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248 Chapter 22
component of capacitors is vertical and below the true component, e.g., as BD is the wattless component of the capacitor. Line AF is the true-power component of the capacitor, FG the true component of the inductance, and GB is the resistance of R. The wattless component of inductance is vertical (BC) and above the true component, FG.
The common expressions wattless power or wattless current are used with AC. There is actually no wattless current, since wherever current flows there must be an expenditure of energy. These currents are out of phase with the applied voltage, and the consideration of power involved is based on energy components and wattless components of these currents. Figure 22-7 illustrates the relationship of the three components in an inductive circuit.
Figure 22-7 Vector diagram showing typical relationships of current components in an AC circuit.
Taking the values of I in Figure 22-7, and using 1000 volts, the line AB (1000 A) times 1000 V = 1,000,000 VA apparent power, or 1000 kVA. BC (500 A) times 1000 V = 500,000 VA as the wattless power component, and AC (866A) times 1000 V = 866,000 watts or 866 kW of true power.
This tells us that the vector diagram in Figure 22-7 is an inductive circuit with an angle lag of 30° or a power factor of 86.6%. Thus, the circuit must be designed for 1000 amperes due to the 86.6% PF, while 866 amperes would be all the actual current that would be required to do the work if we had unity (100%) power factor.
Parallel LC Circuit
Inductance and capacitance are not often found in series; they are usually in parallel. There is a capacitance between conductors of circuits; the conductors are the plates and the insulation, and air the dielectric.
POWER COMPONENT OF 1=866 A
Resistance, Capacitance, and Inductance in Series and Parallel 249
Take a practical example, the circuit shown in Figure 22-8. The system has a 2-^F capacitor and the load is 0.5 H, voltage 2000 V, and frequency 130 Hz.
v 1 1 ,
Xc = 6.28fC = 6.28 X 130 X 0.000002 = 613 ohms
Figure 22-8 Capacitive effect of conductors.
The energy components will be neglected in this example, so C is all Xc or Zc, and L is all XL or ZL Thus,
Ic = E/Zc = 2000/613 = 3.26 amperes in C (leading)
IL = E/Xc = 2000/408 = 4.9 amperes in L (lagging)
Ic is thus out of phase 90°, so is all wattless component IL is also out of phase 90°, so is likewise wattless component
Il - Ic = 4.9 - 3.26 = 1.64 amperes is the total current supplied by the acternator. Now, Z = E/I = 2000/1.64 = 1219 ohms combined impedance.
Since we have neglected the energy losses, the power factor is zero.
The current required is considerably less than that required by either device, due to the fact that c and L are diametrically opposed to each other. The capacitor acts as sort of a generator, storing energy in one alternation and releasing it in the next alternation.
With capacitance and inductance in series, Xc and XL are in series, and as XL = 6.28fL approaches the value of Xc = 1/6.28fc, the line voltage and current will increase. When Xc ana XL are in parallel (as in Figure 22-8), the line current diminishes as Xc approaches XL.
Circuits possessing Xc and XL result in resonance. Resonance can’t exist without inductance and capacitance. Perfect resonance results when Xc = XL.
250 Chapter 22
To illustrate, the alternators in Figure 22-9 and the capacitors are such that voltage builds up as it does when XC and X, balance. In Figure 22-9, alternator A charges C to 2000 volts. In Figure 22-9B, the alternator has reversed polarity. Capacitor C, in series with A, sends its emf back so that at the end of that alternation the two voltages have added, and C is now charged with 4000 volts, in Figure 22-9C. In Figure 22-9D the polarity of the alternator reverses, and the action in Figure 22-9B is repeated, except the capacitor receives a charge of 6000 volts, and so on. (In reality, this circuit would have to be modified to behave as shown in Figure 22-9.)
(E) C charges to 6000 volts.
Figure 22-9 Voltage accumulation.
Resonance occurs at a certain frequency. Should the frequency be increased, the value of XL will rise; but at the same time that an increase in frequency increases XL, it reduces XC. Where resonance
Resistance, Capacitance, and Inductance in Series and Parallel 251
exists, altering the frequency will check it. The smaller the resistance in a series circuit, the greater is the local voltage set up across the capacitor and inductance.
Voltage at resonance is very dangerous and undesirable, because of the damage and possible breakdown of insulation due to the increase in voltage. Ferroresonance in power systems must be avoided by proper designs.
Current resonance is highly desirable because it relieves the alternator of the necessity of furnishing the wattless components of the current and tends to correct the power factor toward unity, or 100%.
System Power Factor
The power factor of a system is of prime importance. A 25,000-kVA system at 80% PF would mean that the actual energy received is