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The Art of Protecting Electrical Systems, Part 3: System Analysis By GEORGE FARRELL and FRANK VALVODA, P.E. Editor’s Note: Last year, we decided to reprise a classic: “The Art of Protecting Electrical Systems,” a series of articles by long-time CSE contributors George Farrell and Frank Valvoda, P.E. The series first ran from 1965 through 1970. Due to the immense popularity of the 31 installments in the series, an updated version was run beginning in Feb. 1989. Over the years since the last installment ran in the late ’90s, we received many requests to re-run the series. Mr. Valvoda passed away in Dec. 2001, and his long-time friend and editorial partner, Mr. Farrell, was planning to review and update the series. We must sadly report of Mr. Farrell’s recent death. In tribute to these authors, we will continue our republishing of the series in monthly installments at www.csemag.com, and will let our readers “update” the series by publishing their commentary in an online forum in conjunction with the articles. After reading an article in the series, please send up your comments to ssiddens@reedbusiness.com. In the previous article, the single-phase equivalent of a typical power distribution system was discussed. It was said to consist of a voltage source with a load, plus a series impedance, i.e., resistance and reactance connecting the source and the load. Capacitance is not usually a significant factor in the typical distribution system, except in long transmission lines and when introduced to improve power factor. In consequence, even with intentionally introduced capacitance, almost every system has a net inductive balance, that is, a resistance-inductance (R-L) circuit with current lagging the voltage. The effect of capacitance may be important, however, in certain individual pieces of equipment. This will be discussed later in the series. Figure 3.1 illustrates such a resistance-inductance circuit. Switch No.1 represents a circuit breaker or other device used to energize and de-energize the system. Switch No. 2 connects the system to ground and represents a short circuit condition. The sudden change in circuit impedance when switch No. 1 is opened or closed, or when switch No. 2 is closed, results in transient overvoltages and overcurrents. These transients, although usually lasting for only short periods of time, may be severe enough to subject all system components to great stress. The sudden change in circuit impedance when switch No. 1 is opened or closed, or switch No. 2 is closed, results in transient overvoltages and overcurrents. This basic equation of the short-circuited system can be used to develop three multiplying factor values. The most significant overcurrent transients occur when switch No. 2 is closed, that is, when the phase conductors are suddenly connected to ground. This discussion examines the conditions existing at that time. Although focused on single-phase faults, the principles may be readily extended to three-phase faults. At the time of the fault, a current define by I = e/Z flows in the system, where Z is the Thevenin impedance of the fault. (The utility and each machine source contributes energy to the fault point.) This current is referred to as the steady state or alternating current or symmetrical component of the fault current. However, another component must be considered. Energy is stored in every system inductance, e.g., conductors, motors, transformers. This stored energy is discharged into the system when a short circuit occurs. Since it is unidirectional, it appears as a direct current and is termed the direct-current component of the fault current. The total fault current decreases as the inductance releases its stored energy through the resistance of the system, usually reaching steady-state values within two to six cycles. The direct-current component decreases with time in accordance with the equation: idc = I0ε-Rt/L Where Rt/L is the time constant of the circuit, with I0 being the maximum direct-current that can flow at the instant the switch is closed. The total fault current at any instant is the sum of the alternating- and direct-current components, adding in accordance with the Superposition Theorem. The equation describing the total current is shown in Figure 3.2, where mathematical manipulation of the dc component’s time constant permits introducing the faulted circuit’s X/R ratio ( = tan φ) for use in plotting and evaluation. Determining multiplying factors Figure 3.2 presents the basic equation of the short-circuited system. Three multiplying factor values are developed using this equation: maximum instantaneous peak amperes (Mp); maximum one-phase rms amperes at ½ cycle (Mm); and average three-phase rms amperes at ½ cycle (Ma). These values are defined in the figure. The Mp multiplying factor is important as a measure of the largest peak value of current that can occur in a circuit for a given X/R ratio. It is utilized in determining the application of current-limiting circuit breakers and fuses. The Mm multiplying factor is used in converting from symmetrical fault current (rms) amperes to asymmetrical (rms) amperes for momentary (½ cycle) and other time interval interrupting currents (3, 5 and 8 cycles) as a guide to selecting equipment. The Ma multiplying factor is the average asymmetrical multiplier for a three-phase fault. Maximum asymmetry can occur in only one phase since degree of asymmetry depends on the closing angle, i.e., point in the cycle when the fault occurs. The other phases, being 120 degrees apart, cannot have the same degrees of asymmetry. This multiplier has fallen into general disuse in 600-volt and below systems since equipment interrupting and withstand ratings were changed from average asymmetrical currents to symmetrical currents in the 1960s. Closing a circuit breaker into a fault at a random time (when one phase may reach the maximum, for example) may subject its contacts to very strong electromagnetic forces, which could damage circuit breaker parts. The maximum multiplying factors are used to verify that the breaker can withstand the maximum current which it may see. Demonstration of how the multiplying factors may be calculated from the actual current waves, such as on a test oscilloscope. The table of multiplying factors in Figure 3.3 has been calculated on a digital computer using the formulas discussed. Intermediate values of multiplying factors may be approximated with sufficient accuracy by interpolation. Figure 3.4 demonstrates how the multiplying factors may be calculated from the actual current waves (such as on a test oscilloscope or by a hard-copy plot of the equation in Figure 3.2). The figure shows the fault current components at a system short-circuit power factor of 15% (X/R = 6.59). Measuring and then calculating the ac and dc components at ½ cycle gives the associated values of Mp = 2.309 and Mm = 1.331. For most applications these multipliers may be rounded to 2.3 and 1.3, respectively. Note that for a short-circuit power factor of 15%, a direct-current component may exist for over four cycles. With increasing short-circuit power factor (decreasing short-circuit X/R ratio) the dc component reaches low values in shorter times. This is important for the selection of equipment. Defining interrupting current The impact of these multiplying factors may be seen in Figure 3.5, a simulated, computer-generated fault current report. Interrupting current in a system is the symmetrical rms fault current multiplied by the factor related to the X/R ratio. The interrupting current at ½ cycle is called the momentary asymmetrical current. This is the most symmetrical rms current for the severe type of fault multiplied by the factor obtained by using Equation 1 (Or Equation 6 for the X/R ration associated with that fault). In Figure 3.5, the most severe fault at bus 15 is line-to-ground: 23,550 amperes at X/R = 26.523. The 1.606 asymmetrical multiplying factor is, therefore, applied to the line-toground fault at ½ cycle, obtaining 37,821 rms asymmetrical amperes. At bus 22 the most severe fault is three-phase: 35,060 amperes at X/R = 6.591. The 1.331 asymmetrical multiplying factor is applied to the three-phase fault at ½ cycle, obtaining 46,665 rms asymmetrical amperes. The computer-calculated values of asymmetrical current are, of course, rounded off for practical applications to equipment ratings. The interrupting current at three and five cycles also has been calculated for buses 15 and 22, where the multiplying factor is based on the time in cycles and the X/R ration of the most severe fault. These calculations illustrate how quickly the dc component decreases when the X/R ratio is small. This calculated current at three and five cycles must not be used to select equipment withstand ratings. Other factors that must be considered will be addressed in a future article. Equipment test power factors Underwriters Laboratories Inc. (UL) has established that the power factors shown in Figure 3.6 be used in test circuits for testing certain low-voltage, over-current protective devices. Because of the way circuit breakers are constructed and tested, they must be derated if system short-circuit power factor is less than the test short-circuit power factor (if system X/R is greater than test X/R). For example, if a molded-case circuit breaker tested at 20% power factor (X/R = 4.9) and rated 22000 amperes is applied in a circuit where X/R = 9.0, the breaker must be derated by 12%. The rating of the breaker, if used in that circuit, would be 22,000 ÷ 1.12 = 19,600. All electrical equipment is tested and rated for operation at specific power factors. In an upcoming issue, these will be discussed in detail, giving characteristics of the individual equipment and including a demonstration of the application of the multiplying factors. Computer-calculated values are rounded off for practical applications to equipment ratings. These power factors have been established by UL for testing low-voltage overcurrent protection devices. Related Stories: The Art of Protecting Electrical Systems, Part 1: Introduction and Scope The Art of Protecting Electrical Systems, Part 2: System Analysis