In ac circuits excited by sinusoidal signal, the phase difference between the source voltage and the source current is called the power factor angle. To explain this, we use the sketch in Fig. 9. In this sketch, a sinusoidal voltage source is connected to a circuit, represented by a block. Let the source voltage and current be defined as follows.
Equation (41) defines the source voltage, and equation (42) defines the source current. Given the values as shown now, the waveforms of source voltage and source current can be displayed, as shown below.
From the waveforms shown, it is seen that the source current lags the source voltage. When the source current lags the source voltage, the circuit has lagging power factor angle. In this case, the power factor angle can be defined as shown below, by equations (43) and (44).
The power factor angle is the phase difference between the source voltage and the source current. For the case considered, the power factor angle is the sum of a and b. Cosine of the power factor angle is called the power factor. We can express the source voltage and the source current by a phasor diagram, shown next.
The sketch in Fig. 10 shows the phasor diagram. It is seen that the source voltage leads the reference phasor, and the source current lags the reference phasor. The phase difference is the sum of the angles, a and b. The source voltage and the current can be expressed as phasors, as shown by equations (45) and (46).
Equations (45) and (46) show the phasor expressions for source voltage and source current. Note that they are qualified by the amplitudes. Hence the apparent power, the active power and the reactive power of the source can be obtained as follows.
Since the phasors are qualified by the peak values, the factor of half is to be used when calculating the apparent power of the source. If the root mean square values of source voltage and source current are used, then the use of this factor is not necessary, since the root mean square value of a sinusoidal signal is the peak value over square root of two. We need to use the minus sign, since current is flowing out of the positive terminal of source, and this procedure is in keeping with the passive sign convention. According to passive sign convention, power associated with an element is positive if current flows into the positive terminal of that element.
Equations (47) to (50) state the expressions for the apparent power, active power and the reactive power of the source.
Now let us concentrate on the expression for active power, expressed by equation (48). Let us assume here that the peak values of source voltage and source current are fixed. It is seen that active power varies with the cosine of the power factor angle. More often than not, a physical source is defined by its rated voltage and its current, and it is desirable that it should be possible to extract maximum active power from the source, keeping in mind that this power delivered by the source is both safe and feasible. It can be seen from equation (48) that the power delivered by the source is maximum, when the power factor angle of the circuit is zero. In other words, the power factor due to the load is unity, since the cosine of the power factor angle is the power factor. Improving the power factor enables more active power to be extracted from the source. Now the question arises how we can improve the power factor. We will explain this next, with the help of a circuit.
More often than not, the load connected to an ac source tends to consume both active power and reactive power. In other words, the load contains some resistance and inductance. We can express the load, as shown by the circuit in Fig. 11. For this circuit, the phasor expression for source current is obtained as follows.
The phasor expression for source current is obtained, as shown by equations from (51) to (55). Equation (51) expresses the source voltage phasor. Equation (52) expresses the impedance in the Cartesian form, and the expression in polar form is shown by equation (53). Since current is the ratio of voltage over impedance, we get equation (54) for the source current. Equation (55) shows the expressions for the amplitude and angle of the source current.
Since the power factor angle is the phase difference between the voltage and current phasors of the source, we get the power factor angle as expressed by equation (56). Here d is known as the load angle, and in this case the power factor angle and the load angle are the same. Let us assume that the load resistance and the load reactance are fixed. Let us see how we can improve the power factor. Improvement in power factor is brought about by adding a capacitor to the circuit, as shown below.
We know that an inductor is a consumer of reactive power and that a capacitor is a source of reactive power. Hence a capacitor can be added, as shown in Fig. 12, to supply either a part or the whole of reactive power due to inductive reactance of the load, where the load is made up of resistance R and inductance L. Normally the capacitance added is restricted, such that the power factor angle seen by the source does not become leading. In other words, it is preferable that the capacitor does not feed reactive power into the source. Not all sources are capable of being a consumer of reactive power. Now let use see what the power factor angle is, as a result of adding a capacitor.
For the circuit in Fig. 12, the admittance seen by the source is the sum of capacitive admittance, and the admittance of the RL load, as shown by the first part of equation (58). The admittance of the RL load is the reciprocal of its impedance. We can multiply both the denominator and the numerator of the expression for the admittance of the RL load by the conjugate of its impedance, and get the second part of equation (58). We can group the imaginary parts of the admittance seen by the source, and we get equation (59). Power factor angle is obtained from the real and imaginary parts of the equivalent admittance, as shown by equation (60). When the imaginary part is zero, the power factor angle is zero, and the power factor is unity. The highest value of capacitance that can be used is shown by equation (61). For larger values of capacitance, the capacitor feeds reactive power back to the source. Normally, it is preferable to avoid this situation. Usually, power factor improvement is not as simple as this. Let us deal with a slightly more difficult example next.
Given a circuit where the load is variable, it is not easy to pick capacitance needed to improve the power factor. The circuit in Fig. 13 has a load and the inductance of the load is variable. Let the minimum inductance of the load be L1. Then the capacitor to be used to improve the power factor may be selected based on the the minimum inductance, as shown next.
If only one capacitor is used for power factor improvement, then its maximum value is expressed by equation (103). In practice, capacitors are added to the circuit in stages, depending on the load.
An applet is presented below, to illustrate how the power factor of a circuit can be improved
The next page is how we can convert star- connected network to a delta-connected networks and vice versa.