7 presents the principles of operation of the brushless dc motor (recently matured technology) and the commutated reluctance motor (emerging technology). The spread of electric machine applications closely followed the expansion of electric utility networks.

## CHAPTER 1 INTRODUCTION 3

Each of the quantities in brackets [2.3] is a complex number known as a phasor associated with the time functions vl(t) and v,(t). It is obvious from 12.31 that the summation of the phasors V and V2 allows the representation of the sum v,(t) + v2(t) as a single sinusoidal time function.

The transition from [2.2] to [2.3] used the real part of the sum of two complex numbers to be equal to the sum of the real parts and the law of exponents. The development could have been carried out in terms of sin ( w t ) using the imaginary part for [2.2] and [2.3]; however, a consistency must be maintained in the choice of the reference time function.

### PHASOR AND IMPEDANCE PRINCIPLES

Transforming i(t), vR(t) and v,(t) into the phase domain and applying [2.6] gives the individual element impedances as.

## CHAPTER 2 SINUSOIDAL STEADY-STATE CIRCUITS

### SINGLE-PHASE NETWORK ANALYSIS

With few exceptions, electricity supply grid generation and transmission systems are three-phase grids. Three-phase power, as well as all polyphase power under balanced operation, has the property that the instantaneous power is constant in value.

## CHAPTER 2 SINUSOIDAL STEADY-STATE CIRCUITS 15

The screen display from execution of (de1wye.m) is shown below, where the required delta equivalent impedance values are.

## CHAPTER 2 SINUSOIDAL STEADY-STATE CIRCUITS 23

The results of this section have shown that the analysis of the somewhat general three-phase circuits of Fig. The MATLAB program (Tphckt.m) was developed to handle one of the circuits of Fig.

## CHAPTER 2 SINUSOIDAL STEADY-STATE CIRCUITS 27

The adjacent side of the power triangle is identified as the average power to the circuit in Fig. 2.2, while the opposite side is clearly the reactive power current. The sum of the average power flow to each branch of a network must give the total average power flow into the network.

Example 2.1 7 1 Determine the total three-phase average power delivered to the unbalanced load of Example 2.13. The total three-phase power would then be the sum of the three wattmeter readings.

Use the MATLAB program (csum.m) as an aid in determining the numerical values of V, and 4. A four-wire, wye-connected load is balanced with each phase of the load consisting of a series connection of R. = 5 S1 and L = 20 rnH.

If a conductor is grasped in the right hand with the thumb extended in the direction of the current, then the fingers curl around the conductor in the direction of the established B or H field. If the dA plane is perpendicular to B, then the right-hand rule also gives the direction of flow of magnetic flux 4.

## CHAPTER 3 MAGNETIC CIRCUITS AND ENERGY CONVERSION

3.6, assume that the H field is uniform over each of the four signed lengths e; then application of [3.11] leads to. Practical electromechanical energy conversion devices are constructed with ferromagnetic metals that form almost the complete magnetic field paths, resulting in devices with high torque or power volume density. Although electromechanical energy conversion devices could theoretically be designed with non-magnetic material for field paths, the value of the achievable magnetic field intensity is practically limited for heating reasons by the allowable current density of the conductors supporting the magnetic field.

Air-cooled conductors are limited to current densities of 1.5 to 10 MA/m2 (1 to 6.5 kAlin2) depending on the availability of cooling air for the conductors and natural convection vs.

## CHAPTER 3 MAGNETIC CIRCUITS AND ENERGY CONVERSION 55

After the domain walls have completely collapsed, the magnetic moments have only one component in the direction of the applied external magnetic field. Grain-oriented ESS is used in transformers where the direction of the magnetic field is fixed. Except when operating in the saturation region of the ferromagnetic material, the mmf drop of the air gap (λ 1 = 4 δ/pA 1 ) is dominant.

A reasonable calculation of the air gap permeability for the air gap of the opposite side of the plane shown in fig.

## CHAPTER 3 MAGNETIC CIRCUITS AND ENERGY CONVERSION 65

The expanded graph of air gap flux and leakage flux is shown by Fig. For the Type 2 problem of a serial magnetic circuit with a uniform core cross-sectional area and an air gap with negligible fringes, the operating point can be determined by a graphical technique known as the load line method. If the I Example 3.5 coil current I = 5 A, determine the value of the core flux with leakage and air gap edges.

If the coil current I = 15 A, determine the value of the air gap flux if the leakage flux is ignored.

CHAPTER 3 MAGNETIC CIRCUITS AND ENERGY CONVERSION 69

Optionally, MATLAB was used to plot the flux-mmf characteristics of the center element shown in the figure. Analysis of this parallel magnetic circuit required the construction of an intermediate flux-mmf plot to graphically determine the current through the center memo - ber. After carrying out the above example, only one operating point is still known without a wide range according to the degree of saturation of the magnetic circuit.

MATLAB program (dwckt1.m) was developed to perform a broad analysis of this particular parallel magnetic circuit of Fig.

## CHAPTER 3 MAGNETIC CIRCUITS AND ENERGY CONVERSION 75

The screen display is the result of running (dwleak2.m) using the coil and dimensional data from Example 3.10 for the case of a coil current I = 7.5 A. Inductance is a characteristic of an electrical circuit that has variables of relates the magnetic field to electrical circuit variables; hence its potential utility in quantitative analysis of this energy conversion process is obvious.

## CHAPTER 3 MAGNETIC CIRCUITS AND ENERGY CONVERSION 77

2 3 ~ when the current increases from 0 to I must be equal to the energy (Wf) stored in the magnetic field of the ferromagnetic core. It is seen from [3.32] that this stored magnetic energy is given by the area to the left of the magnetization curve and below A = A,. Since CA is the volume of the ferromagnetic core, a valid interpretation of [3,33] is that the area to the left of the B-H curve of the core material is the energy density of the core.

Thus, for a fixed flux density, the volume of the ferromagnetic core is directly proportional to the energy stored in its magnetic field.

If a PM undergoes the B-H transition while in the magnetic moment stretching region, that is, to the right of the knee of the descending hysteresis loop, the B-H location remains in the initial main hysteresis loop. Second, if the PM is part of a magnetic circuit with other mmf sources capable of lowering the H field of the PM material during operation, care must be taken in design and operation to ensure that the B-H trajectory during operation remains above the knee. of an acceptable descending hysteresis loop. Determine the ratio of the ferromagnetic cross-sectional area to that of the PM material as.

Rearrangement of [3.38] yields the external B,-Hrn relationship of the PM material which must also simultaneously satisfy the material internal Bm-H, relationship of the demagnetization curve.

The mass and inertia of the movable elements are also modeled outside the conservative region; therefore there is no mechanical energy storage in the area.

## CHAPTER 3 MAGNETIC CIRCCITS AND ENERGY CONVERSION

The stored magnetic energy is the area to the left of the applicable graph A-i, or. The area to the left of graph A was shown to be equal to the stored magnetic energy by [3.32]. The dimension t?, is the length of the central flux path of the ferromagnetic material if the air gap is closed.

Extending (c0en~e.m) to handle other magnetic circuits requires only changing the magnetic circuit description in (magckt l .m).

The required holding force largely dictates the basic cross-section of the ferromagnetic core components. Base or pick-up current I, for the case of use, the magnetic circuit is linear and the current is specified independently of the coil flux coupling. The base area A is determined by application of [3.81] using the specified holding force and saturation flux density of the chosen ferromagnetic material.

The air gap clearance is placed between the axial center of the stop post and a radial point outward on the frame for the left side of the coil and the same distance to the left of the right end plate.

CHAPTER 3 MAGNETIC CIRCUITS AND ENERGY CONVERSION 115

## CHAPTER 3 MAGNETIC CIRCUITS AND ENERGY CONVERSION 125

The non-linearity of magnetic circuits can lead to low-order odd harmonics for the case of sinusoidal excitation. Permanent magnets can be used for generating magnetic circuits in place of current carrying coils offering the advantage of improved efficiency due to the absence of the ohmic losses. Determine the percent difference in air gap permeance for the two air gap configurations of Fig.

With this expression for the effective area of the air gap known, derive an expression similar to [3.19] that allows the line-load method to be extended to include the edges of the air gap.

Find an expression for the developed force acting on the piston as a function of air gap length x. If the inductance of the coil on the stationary member is given by L , = La + L, c o d , find an expression for the torque acting on the sewing. If voltage is increased by a factor of 10 while current is decreased by a factor of 10, the cross-sectional area (thus volume) of the transmission line conductor is decreased by a factor of 10 for a given level of current density.

It will be clear later that the latter performance characteristic can only be fulfilled if a high percentage of the magnetic flux established by one coil connects all the other coils.

### VOLTAGE AND CURRENT RELATIONSHIPS

8 ~ can be determined from Faraday's law using the mutual flux +,(t) perfectly coupling the two coils. Or it can be said that the terminal voltages of the ideal transformer vary directly with the turns ratio of the coil regardless of the nature of the time-varying flux $J,(t).

CHAPTER 4 TRANSFORMERS

## 4,303 POWER AND IMPEDANCE RELATIONSHIPS

*CHAPTER 4 a TRANSFORMERS**CHAPTER 4 0 TRANSFORMERS 147**CHAPTER 4 TRANSFORMERS 151**CHAPTER 4 0 TRANSFORMERS 155*

Let an impedance Z be connected across the output terminals of the ideal transformer of Fig. The dominant use of the transformer is in applications where sinusoidal steady-state analysis is relevant. The knee of the hysteresis loops is determined by specifying the saturation flux density (puFsat) and the associated saturation excitation current (pulsar).

First, the magnitude of the load current percentage for the two most important harmonics (third and fifth) is determined.

## 4.404 NAMEPLATE AND COIL POLARITY

### CHAPTER 4 TRANSFORMERS 163

*PERFORMANCE ASSESSMENT*

Losses in a transformer occur in two places4hrnic losses of the windings (R,R2) and losses in the ferromagnetic core (R,). The efficiency of a transformer can vary depending on the apparent power handled and the power factor of the connected load. Based on the equivalent circuit values of the transformer and the rating of the transformer, the program calculates the efficiency vs.

The increase in material will lead to an increase in the initial capital cost of the transformer.

### CHAPTER 4 TRANSFORMERS 177

An apparent power rating of the autotransformer in terms of the apparent power rating of the original two-winding transformer can be formed. Example 4.1 6 1 Determine the apparent power rating of the autotransformer of Example 4.15 and estimate the full-load, unit-PF efficiency of the autotransformer if the original two-winding transformer had a full-load efficiency of 98 percent for a unit PF. The autotransformer windings carry identical currents as for the case of the two-winding transformer, and the core flux density and frequency are unchanged.

Example 4.16 vividly introduced the power and efficiency benefits of the autotransformer when configured for a small step-up or step-down voltage.

The initial study of a three-phase transformer is usually an analysis of the transformer as ideal. Since the phase sequence a-b.c and balanced conditions exist, the remainder of the output voltage and current three-phase circuits can be immediately written based on the given quantities. Determine the rated voltage and current and turns ratio A-Y of the connected 100 MVA transformer in this substation.

Typically, the transformer's primary viking simply has connection access to points in the winding as shown in fig.

CHAPTER 4 TRANSFORMERS 185

4.1 0.2 TAP CHANGE UNDER LOAD