In this article we will discuss about:- 1. Introduction to Single Phase AC Circuit 2. Purely Resistive Circuit 3. Purely Inductive Circuit 4. Purely Capacitive Circuit 5. Resistance — Capacitance (R-C) Series Circuit 6. Apparent Power, True Power, Reactive Power and Power Factor.
Contents:
- Introduction to Single Phase AC Circuit
- Purely Resistive Circuit
- Purely Inductive Circuit
- Purely Capacitive Circuit
- Resistance — Capacitance (R-C) Series Circuit
- Apparent Power, True Power, Reactive Power and Power Factor
1. Introduction to Single Phase AC Circuit:
In a dc circuit the relationship between the applied voltage V and current flowing through the circuit I is a simple one and is given by the expression I = V/R but in an a c circuit this simple relationship does not hold good. Variations in current and applied voltage set up magnetic and electrostatic effects respectively and these must be taken into account with the resistance of the circuit while determining the quantitative relations between current and applied voltage.
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With comparatively low-voltage, heavy- current circuits magnetic effects may be very large, but electrostatic effects are usually negligible. On the other hand with high-voltage circuits electrostatic effects may be of appreciable magnitude, and magnetic effects are also present.
Here it has been discussed how the magnetic effects due to variations in current do and electrostatic effects due to variations in the applied voltage affect the relationship between the applied voltage and current.
2. Purely Resistive Circuit:
A purely resistive or a non-inductive circuit is a circuit which has inductance so small that at normal frequency its reactance is negligible as compared to its resistance. Ordinary filament lamps, water resistances etc., are the examples of non-inductive resistances. If the circuit is purely non-inductive, no reactance emf (i.e., self- induced or back emf) is set up and whole of the applied voltage is utilised in overcoming the ohmic resistance of the circuit.
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Consider an ac circuit containing a non-inductive resistance of R ohms connected across a sinusoidal voltage represented by v = V sin wt, as shown in Fig. 4.1 (a).
As already said, when the current flowing through a pure resistance changes, no back emf is set up, therefore, applied voltage has to overcome the ohmic drop of i R only:
And instantaneous current may be expressed as:
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i = Imax sin ωt
From the expressions of instantaneous applied voltage and instantaneous current, it is evident that in a pure resistive circuit, the applied voltage and current are in phase with each other, as shown by wave and phasor diagrams in Figs. 4.1 (b) and (c) respectively.
Power in Purely Resistive Circuit:
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The instantaneous power delivered to the circuit in question is the product of the instantaneous values of applied voltage and current.
Where V and I are the rms values of applied voltage and current respectively.
Thus for purely resistive circuits, the expression for power is the same as for dc circuits. From the power curve for a purely resistive circuit shown in Fig. 4.1 (b) it is evident that power consumed in a pure resistive circuit is not constant, it is fluctuating.
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However, it is always positive. This is so because the instantaneous values of voltage and current are always either positive or negative and, therefore, the product is always positive. This means that the voltage source constantly delivers power to the circuit and the circuit consumes it.
3. Purely Inductive Circuit:
An inductive circuit is a coil with or without an iron core having negligible resistance. Practically pure inductance can never be had as the inductive coil has always small resistance. However, a coil of thick copper wire wound on a laminated iron core has negligible resistance arid is known as a choke coil.
When an alternating voltage is applied to a purely inductive coil, an emf, known as self-induced emf, is induced in the coil which opposes the applied voltage. Since coil has no resistance, at every instant applied voltage has to overcome this self-induced emf only.
From the expressions of instantaneous applied voltage and instantaneous current flowing through a purely inductive coil it is observed that the current lags behind the applied voltage by π/2 as shown in Fig. 4.2 (b) by wave diagram and in Fig 4.2 (c) by phasor diagram.
Inductive Reactance:
ωL in the expression Imax = Vmax/ωL is known as inductive reactance and is denoted by XL i.e., XL = ω L
If L is in henry and co is in radians per second then XL will be in ohms.
Power in Purely Inductive Circuit:
Instantaneous power, p = v × i = Vmax sin ω t Imax sin (ωt – π/2)
Or p = – Vmax Imax sin ω t cos ω t = Vmax Imax/2 sin 2 ωt
The power measured by wattmeter is the average value of p which is zero since average of a sinusoidal quantity of double frequency over a complete cycle is zero. Hence in a purely inductive circuit power absorbed is zero.
Physically the above fact can be explained as below:
During the second quarter of a cycle the current and the magnetic flux of the coil increases and the coil draws power from the supply source to build up the magnetic field (the power drawn is positive and the energy drawn by the coil from the supply source is represented by the area between the curve p and the time axis). The energy stored in the magnetic field during build up is given as Wmax = 1/2 L I2max.
In the next quarter the current decreases. The emf of self-induction will, however, tends to oppose its decrease. The coil acts as a generator of electrical energy, returning the stored energy in the magnetic field to the supply source (now the power drawn by the coil is negative and the curve p lies below the time axis).
The chain of events repeats itself during the next half cycles. Thus, a proportion of power is continually exchanged between the field and the inductive circuit and the power consumed by a purely inductive coil is zero.
4. Purely Capacitive Circuit:
When a dc voltage is impressed across the plates of a perfect condenser, it will become charged to full voltage almost instantaneously. The charging current will flow only during the period of “build up” and will cease to flow as soon as the capacitor has attained the steady voltage of the source. This implies that for a direct current, a capacitor is a break in the circuit or an infinitely high resistance.
In Fig. 4.4 a sinusoidal voltage is applied to a capacitor. During the first quarter-cycle, the applied voltage increases to the peak value, and the capacitor is charged to that value. The current is maximum in the beginning of the cycle and becomes zero at the maximum value of the applied voltage, so there is a phase difference of 90° between the applied voltage and current. During the first quarter-cycle the current flows in the normal direction through the circuit; hence the current is positive.
In the second quarter-cycle, the voltage applied across the capacitor falls, the capacitor loses its charge, and current flows through it against the applied voltage because the capacitor discharges into the circuit. Thus, the current is negative during the second quarter-cycle and attains a maximum value when the applied voltage is zero.
The third and fourth quarter-cycles repeat the events of the first and second, respectively, with the difference that the polarity of the applied voltage is reversed, and there are corresponding current changes.
In other words, an alternating current flows in the circuit because of the charging and discharging of the capacitor. As illustrated in Figs. 4.4 (b) and (c) the current begins its cycle 90 degrees ahead of the voltage, so the current in a capacitor leads the applied voltage by 90 degrees – the opposite of the inductance current-voltage relationship.
Let an alternating voltage represented by v = Vmax sin ω t be applied across a capacitor of capacitance C farads.
The expression for instantaneous charge is given as:
q = C Vmax sin ωt
Since the capacitor current is equal to the rate of change of charge, the capacitor current may be obtained by differentiating the above equation:
From the equations of instantaneous applied voltage and instantaneous current flowing through capacitance, it is observed that the current leads the applied voltage by π/2, as shown in Figs. 4.4 (b) and (c) by wave and phasor diagrams respectively.
Capacitive Reactance:
1/ω C in the expression Imax = Vmax/1/ω C is known as capacitive reactance and is denoted by XC i.e., XC = 1/ω C
If C is in farads and ω is in radians/s, then Xc will be in ohms.
Power in Purely Capacitive Circuit:
Hence power absorbed in a purely capacitive circuit is zero. The same is shown graphically in Fig. 4.4 (b). The energy taken from the supply circuit is stored in the capacitor during the first quarter- cycle and returned during the next.
The energy stored by a capacitor at maximum voltage across its plates is given by the expression:
This can be realized when it is recalled that no heat is produced and no work is done while current is flowing through a capacitor. As a matter of fact, in commercial capacitors, there is a slight energy loss in the dielectric in addition to a minute I2 R loss due to flow of current over the plates having definite ohmic resistance.
The power curve is a sine wave of double the supply frequency. Although it raises the power factor from zero to 0.002 or even a little more, but for ordinary purposes the power factor is taken to be zero. Obviously the phase angle due to dielectric and ohmic losses decreases slightly.
5. Resistance — Capacitance (R-C) Series Circuit:
Consider an ac circuit consisting of resistance of R ohms and capacitance of C farads connected in series, as shown in Fig. 4.18 (a).
Let the supply frequency be of fHz and current flowing through the circuit be of I amperes (rms value). Voltage drop across resistance, VR = I R in phase with the current.
Voltage drop across capacitance, VC = I XC lagging behind I by π/2 radians or 90°, as shown in Fig. 4.18 (b).
The applied voltage, being equal to phasor sum of VR and VC, is given in magnitude by-
The applied voltage lags behind the current by an angle ɸ:
If instantaneous voltage is represented by:
v = Vmax sin ω t
Then instantaneous current will be expressed as:
i = Imax sin (ω t + ɸ)
And power consumed by the circuit is given by:
P = VI cos ɸ
Voltage triangle and impedance triangle Fig. 4.19 are shown in Figs. 4.19 (a) and 4.19 (b) respectively.
6. Apparent Power, True Power, Reactive Power and Power Factor:
The product of rms values of current and voltage, VI is called the apparent power and is measured in volt-amperes or kilo-volt amperes (kVA).
The true power in an ac circuit is obtained by multiplying the apparent power by the power factor and is expressed in watts or kilo-watts (kW).
The product of apparent power, VI and the sine of the angle between voltage and current, sin ɸ is called the reactive power. This is also known as wattless power and is expressed in reactive volt-amperes or kilo-volt amperes reactive (kVA R).
The above relations can easily be followed by referring to the power diagram shown in Fig. 4.7 (a).
Power factor may be defined as:
(i) Cosine of the phase angle between voltage and current,
(ii) The ratio of the resistance to impedance, or
(iii) The ratio of true power to apparent power.
The power factor can never be greater than unity. The power factor is expressed either as fraction or as a percentage. It is usual practice to attach the word ‘lagging’ or ‘ leading’ with the numerical value of power factor to signify whether the current lags behind or leads the voltage.
Active Component of Current:
The current component which is in phase with circuit voltage (i.e., I cos ɸ) and contributes to active or true power of the circuit is called the active (wattful or in-phase) component of current.
Reactive Component of Current:
The current component which is in quadrature (or 90° out of phase) to circuit voltage (i.e., I sin ɸ) and contributes to reactive power of the circuit, is called the reactive (or wattless) component of current.
Q-Factor of Coil:
Reciprocal of power factor is known as Q-factor of the coil. It is also called the quality factor or figure of merit of a coil.
Power in Iron-Cored Choking Coil:
The power drawn by a choking coil is used up to supply the iron losses in the core in addition to that used up for heating the coil.
Power consumed by a choking coil, P = Iron losses (hysteresis and eddy current loss), Pi + power loss in ohmic resistance i.e., I2 R (R being the dc resistance or true resistance of the coil).