends on the chosen phase sequence and current reference. The total active power is

Given:

For a star connection,

Therefore,

The phase current is

For a star connection,

The total active power is

Thus,

Therefore:

• Phase voltage:
• Line current:
• Total active power:

Given:

For a delta connection,

The phase current is

The line current magnitude is

The impedance angle is , so the power factor is

The apparent power is

The active power is

The reactive power is

Therefore:

• Phase current:
• Line current magnitude:
• Power factor: lagging
• Active power:
• Reactive power:
• Apparent power:

Comparison of star and delta connections:

• In a star connection:
  
• In a delta connection:
  
• A star connection can provide a neutral point and two voltage levels.
• A delta connection has no neutral point and provides a closed path for circulating currents.
• For the same line voltage and the same phase impedance, a delta load draws three times the line current and consumes three times the power of a star load.

For either balanced connection, power per phase is

Thus, total active power is

Using the star or delta line-to-phase relations, this becomes

Similarly, total reactive power is

Total apparent power is

The powers satisfy

These expressions apply to balanced three-phase loads whether they are connected in star or delta.