Voltage Drop and Short Circuit

Voltage Drop and Short Circuit

Voltage Drop and Short Circuit

Three of the everyday calculations of a building electrical design under one select: the voltage drop of a feeder or branch circuit, the available fault current at every bus of a radial low-voltage system, and the full-load, inrush and locked-rotor currents of transformers and motors. All three are closed-form methods on the circuit equations. The values a code or a vendor owns (allowable ampacities, raceway reactances, interrupting ratings, code-letter kVA per hp) are entered inputs with sourced defaults, and the page says where the analyst reads them. The conductor data come from the NBS copper and aluminum wire tables through the Lighting and Circuit Sizing library, which also holds the branch-circuit version of the drop and fault estimate; this tool adds parallel conductors, the sending-end solve, motor starting, the radial system with motor contribution, sequence impedances and the asymmetry factors.

Conductor impedance

The cross section follows from the American Wire Gage definition (diameter 5 × 92^((36 − n)/39) mils, area its square in circular mils) or the kcmil designation. The resistance of one conductor is ρ₂₀ (1 + α₂₀ (T − 20)) / cmil per foot with ρ₂₀ = 10.371 Ω·cmil/ft and α₂₀ = 0.00393 per °C for annealed copper of 100 percent IACS conductivity (NBS Handbook 100), 17.002 Ω·cmil/ft and 0.00403 per °C for EC-H19 aluminum of 61 percent conductivity (NBS Handbook 109), at the entered operating temperature (75 °C by default, the usual ampacity column; a lower temperature gives a higher fault current), with the 2 percent lay allowance of a stranded conductor. The reactance of one conductor is entered (the NEC Chapter 9 Table 9 value for the raceway) or, when the field is blank, estimated from the spacing: X = 2π f (μ₀/2π) ln(D / (0.7788 r)) per unit length, the two-wire line formula with the geometric mean radius of a solid round conductor, times 1.25 for a steel raceway (the ratio of the steel to the non-magnetic column of that table, a single stated fact). Parallel conductors of one size divide the run's resistance and reactance by their count; the NEC allows parallel sets of 1/0 AWG and larger (310.10(G)).

Voltage drop

With the load voltage held at the nominal value V and the load current I lagging by φ, the sending-end voltage is the phasor sum V_s = V + I (cos φ − j sin φ)(R + jX) over the loop: both conductors for a single-phase two-wire circuit, one conductor with the line-to-neutral voltage and the √3 factor for three-phase. The exact drop is |V_s| − V, and the page reports it as a percentage of the nominal voltage against the entered limit. The NEC informational notes recommend 3 percent for a branch circuit or feeder and 5 percent for both together (210.19, 215.2); UFC 3-501-01 3-2.7 sets 2 percent for feeders and 3 percent for branch circuits at rated capacity. Three other forms are printed beside the exact drop. The IEEE Std 141 approximation keeps the in-phase part, I (R cos φ + X sin φ) times 2 or √3, which is within a fraction of a percent of the exact value at ordinary angles; the error is printed. The resistive form drops X. TM 5-811-1 Figure 5-2 writes the same approximation as kVA (R cos φ + X sin φ) / (10 kV²) percent with three-phase kVA and line-to-line kV (twice that for single-phase kVA); its worked example, 15,000 kVA at 13.8 kV and 0.90 power factor through one mile of 4/0 copper at 0.278 + j0.674 Ω, gives 4.28 percent and is reproduced by the test script. Holding the sending end at the nominal value instead and solving the quadratic |V_r + I(cos φ − j sin φ)(R + jX)| = V for the receiving-end magnitude gives the drop the load actually sees, which differs from the first form only in the third digit for ordinary circuits but matters for the dip at motor starting. The drop at starting is the same phasor sum with the locked-rotor current at its low power factor (0.2 to 0.4 for induction motors), where the reactive term I X sin φ dominates; TM 5-811-1 5-4 treats motor starting separately from the running-load allocation, so the value is reported, not checked against the running limit.

The smallest catalogue size that meets the limit by the exact drop with the entered parallel count is reported, and a table and two charts give the drop of every size and the drop of the entered size against the length. Ampacity: the table value entered (NEC Table 310.16, a code value for one conductor at its temperature rating and 30 °C ambient) times the parallel count is corrected for ambient by √((T_c − T_a)/(T_c − 30)), the relation that follows from I²R heating against a fixed temperature rise, and by the entered adjustment factor for more than three current-carrying conductors (310.15(C)); the corrected ampacity must carry the load and the table ampacity must reach the design current, 125 percent of a continuous load (210.19, 215.2).

Short-circuit current of a radial system

The ohmic method of IEEE Std 141, written in complex arithmetic on the secondary voltage. The transformer impedance on its own base is Z_t = (%Z/100) V²/S, split into R and X by the nameplate or test X/R; the utility is the impedance V²/S_sc of its available short-circuit MVA referred to the secondary, split by its X/R (TM 5-811-1 Figure 4-1 converts MVA to kA at the primary as MVA/(√3 kV): 1,200 MVA at 69 kV gives 10 kA); an infinite bus is the limit of no utility impedance, where the terminal current is the transformer full-load current times 100/%Z. The connected motors are a source too: an induction motor group is represented by its subtransient reactance on its own kVA, 0.25 per unit in the TM 5-811-14 examples (a contribution of four times the motor full-load current), with an entered X/R. At the main bus the Thevenin impedance is the utility-transformer branch in parallel with the motor branch, and the bolted three-phase current is V_ph / |Z₁|; each run then adds its impedance (R and X of one conductor over the parallel count, times the length) and the current at the end of run k is V_ph / |Z₁(k)|. The page prints the per-unit values on the transformer base for the main bus (base ohms kV² · 1000 / kVA, base current kVA / (√3 kV)), which is how TM 5-811-14 Appendix G works its Example 5: the per-unit conversions of its equations G-14 to G-42 and the reduction of its Figure G-21 at the MP 4 bus (I_base = 2,405.6 A at 480 V on 2,000 kVA, X″ = 0.249 per unit, I″ = 9,661 A) are reproduced by the test script.

The line-to-ground fault uses the sequence networks: I = 3 V_ph / |Z₁ + Z₂ + Z₀| with Z₂ = Z₁ and Z₀ built from the entered ratios, the transformer's Z₀/Z₁ (1.0 for a delta-wye solidly grounded unit, whose delta primary blocks the utility's zero sequence) plus the runs' Z₀/Z₁ along the path; the motors are ungrounded and contribute nothing to the zero-sequence network. With both ratios at 1.0 the line-to-ground current at the transformer terminals equals the three-phase value, the classic result for a delta-wye unit, and falls below it along the runs where the ground-return impedance adds.

Asymmetry. A fault in an R-L circuit at the zero of the voltage wave carries a dc offset that decays with the time constant τ = (X/R)/(2π f); the first-cycle peak is √2 (1 + e^(−π R/X)) times the symmetrical rms, and the rms current at half a cycle, the momentary duty, is √(1 + 2 e^(−2π R/X)) times it (the total rms with the full offset present). TM 5-811-14 2-8 describes the waveforms and its equation 4-3 uses the fixed factor 1.6 on a motor's locked-rotor current, which the closed form gives at X/R ≈ 25; its paragraph 3-5f notes that a device's symmetrical rating is tested at a standard X/R and must be derated above it. The tool applies the ratio of the half-cycle asymmetry factors at the bus X/R and at the entered test X/R (6.6, power factor 0.15, for molded-case breakers rated above 20 kA under UL 489; 4.9 for 10 to 20 kA) to the entered device rating and compares the derated rating with the bus current; UFC 3-520-01 3-2.11 sets the minimum ratings at 10 kA symmetrical for 240 V and below and 14 kA for 480 V. The point-to-point form, f = |Z_run| I_prev / V_ph, M = 1/(1 + f), I = M I_prev, adds the run impedance to the upstream one as a magnitude (the published C constant of that method is 1/|z| per unit length); it is printed for comparison and runs close to the complex value because the runs are mostly resistive and the upstream X/R falls along the feeder. The conductor withstand, (I/A)² t = K log₁₀((T₂ + C)/(T₁ + C)) with A in circular mils, K = 0.0297 and C = 234 for copper (0.0125 and 228 for aluminum), the adiabatic form of ICEA P-32-382, is evaluated from the operating temperature to 200 °C at the entered clearing time for a fault at the end of each run. A chart gives the fault current along the feeder and a sketch the one-line with the currents at every bus. Arc-flash incident energy and boundaries (IEEE Std 1584, NFPA 70E) are not computed: the fault currents here are the inputs such a study starts from.

Transformer and motor currents

Transformer full-load current I = S / (√3 V) on each side with the line-to-line voltages (S / V single-phase); the magnetizing inrush is the entered multiple of the full-load current, 8 to 12 times for 0.1 second being the coordination value of TM 5-811-14 4-6c. Motor input power is hp × 745.7 W divided by the efficiency, the full-load current P_in / (√3 V pf) with the nameplate efficiency and power factor (this replaces the NEC 430.250 table, which is a conservative tabulation for the same purpose); 125 percent of it is the branch-circuit conductor minimum (430.22) and 250 percent the maximum inverse-time breaker setting (430.52). The locked-rotor current follows from the code-letter kVA per hp of NEC Table 430.7(B): LRA = (kVA/hp) hp / (√3 V), the form of TM 5-811-14 equation G-1 (code letter G, 6.29 kVA/hp, 15 hp at 460 V: 119 A). The manual's own approximations for coordination, kVA ≈ hp, LRA = FLA / X″ with X″ = 0.28 and the 1.6 first-cycle factor (equations 4-1 to 4-3, 200 hp at 480 V: 241, 861 and 1,378 A), are printed beside for comparison.

Limits

Balanced three-phase loads and faults on a radial system with one transformer; no loops, no second source, no generators (a generator is entered as part of the utility or as a motor-like branch at its own reactance), no current-limiting devices, no arc impedance; motors lumped at the main bus; the conductor temperature entered rather than computed; the sequence ratios entered rather than derived from the raceway and grounding geometry. Single-phase systems are handled for voltage drop only; the fault sub-tool is three-phase. The asymmetry factors are the closed forms for the worst point on the wave and the device derating a ratio of them, which is the preliminary practice of IEEE Std 141; a coordination study follows IEEE Std 242 with the manufacturers' curves.