AC CIRCUIT ANALYSIS
ALTERNATING QUANTITY: An alternating quantity is that which acts in alternate directions and whose magnitude undergoes a definite cycle of changes in definite intervals of time.
ALTERNATING VOLTAGE: Alternating voltage may be generated by
ADVANTAGES OF SINE WAVE
CYCLE: A cycle may be defined as one complete set of positive and negative values of an alternating quantity repeating at equal intervals. Each complete cycle is spread over 360° electrical.
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PERIODIC TIME: The time taken by an alternating quantity in seconds to trace one complete cycle is called periodic time or timeperiod. It is usually denoted by symbol T.
FREQUENCY: The number of cycles per second is called frequency and is denoted by symbol f.
f = 1/T
If the angular velocity w is expressed in radians per second, then
w= 2π/T
w = 2πf
Peak Factor: Ratio of maximum value to the RMS value is known as crest or peak.
factor or amplitude factor.
Peak factor = Maximum Value/RSM Value
Form Factor: Ratio of RMS Value to average value is known as form factor.
Form Factor = RMS Value/Average Value
PHASE DIFFERENCE
ACTIVE, REACTIVE & APPARENT POWER OF AC CIRCUIT
Impedence in AC circuit Z = R ± jX = Z∠Φ= Z cosv + jZ sinΦ
where X = j(X_{L}X_{C})
where Z = √R^{2 }+ X^{2}^{ }
R= Z cosΦ
X= Z sinΦ
Power Factor of the circuit ⇒ cosΦ = R/Z
Current in the circuit = E/Z
This current has two components I cosΦ and I sinΦ. The component I cosΦ is called in phase or watt full component and I sinΦ is perpendicular to E and is called wattless component.
Active (Real) Power = Voltage x Current x cosΦ watts
The total power EI in voltamperes supplied to a circuit consists of two components
(a) Active power = EI cosΦ watts
(b) Reactive power = EI sinΦ voltamperes reactive or simply VAR.
OA = Active power = EI cosΦ presented by watts
AB = Reactive power = EI sin Φ expressed by VAR
OB = Total power = EI expressed by VA
4. RESONANCE IN AC CIRCUIT
SERIES RESONANCE
The circuit, with resistance R, inductance L, and a capacitor, C in series is connected to a singlephase variable frequency (f) supply.
The total impedance of the circuit is
Z∠Φ = R+j(X_{L}X_{C})
where X_{L} = jwL
X_{C} = 1/jwC
The current in the circuit is maximum, if
The frequency under the above condition is
The magnitude of the impedance under the above condition is Z = R, with the reactance X = 0, as the inductive reactance is equal to capacitive reactance. The phase angle is φ = 0, and the power factor is unity (cos φ= 1), which means that the current is in phase with the input (supply) voltage. So, the magnitude of the current I = V/R.
The magnitude of the voltage drop in the inductance L/capacitance C, both are equal as the reactance are equal is⋅= (I.w_{o}L = I.1/(w_{o}C)
Quality Factor Q = w_{o}L/R = 2πf_{o}L/R = 1/R √L/C
The impedance of the circuit with the constant values of inductance L, and capacitance C is minimum at resonant frequency (f_{o}), and increases as the frequency is changed, i.e. increased or decreased, from the above frequency. The current is maximum at f=f_{o} , and decreases as frequency is changed (f>f_{o} or f<f_{o}) i.e. f≠f_{o.}
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PARALLEL RESONANCE
The circuit, with resistance R, inductance L, and a capacitor, C in parallel is connected to a singlephase variable supply frequency (f).
The total admittance of the circuit is
Y∠Φ = 1/R + J(w_{C}1/w_{L})
The current in the circuit is maximum, if
The frequency under the above condition is
The magnitude of the impedance under the above condition is (Z=R), while the magnitude of the admittance is (Y = (1/R)= G). The reactive part of the admittance is B =0, as the susceptance (inductive) B_{L }= (1/ωL) is equal to the susceptance (capacitive) B_{C }= ωC. The phase angle is φ = 0, and the power factor is unity (cos φ=1).
The input current increases as the frequency are changed, i.e. increased or decreased from the resonant frequency (f>f_{o} or f<f_{o}) i.e.f≠f_{o.}
POWER IN AC CIRCUIT
We saw in our tutorial about Electric Power that AC circuits which contain resistance and capacitance or resistance and inductance, or both, also contain real power and reactive power. So in order for us to calculate the total power consumed, we need to know the phase difference between the sinusoidal waveform of the voltage and current.
The phase angle is given by,
φ = arg(V) − arg(I)
i.e. the angle from the current to the voltage. Therefore, positive phase angles mean that the current lags the voltage, and thus are called lagging, and negative phase angles mean that the current leads the voltage, and are called leading.
The power factor is defined as,
power factor = P/S = cos(φ)
As this is always a positive number the tag ‘leading’ or ‘lagging’ is usually added to describe the phase difference.
where P (=VI cosφ )is the Real Power
S(= VI) is the Apparent Power.
Real Power in AC Circuit
Real Power P = I^{2}R = V*I*cos(θ) Watts, (W)
P = V_{rms} x I_{rms} x Cosθ ⇒ since cos 0^{0}=1
So P = V_{rms} x I_{rms }x 1
or {P = V_{rms} x I_{rms}}
Reactive Power in AC Circuit:
Reactive Power Q = I^{2}X = V*I*sin(θ) voltamperes reactive, (VAr’s)
Q = V_{rms} x I_{rms} x Sinθ ⇒ since from phasor, sin 90^{0}=1
Q = V_{rms} x I_{rms} x 1
Q = V_{rms} x I_{rms}
Thus reactive power is the I^{2}X reactive element that has units in voltamperes reactive (VAr), Kilovoltamperes reactive (kVAr), and Megavoltamperes reactive (MVAr).
Apparent Power in AC Circuit
Where:
The power factor is calculated as the ratio of the real power to the apparent power because this ratio equals cosθ.
Power Factor in AC Circuit
So the power factor will be:
Power Factor, pf = cos 0^{o} = 1.0
Which means that the number of watts consumed is the same as the number of voltamperes consumed producing a power factor of 1.0, or 100%. In this case, it is referred to a unity power factor.
In a purely reactive circuit, the current and voltage outofphase with each other by 90^{o}.
the power factor, in this case, will be:
Power Factor, pf = cos 90^{o} = 0
Which means that the number of watts consumed is zero but there is still a voltage and current supplying the reactive load.
Real Power (P) = Apparent power (S) x Power factor (p.f)
The disadvantage of Low Power Factor:
Poor power Factor or Low Power Factor less than unity has the following disadvantage:
Where KVA = KW/power factor
i.e The KVA is inversely proportional to KVA for a given KW. For lower power factor the electrical machinery should be having higher KVA rating to drive constant KW load. Then the size and cost of the electrical machine become expensive.
How to avoid low power factor? (or) What are the various methods to avoid low power factor?
Various methods of power factor improvement are given below;
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