2.7.1 System Grounding

System and circuit conductor grounding is an intentional connection between the electric system conductors and grounding electrodes that provides an effective connection to ground (earth). The basic objectives being sought are to

• limit overvoltages due to lightning,

• limit line surges, or unintentional contact with high-voltage lines,

• stabilize the voltage to ground during normal operation, and

• facilitate overcurrent device operation in case of ground faults.

Definitions

• Solidly grounded: no intentional grounding impedance.

• Effectively grounded: grounded through a grounding connection of sufficiently low impedance (inherent or intentionally added or both) such that ground faults that may occur cannot build up voltages in excess of limits established for apparatus, circuits, or systems so grounded. R₀ < X₁, X₀ ≤ 3X₁, where X₁ is the leakage reactance, and X₀ and R₀ are the grounding reactance and resistance, respectively.

• Reactance grounded: X₀ ≤ 10X₁.

• Resistance grounded: R₀ ≥ 2X₀.

• High-resistance grounded: ground fault up to 10 A is permitted.

• Grounded for serving line-to-neutral loads: Z₀ ≤ Z₁.

• Ungrounded: no intentional system grounding connection.

Service Continuity

For many years a great number of industrial plant distribution systems have been operated ungrounded (Fig. 2.45) at various voltage levels. In most cases this has been done with the thought of gaining an additional degree of service continuity. The fact that any one contact occurring between one phase of the system and ground is unlikely to cause an immediate outage to any load may represent an advantage in many plants (e.g., submarines), varying in its importance according to the type of plant.

Grounded Systems

In most cases systems are designed so that circuit protective devices will remove a faulty circuit from the system regardless of the type of fault. A phase-to-ground fault generally results in the immediate isolation of the faulted circuit with the attendant outage of the loads on that circuit (Fig. 2.46). However, experience has shown in a number of systems that greater service continuity may be obtained with grounded-neutral than with ungrounded-neutral systems.

Multiple Faults to Ground

Although a ground fault on one phase of an ungrounded system generally does not cause a service interruption, the occurrence of a second ground fault on a different phase before the first fault is cleared does result in an outage (Fig. 2.47). If both faults are on the same feeder, that feeder will be opened. If the second fault is on a different feeder, both feeders may be deenergized.

The longer a ground fault is allowed to remain on an ungrounded system, the greater is the likelihood of a second ground fault occurring on another phase resulting in an outage. The advantage of an ungrounded system in not immediately dropping load upon the occurrence of a ground fault may be largely destroyed by the practice of ignoring a ground fault until a second one occurs and repairs are required to restore service. With an ungrounded system it is extremely important that an organized maintenance program be provided so that first ground fault is located and removed as soon as possible after its occurrence.

Safety

Many of the hazards to personnel and property existing in some industrial electrical systems are the result of poor or nonexisting grounding of electrical equipment and metallic structures. Although the subject of equipment grounding is treated in the next section, it is important to note that regardless of whether or not the system is grounded, safety considerations require thorough grounding of equipment and structures.

Proper grounding of a low-voltage (600 V or less) distribution system may result in less likelihood of accidents to personnel than leaving the system supposedly ungrounded. The knowledge that a circuit is grounded generally will result in greater care on the part of the workman.

It is erroneous to believe that in an ungrounded system a person may contact an energized phase conductor without personal hazard. As Fig. 2.48 shows, an ungrounded system with balanced phase-to-ground capacitance has rated line-to-neutral voltages existing between any phase conductor and ground. To accidentally or intentionally contact such a conductor may present a serious, perhaps lethal shock hazard in most instances.

During the time a ground fault remains on one phase of an ungrounded system, personnel contacting one of the other phases and ground are subjected to a voltage 1.73 (that is, $\sqrt{3}$) times that which would be experienced on a solidly neutral-grounded system. The voltage pattern would be the same as the case shown in Fig. 2.48c.

Power System Overvoltages

Some of the more common sources of overvoltages on a power system are the following:

• lightning (see Application Examples 2.9 and 2.10),

• switching surges,

• capacitor switching,

• static,

• contact with high-voltage system,

• line-to-ground faults,

• resonant conditions, and

• restriking ground faults.

Lightning

Most industrial systems are effectively shielded against direct lightning strikes (Fig. 2.49). Many circuits are either underground in ducts or within grounded metal conduits or raceways. Even open-wire circuits are often shielded by adjacent metallic structures and buildings. Large arresters applied at the incoming service limit the surge voltages within the plant resulting from lightning strikes to the exposed service lines. Other arrester applications may be necessary within the plant to protect low impulse strength devices such as rotating machines and computers.

Switching Surges

Normal switching operation in the system (e.g., capacitor switching) can cause overvoltages. These are generally not more than three times the normal voltage and of short duration.

2.7.1.2 Application Example 2.9: Propagation of a Surge through a Distribution Feeder with an Insulator Flashover

This example illustrates the performance of a transmission system when the current of a lightning strike is large enough to cause flashover of tower insulator strings [57]. A lightning strike with the rise time τ₁ = 1.0 μsec, the fall time τ₂ = 5.0 μs, and the peak magnitude of current waveform I₀ = 15.4 kA strikes a 138 kV transmission line at node 1 (see Fig. E2.9.1). The span lengths of this transmission line are nonuniform and are given by NL (normalized length). Both ends of the transmission line are terminated by Z₀ = 300 Ω.

The transmission line has steel towers with 10 porcelain insulators. The tower-surge impedance consists of a 20 Ω resistance in series with 20 μH of inductance. The model used to simulate the first two insulators is shown in Fig. E2.9.1. Note that the transmission line can be simulated by the PSpice (lossless) transmission line model T, where Z₀ = 300 Ω, f = 60 Hz, and NL as indicated in Fig. E2.9.1. The current-controlled switch can be simulated by W, where I_ON = 1 A.

a) Calculate the four span lengths in meters.

b) Calculate (using PSpice) and plot the transmission line voltage (in MV) as a function of time (in μs) at the point of strike, at the first insulator, and at the second insulator.

Solution to Application Example 2.9

a) The normalized length of a distance traveled by a wave (e.g., span) NL is a function of the frequency f, the span length l, and the speed of light c = 2.99792 · 10⁸ m/s [58]: $NL=\frac{l}{c}f$ or the length of a span is $l=c\frac{NL}{f}$. The time to travel the distance NL · λ, where λ is the wavelength, is $TD=\frac{NL}{f}=\frac{l}{c}=NL\cdot \lambda$.
Length of span #1: l₁ = 2500 m, length of span #2: l₂ = 500 m, length of span #3: l₃ = 1000 m, length of span #4: l₄ = 1000 m.

b) The PSpice program is listed in Table E2.9.1. One obtains the results of Figs. E2.9.2 and E2.9.3.

2.7.1.3 Application Example 2.10: Lightning Arrester Operation

This problem illustrates the performance of a lightning arrester [57]. The same basic parameters as for the lightning strike employed in Application Example 2.9 are used in this example. The only changes are in the value of I₀ (in this problem a value of 7.44 kA is assumed) and in the voltage of the second lightning arrester (V_D2 = 600 kV). This value is small enough that flashover of the insulator string will not occur; however, it is large enough to cause the lightning arrester to operate due to V_D2 = 600 kV. The system being simulated is shown in Fig. E2.10.1.

a) Calculate and plot the transmission line voltage (in MV) as a function of time (in μs) at the point of strike, at the insulator, and at the lightning arrester.

b) Calculate and plot the currents at the point of strike, at the insulator, and at the lightning arrester (in kA) as a function of time (in μs).

Solution to Application Example 2.10

a) Calculations are performed using PSpice, and the plots of the transmission line voltage (in MV) as a function of time (in μs) at the point of strike, at the insulator, and at the lightning arrester are shown in Figs. E2.10.2 and E2.10.3. The PSpice program is listed in Table E2.10.1.

b) One obtains the results of Figs. E2.10.4 and E2.10.5 using the PSpice program codes of Table E2.10.1. This is a good example where the various versions of PSpice have different convergence properties similar to the Mathematica software (see Application Example 8.5).

2.7.2 Equipment Grounding

All exposed metal parts of electrical equipment (including generator frames, mounting bases, electrical instruments, and enclosures) should be bonded to a grounding electrode.

2.7.3 Static Grounding

The British standard [59] regarding grounding states that the most common source of danger from static electricity is the retention of charge on a conductor, because virtually all the stored energy can be released in a single spark to earth or to another conductor. The accepted method of avoiding the hazard is to connect all conductors to each other and to earth via electrical paths with resistances sufficiently low to permit the relaxation of the charges. This was stated in the original British standard (1980) and is carried out worldwide [60].

2.7.4 Connection to Earth

Resistance to Earth. The most elaborate grounding system that can be designed may prove to be inadequate unless the connection of the system to the earth is adequate and has a low resistance. It follows, therefore, that the earth connection is one of the most important parts of the whole grounding system. It is also the most difficult part to design and to obtain. The perfect connection to earth should have zero resistance, but this is impossible to obtain. Ground resistance of less than 1 Ω can be obtained, although this low resistance may not be necessary in many cases. Since the desired resistance varies inversely with the fault current to ground, the larger the fault current, the lower must be the resistance [61,62]. Figure 2.50a illustrates the deformation of grounding conductor due to large lightning currents.

2.7.5 Calculation of Magnetic Forces

Considerable work [63,64] has been done concerning the calculation of electromagnetic forces based on closed-form analysis. These methods can be applied to simple geometric configurations only. In many engineering applications, however, three-dimensional geometric arrangements must be analyzed. Based on the theory developed by Lawrenson [65] filamentary conductor configurations can be analyzed. Figures 2.50a,b illustrate the force density distribution for a filamentary 90° bend at a current of I = 100 kA. Three-dimensional arrangements are investigated in [66]. Force calculations based on the so-called Maxwell stress are discussed in Chapter 4.

2.8.2 A 4.5 kVA Three-Phase Transformer Bank #1 Feeding Full-Wave Rectifier

A 4.5 kVA, 240 V/240 V, Δ–Δ connected three-phase transformer (Fig. 2.51) consisting of three single-phase transformers (bank #1, Appendix 3.2) is used to feed a full-wave diode rectifier (see Appendix 3.5) with an LC filter connected across the resistive load (see Fig. 6-5 of [75]). In Table 2.3, measured data are compared with those of linear load condition. The measured wave shapes of input voltage (v_AC), exciting current (i_A – i_aʹ), series voltage drop (v_AC – v_acʹ), and output current (i_a‘) of phase A are shown in Fig. 2.54a,b for linear and nonlinear load conditions. The total harmonic distortion THD_i, K-factor, and harmonic components of the output current are listed in Table 2.4.

Table 2.3

Measured Data of 4.5 kVA Transformer Bank #1

	Linear load (Fig. 2.54a)	Nonlinear load (Fig. 2.54b)
vAC	237.7 Vrms	237.7 Vrms
$i_A-i_a^{\prime }$	1.10 Arms	1.12 Arms
$v_{\mathit{AC}}-v_{ac}^{\prime }$	8.25 Vrms	8.19 Vrms
ia′	11.09 Arms	10.98 Arms
Iron-core loss	100.7 W	103.9 W
Copper loss	161.3 W	157.6 W
Total loss	262.0 W	261.5 W
Output power	4532 W	4142 W
Efficiency	94.55%	94.09%

Table 2.3 compares measured data and shows that the transformer is operated at nonlinear load with about the same losses occurring at linear load (261.3 W). With the apparent power derating definition

$\mathit{deratin}{g}_S=\frac{V_{rms}{I}_{rms}}{V_{rms}^{\mathrm{rated}}{I}_{rms}^{\mathrm{rated}}}\text{,}$

the derating at the nonlinear load of Table 2.3 is 99%. The real power delivered to the nonlinear load is 91.4% of that supplied at linear load.

2.8.3 A 4.5 kVA Three-Phase Transformer Bank #2 Supplying Power to Six-Step Inverter

A 4.5 kVA transformer bank #2 (Appendix 3.3) supplies power to a half-controlled six-step inverter (Appendix 3.6), which in turn powers a three-phase induction motor. The motor is controlled by adjusting the output current and frequency of the inverter. The transformer is operated at rated loss at various motor speeds. Rated loss of bank #2 is determined by a linear (resistive) load at rated operation. The iron-core and copper losses are measured separately and are listed in Table 2.5a. Measured wave shapes of input voltage, exciting current, series voltage drop, and output current of phase A are shown in Fig. 2.55a,b for linear and nonlinear conditions. The output current includes both odd and even harmonics due to the half-controlled input rectifier of the six-step inverter. Dominant harmonics of input voltage and output current are listed for different motor speeds in Table 2.5b. The total harmonic distortion of the input voltage and output current as well as the K-factor are listed in Table 2.5c for the speed conditions of Tables 2.5a and 2.5b.

Table 2.5b

Input Voltage (in rms Volts) and Output Current (in rms Amperes) Harmonics of Phase A (Fig. 2.55a,b)

h		1	2	4	5	7	8	10
Linear load	vACh	236.9	0.42	0.48	2.96	1.48	0.20	0.40
	${i_{ah}}^{\prime }$	11.24	0.04	0.02	0.12	0.08	0.00	0.00
100% speed	vACh	237.2	1.37	0.49	2.88	1.58	0.06	0.30
	${i_{ah}}^{\prime }$	9.05	5.91	1.08	0.56	0.30	0.32	0.20
90% speed	vACh	237.4	1.58	0.53	2.57	1.60	0.10	0.20
	${i_{ah}}^{\prime }$	9.06	6.83	1.87	0.54	0.54	0.16	0.30
80% speed	vACh	237.5	1.67	0.43	2.91	1.63	0.02	0.30
	${i_{ah}}^{\prime }$	8.78	6.82	2.47	0.69	0.67	0.43	0.20
70% speed	vACh	237.6	1.67	0.41	2.78	1.50	0.13	0.20
	${i_{ah}}^{\prime }$	8.64	6.91	2.82	0.99	0.68	0.60	0.20
60% speed	vACh	237.3	1.71	0.28	3.14	1.21	0.14	0.20
	${i_{ah}}^{\prime }$	8.59	6.90	3.20	1.29	0.68	0.74	0.20
50% speed	vACh	237.8	1.73	0.35	3.18	1.30	0.21	0.00
	${i_{ah}}^{\prime }$	8.09	6.57	3.52	1.67	0.54	0.79	0.50
40% speed	vACh	237.7	1.73	0.47	3.08	1.67	0.30	0.30
	${i_{ah}}^{\prime }$	7.78	6.66	3.70	2.17	0.48	0.82	0.80

2.8.4 A 15 kVA Three-Phase Transformer Supplying Power to Resonant Rectifier

A 15 kVA, 240 V/240 V, Δ–Δ connected three-phase transformer (Appendix 3.4) is used to supply power to a resonant rectifier [12] (Appendix 3.7). The transformer is operated with the resonant rectifier load at the same total loss generated by a three-phase linear (resistive) load. Measured data are compared in Table 2.6. Measured wave shapes of input voltage, exciting current, series voltage drop, and output current of phase A are depicted in Fig. 2.56a,b. The fundamental phase shift between output transformer line-to-line voltage v_acʹ and phase current i_aʹ of the resonant rectifier is 67.33°, and the output displacement factor (within transformer phase) is therefore cos(67.33° – 30°) = 0.795. The fundamental phase shift between output line-to-line voltage and phase current for the linear resistive load is 30.95°, and the output displacement factor (within transformer phase) is, therefore, cos(30.95° – 30°) = 1.0. Note that the wave shapes of the output currents of Fig. 2.56a,b are about sinusoidal.

Table 2.6

Measured Data of 15 kVA Three-Phase Transformer with Resonant Rectifier Load (Fig. 2.56a,b)

	Linear load	Resonant rectifier
vAC	215.7 Vrms	215.6 Vrms
$\left(i_A-{i_a}^{\prime }\right)$	2.39 Arms	2.59 Arms
$\left(v_{\mathit{AC}}-{v_{ac}}^{\prime }\right)$	4.42 Vrms	4.8 Vrms
${i_a}^{\prime }$	27.00 Arms	26.94 Arms
Iron-core loss	42.0 W	49.8 W
Copper loss	191.9 W	192.2 W
Total loss	233.9 W	242 W
Output power	10015 W	7859 W
Efficiency	97.72%	97.01%

If the transformer with the resonant rectifier load is operated (see Table 2.6) at about the same loss as linear load (233.9 W), the output current of the transformer with the resonant rectifier load is 26.94 A, which corresponds to the copper loss of 192.2 W = (242 W – 49.8 W). Therefore, the transformer apparent power derating of the transformer for the nonlinear load is 99.7%. The real power supplied to the nonlinear load is 78.5% of that of the linear load.

2.8.5 A 15 kVA Three-Phase Transformer Bank Absorbing Power from a PWM Inverter

The same transformer of Section 2.8.4 absorbs power from a PWM inverter [12] (Appendix 3.8) and supplies power to a resistive load (case #1) and to a utility system (case #2). The transformer losses are also measured when supplying a linear resistive load fed from sinusoidal power supply (linear load). All measured data are compared in Table 2.7a.

Table 2.7a

Measured Data of 15 kVA Three-Phase Transformer Connected to PWM Inverter (Fig. 2.57a,b,c)

	Linear load of transformer	Case #1	Case #2
vAC	234.2 Vrms	234.4 Vrms	248.2 Vrms
$\left(i_A-{i_a}^{\prime }\right)$	2.93 Arms	3.30 Arms	3.82 Arms
$\left(v_{\mathit{AC}}-{v_{ac}}^{\prime }\right)$	3.56 Vrms	3.86 Vrms	4.28 Vrms
${i_a}^{\prime }$	32.4 Arms	33.3 Arms	28.2 Arms
Iron-core loss	232.2 W	254.6 W	280.5 W
Copper loss	177.1 W	172.6 W	130.4 W
Total loss	409.3 W	427.2 W	410.9 W
Output power	13337 W	13544 W	12027 W
Efficiency	97.02%	96.94%	96.07%

Measured wave shapes of input voltage, exciting current, series voltage drop, and output current of phase A are shown in Fig. 2.57a,b,c. The total harmonic distortion THD_i, K-factor, and harmonic amplitudes of the transformer output current are listed in Table 2.7b.

Figure 2.58 summarizes the total harmonic distortion (THD_i), apparent power (kVA) derating, and real power (kW) derating for uncontrolled (a, b), half-controlled (c), and controlled (d, e) converter loads of transformers. In particular the graphs of Fig. 2.58 can be identified [75] as follows:

a) 25 kVA single-phase pole transformer [7,11,15,75,76] feeding uncontrolled full wave rectifier load,

b) 4.5 kVA three-phase transformer feeding uncontrolled full wave rectifier load,

c) 4.5 kVA three-phase transformer feeding half-controlled rectifier load,

d) 15 kVA three-phase transformer absorbing power from PWM inverter (14 kW), and

e) 15 kVA three-phase transformer feeding resonant rectifier load (8 kW).

Calculate the currents flowing in all branches as indicated in Fig. P2.1 if the pole transformer is loaded as shown. For the pole transformer you may assume ideal transformer relations. Note that such a single-phase configuration is used in the pole transformer (mounted on a pole behind your house) supplying residences within the United States with electricity. What are the real (P), reactive (Q), and apparent (S) powers supplied to the residence?

Problem 2.2 Analysis of an Inductor with Three Coupled Windings

a) Draw the magnetic equivalent circuit of Fig. P2.2.

b) Determine all self- and mutual inductances. Assuming that the winding resistances are known draw the electrical equivalent circuit of Fig. P2.2.

Problem 2.3 Derating of Transformers Based on K-Factor

A substation transformer (S = 10 MVA, V_L-L = 5 kV, %P_EC-R = 10) supplies a six-pulse rectifier load with the harmonic current spectrum as given by Table X of the paper by Duffey and Stratford [82].

a) Determine the K-factor of the rectifier load.

b) Calculate the derating of the transformer for this nonlinear load.

Problem 2.4 Derating of Transformers Based on F_HL-Factor

A substation transformer (S = 10 MVA, V_L-L = 5 kV, %P_EC-R = 10) supplies a six-pulse rectifier load with the harmonic current spectrum as given by Table X of the paper by Duffey and Stratford [82].

a) Determine the F_HL-factor of the rectifier load.

b) Calculate the derating of the transformer for this nonlinear load.

Problem 2.5 Derating of Transformers Based on K-Factor for a Triangular Current Wave Shape

a) Show that the Fourier series representation for the current waveform in Fig. P2.5 is

$i\left(t\right)=\frac{8I_m}{{\pi }^2}{\displaystyle \sum _{h=1,3,5,\dots }^{\infty }\left(\frac{1}{h^2}\right)}sin\left(\frac{h\pi }{2}\right)sinh{\omega }_{0}t\text{.}$

b) Approximate the wave shape of Fig. P2.5 in a discrete manner, that is, sample the wave shape using 36 points per period and use the program of Appendix 2 to find the Fourier coefficients.

c) Based on the Nyquist criterion, what is the maximum order of harmonic components h_max you can compute in part b? Note that for h_max the approximation error will be a minimum.

d) Calculate the K-factor and the derating of the transformer when supplied with a triangular current wave shape, and %P_EC-R = 10. The rated current I_R corresponds to the fundamental component of the triangular current.

Problem 2.6 Derating of Transformers Based on K-Factor for a Trapezoidal Current Wave Shape

a) Show that the Fourier series representation for the current waveform in Fig. P2.6 is

$i\left(t\right)=\frac{12I_m}{{\pi }^2}{\displaystyle \sum _{h=1,3,5,\dots }^{\infty }\left(\frac{1}{h^2}\right)}sin\left(\frac{h\pi }{3}\right)sinh{\omega }_{0}t\text{.}$

b) Approximate the wave shape of Fig. P2.6 in a discrete manner, that is, sample the wave shape using 36 points per period and use the program of Appendix 2 to find the Fourier coefficients.

c) Based on the Nyquist criterion, what is the maximum order of harmonic components h_max you can compute in part b? Note that for h_max the approximation error will be a minimum.

d) Calculate the K-factor and the derating of the transformer when supplied with a trapezoidal current wave shape, and % P_EC-R = 10. The rated current I_R corresponds to the fundamental component of the trapezoidal current.

Problem 2.7 Force Calculation for a Superconducting Coil for Energy Storage

Design a magnetic storage system which can provide for about 6 minutes a power of 100 MW, that is, an energy of 10 MWh. A coil of radius R = 10 m, height h = 6 m, and N = 3000 turns is shown in Fig. P2.7. The magnetic field strength H inside such a coil is axially directed, essentially uniform, and equal to H = Ni/h. The coil is enclosed by a superconducting box of 2 times its volume. The magnetic field outside the superconducting box can be assumed to be negligible and you may assume a permeability of free space for the calculation of the stored energy.

a) Calculate the magnetic field intensity H and the magnetic flux density B of the coil for constant current i = I₀ = 8000 A.

b) Calculate the energy E_stored stored in the magnetic field (either in terms of H or B).

c) Determine the radial pressure p_radial acting on the coil.

d) Devise an electronic circuit that enables the controlled extraction of the energy from this storage device. You may use bidirectional superconducting switches S₁ and S₂, which are controlled by a magnetic field via the quenching effect. S₁ is ON and S₂ is OFF for charging the superconducting coil. S₁ is OFF and S₂ is ON for the storage time period. S₁ is ON and S₂ is OFF for discharging the superconducting coil.

Problem 2.8 Measurement of Losses of Delta/Zigzag Transformers

Could the method of Section 2.8 also be applied to a Δ/zigzag three-phase transformer as shown in Fig. P2.8? This configuration is of interest for the design of a reactor [67] where the zero-sequence (e.g., triplen) harmonics are removed or mitigated, and where through phase shifting the 5th and 7th harmonics [83] are canceled.

Problem 2.9 Error Calculations

For a 25 kVA single-phase transformer of Fig. P2.9 with an efficiency of 98.46% at rated operation, the total losses are P_loss = 390 W, where the copper and iron-core losses are P_cu = 325 W and P_fe = 65 W, respectively. Find the maximum measurement errors of the copper P_cu and iron-core P_fe losses as well as that of the total loss P_loss whereby the instrument errors are given in Table P2.9.

Table P2.9

Instruments and Their Error Limits of Fig. P2.9

Instrument	Full scale	Full-scale errors
PT1	240 V/240 V	PT1 = 0.24 V
PT2	7200 V/240 V	PT2 = 0.24 V
CT1	100 A/5 A	CT1 = 5 mA
CT2	5 A/5 A	CT2 = 5 mA
V1	300 V	V1 = 0.3 V
V2	300 V	V2 = 0.3 V
A1	5 A	A1 = 5 mA
A2	5 A	A2 = 5 mA

Problem 2.10 Delta/Wye Three-Phase Transformer Configuration with Six-Pulse Midpoint Rectifier Employing an Interphase Reactor

a) Perform a PSpice analysis for the circuit of Fig. P2.10, where a three-phase thyristor rectifier with filter (e.g., capacitor C_f) serves a load (R_load). You may assume ideal transformer conditions. For your convenience you may assume (N_1/N₂) = 1, R_syst = 0.01 Ω, X_syst = 0.05 Ω @ f = 60 Hz, $v_{\mathit{AB}}\left(t\right)=\sqrt{2}600Vcos\omega t$, $v_{BC}\left(t\right)=\sqrt{2}600\mathrm{Vcos}\left(\omega t–120°\right)$, and $v_{CA}\left(t\right)=\sqrt{2}600Vcos\left(\omega t–240°\right)$. Each of the six thyristors can be modeled by a self-commutated electronic switch and a diode in series, as is illustrated in Fig. E1.1.2 of Application Example 1.1 of Chapter 1. Furthermore, C_f = 500 μF, L_left = 5 mH, L_right = 5 mH, and R_load = 10 Ω. You may assume that the two inductors L_left and L_right are not magnetically coupled, that is, L_leftright = 0. Plot one period of either voltage or current after steady state has been reached for firing angles of α = 0°, 60°, and 120°, as requested in parts b to e. Print the input program.

b) Plot and subject the line-to-line voltages v_AB (t), v_ab(t), and v′_ab(t) to Fourier analysis. Why are they different?

c) Plot and subject the input line current i_AL (t) of the delta primary to a Fourier analysis. Note that the input line currents of the primary delta do not contain the 3rd, 6th, 9th, 12th,…, that is, harmonic zero-sequence current components.

d) Plot and subject the phase current i_Aph(t) of the delta primary to a Fourier analysis. Note that the phase currents of the primary delta contain the 3rd, 6th, 9th, 12th,…, that is, harmonic zero-sequence current components.

e) Plot and subject the output voltage v_load(t) to a Fourier analysis.

f) How do your results change if the two inductors L_left and L_right are magnetically coupled, that is, L_leftright = 4.5 mH?

Problem 2.11 Determination of Measurement Uncertainty

Instead of maximum errors, the U.S. Guide to the Expression of Uncertainty in Measurement [84] recommends the use of uncertainty. As an example the uncertainties and the associated correlation coefficients of the simultaneous measurement of resistance and reactance of an inductor having the inductance L and the resistance R are to be calculated. The five measurement sets of Table P2.11 are available for an inductor consisting of the impedance Z = R + jX, where terminal voltage |Ṽ_t|, current |Ĩ_t|, and phase-shift angle Φ are measured.

Table P2.11

Measurement Values of the Input Quantities |_t|, |Ĩ_t|, and Φ Obtained from Five Sets of Simultaneous Observations

	Input quantities
Set number k	$\left|{\tilde{\mathit{V}}}_{\mathit{t}}\right|$ (V)	$\left|{\tilde{\mathit{I}}}_{\mathit{t}}\right|$ (mA)	Φ (radians)
1	5.006	19.664	1.0457
2	4.993	19.638	1.0439
3	5.004	19.641	1.0468
4	4.991	19.684	1.0427
5	4.998	19.679	1.0433

With R = ($\left|{\tilde{V}}_t\right|$/ $\left|{\tilde{I}}_t\right|$)cosΦ, X = ($\left|{\tilde{V}}_t\right|$/ $\left|{\tilde{I}}_t\right|$)sinΦ, Z = ($\left|{\tilde{V}}_t\right|$/ $\left|{\tilde{I}}_t\right|$), and Z² = R² + X² one obtains two independent output quantities – called measurands.

a) Calculate the arithmetic means of $\left|{\tilde{V}}_t\right|$, $\left|{\tilde{I}}_t\right|$, and Φ.

b) Determine the experimental standard deviation of mean of s($\overline{V}$), s(Ī), and s($\overline{\mathrm{\Phi }}$)

c) Find the correlation coefficients r(|Ṽ_t|, |Ĩ_t|), $r\left(\left|{\tilde{V}}_t\right|,\mathrm{\Phi }\right)\text{,}$ and $r\left(\left|{\tilde{I}}_t\right|,\mathrm{\Phi }\right)$.

d) Find the combined standard uncertainty μ ≡ u_c(y_ℓ) of result of measurement where ℓ is the measurand index.

e) Determine the correlation coefficients r(R/X), r(R/Z), and r(X/Z).