CHAPTER 2 D-Q AXES FLUX MEASUREMENT IN SYNCHRONOUS MACHINES
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1 22 CHAPTER 2 D-Q AXES FLUX MEASUREMENT IN SYNCHRONOUS MACHINES 2.1 INTRODUCTION For the accurate analysis of synchronous machines using the two axis frame models, the d-axis and q-axis magnetic characteristics are required. Due to saliency, the q-axis magnetic characteristics are different from the d- axis characteristics. The main reason for the absence of a single direct method to find the saturation in d- and q- axes is the lack of uniformity in flux distribution between them. The effect of saturation is directly reflected in the air gap flux density distribution. Therefore by measuring the flux in d- and q- axes, the corresponding magnetic characteristics are determined. 2.2 NEED FOR D-Q AXES FLUX MEASUREMENT The air gap is very small in the d-axis and the reluctance offered to the flux is small. Hence the flux in d-axis is large, whereas it is small in q-axis due to the large air gap and high reluctance. For this reason, a synchronous machine produces more magnetic flux in the d-axis than the q-axis. The q- axis saturates much earlier than the d-axis. On moving from the d-axis towards the q-axis, the air gap increases continuously. Therefore, analyzing the performance of the machine with a single value of armature reactance, X a does not yield satisfactory results. The armature reaction affects the synchronous reactance of the machine, which in turn affects the synchronous impedance and is inversely proportional to the synchronizing power. The value of load angle of a synchronous machine
2 23 mainly depends upon the q-axis emf. Thus the q-axis characteristic is also important in order to calculate the performance of the machine precisely, such as reluctance power, synchronizing power, power angle and regulation. 2.3 BASIC MODEL AND MAGNETIC CHARACTERISTICS OF A SYNCHRONOUS MACHINE Basic Synchronous machine model The model of a salient pole synchronous machine may be given in the rotor reference frame (rotor electrical speed is ) with the following relations given in set of Equations (2.1). v ds = R s i ds + d ds / dt - qss v qs = R s i qs + d qs / dt + ds v f = R f i f + d f / dt 0 = R dr i dr + d dr / dt 0 = R qr i qr + d qr / dt ds = L s i ds + dm qs = L s i qs + qm f = L f i f + dm (2.1) dr = L dr i dr + dm qr = L qr i qr + qm dm = L dm i dm qm = L qm i qm i dm = i ds + i f + i dr i qm = i qs + i qr
3 24 where v, i and denote voltage, current and flux linkage respectively. Indices s, f and r stand for stator, excitation and damper winding. Torque and mechanical equations are neglected. Index m denotes parameters and variables associated with the magnetizing flux, while index describes constant leakage inductances. Inductance terms L dm and L qm denote saturated values of mutual inductances along d and q-axis Magnetic characteristics of a Synchronous machine In both the Projected pole stationary field type and the Projected pole rotating field type Synchronous machines, there exists two magnetic axes known as d-axis and q-axis. Magnetic flux distribution in a Projected pole stationary field type synchronous machine is shown in Figure 2.1. Due to saliency, the q-axis magnetic characteristics are different from the d-axis characteristics. The direct (d-axis) characteristics can be obtained by conducting the open circuit (OC) test. But the quadrature (q-axis) characteristics cannot be determined by applying conventional methods. Figure 2.1 Magnetic Flux Distribution
4 MEASUREMENT OF MAGNETIC FLUX USING THE EFFECT OF MAGNETIC FIELD ON SOLID STATE DEVICES Germanium diodes are used for the measurement of magnetic field as their property change due to the application of magnetic field. If a forward biased diode is placed in the magnetic field, its forward current decreases which is directly proportional to the magnetic field strength. The measurement of variation in this current, leads to the measurement of magnetic field strength. The effect of the magnetic field on these diodes is more sensitive to magnetic fields when forward biased than reverse biased. Hence, the forward current bias is made good for the measurement of magnetic field. In developing a device for flux measurement using the effect of magnetic field on diodes and to make the results applicable for practical measurement of the magnetic field, the direct quiescent current flowing in diode has to be eliminated in the final measurement of bridge output voltage. This requirement can be met by connecting the diode in bridges as Single Diode Bridge/ Double Diode Bridge/ Three Diode Bridge. Among the three bridges, double diode bridge is more appropriate and the reason of which is discussed in the succeeding sections. 2.5 SELECTION OF BRIDGE CIRCUIT In the single diode bridge, it is observed that there is a slight and steady one-way change on the output voltage of the bridge because of slow heating of the diode by its own current. Single diode bridge is not selected due to this reason. In the case of a three diode bridge, an output of few microamperes (20µA) is obtained under strong magnetic field. But in the case of the two diode bridge, the output obtained under the same condition is
5 26 nearly two milli Volt. A milli Volt is far better quantity for amplification than microampere. Hence the two diode bridge is used in the construction of the flux sensor. The maximum change in sensitivity of the two diode bridge at a strong magnetic field is found to be not more than 0.2 % per C. For a variation of 10 C in the room temperature, the error is quite small (about 2%) Functioning of Two Diode Bridge In the two diode bridge shown in Figure 2.2, the Germanium sensor diode (1N 60) is placed in one of the arms of the bridge. The specifications and characteristics of the Germanium diode used are given in Appendix 1. A 12 resistor is used in series with the bridge compensation diode. In the other two arms, a 2 k potentiometer and a 560 resistor are used. The resistors and potentiometer used in the bridge circuit have an accuracy of 1%. The bridge sensitivity is found to be maximum, when operating at a voltage of 5 V. For a magnetic field strength of about 0.7 µwb/m 2, bridge output of nearly 2 mv is obtained. The germanium semiconductor diode connected in one of the arms of the bridge circuit located along d- and q-axes act as the flux sensing device. To operate the bridge, it is first balanced by adjusting the resistance arms without the application of magnetic field. As the magnetic field is applied, the forward current is decreased in the forward biased germanium diode and this change causes an unbalance in the bridge. This results in a change in the bridge output voltage, which is directly proportional to the magnetic field strength. The bridge output voltage is calibrated to the corresponding flux using a standard flux meter.
6 27 Figure 2.2 Two-diode bridge circuit used for flux measurement 2.6 EXPERIMENTAL SET-UP FOR FLUX MEASUREMENT Selection of number of Bridge Circuits As it is evident that the two diode bridge circuits for flux measurement are to be used, sixteen such bridges are designed in a printed circuit board. As all the four stator poles are symmetrical in construction, all the d-axis and also all the q-axis characteristics are symmetrical. Four sensor diodes are placed on a pole for the measurement of d-axis flux. With more emphasis on the measurement of the q-axis flux, six sensor diodes are placed on each of the two symmetrical q-axes. Among the sixteen bridge circuits, twelve account for the measurement of q-axis flux and four for the d-axis flux. A source is used to feed the input voltage to the bridge.
7 Location of sensor diodes inside the machine Photographs in Figure 2.3 and Figure 2.4 show the field poles, armature, air gap along d- and q- axes, casing and capping fixtures used for fixing diodes along q-axis and the diodes placed on one face of a field pole along the d-axis. Four sensor diodes for the d-axis flux measurement are placed, with two each, on both ends of a pole, while six such sensor diodes are placed equidistant on each of the two symmetrical q-axes. Each pair of the d-axis diodes are fixed (pasted) on the pole winding on either side of the pole along the d-axis. Similarly, a fixture using casing and capping arrangement is used for sensor diodes placement along the q-axis. In order to account the ampere circuital law effect, shielded twisted pair wire is used in the diode bridge circuit. Figure 2.3 Placement of sensor diodes for measurement of q- axis flux
8 29 Figure 2.4 Placement of sensor diodes for measuring both d- axis and q- axis flux Instrumentational Amplifier The sensor bridge output voltage is fed to instrumentation amplifier for amplification as the sensor output is only few millivolts. Since there are sixteen sensor diode bridges placed, sixteen instrumentation amplifier circuits are used. The instrumentational amplifier is shown in Figure 2.5 whose gain is given by A i =1+((2xR 2 )/R 1 ). A wide range of gain is obtained by varying the value of R 1. By selecting R 2 =R 3 =R 0 =100 k and R 1 =5 k, the gain of instrumentation amplifier is made to be A i = Summer Circuits Amplified output from twelve q-axis instrumentation amplifiers is fed to the twelve-input terminal Adder circuit as shown in Figure 2.6. Similarly the output of four d-axis instrumentation amplifiers is fed to the four-input terminal adder circuit shown in Figure 2.7. The gain of each of the adder circuit is designed to be A s =(R f /R in )=10, by selecting the values of R f =100 k & R in =10 k.
9 30 Figure 2.5 Instrumentation amplifier Figure 2.6 Twelve-input terminal adder circuit
10 31 Figure 2.7 Four-input terminal adder circuit 2.7 TESTING, CALIBRATION AND ESTIMATION OF FLUX Testing Input voltage is given to all the sixteen bridge circuits in parallel. By adjusting the variable resistor (potentiometer) in one arm of the bridge, the output voltage of a bridge is made zero, i.e., the bridge is balanced. All the bridges are first balanced by adjusting the potentiometers in each bridge before the application of the magnetic field. After balancing, +12 volt power supply circuit for biasing the operational amplifiers is switched on. For obtaining the open circuit characteristics (OCC) of the laboratory synchronous machine (3000 VA, 415 V) an experiment is conducted as per the classical method, in which the synchronous machine is driven by a DC motor at rated speed. The open circuit characteristic is obtained experimentally. The generated line voltage corresponding to various field currents are noted and tabulated in Table 2.1. Per unit values of line voltage and field current are obtained and the characteristic (OCC) in terms of these values is plotted in Figure 2.8. The d-axis magnetic
11 32 characteristic of the synchronous machine is the same as the OCC, obtained from the conventional OC test, with machine excited from its field winding. Table 2.1 Open Circuit Test Readings Alternator Field Line Voltage (V) Current (A) OCC Line Voltage (pu) Field Current (pu) Line Voltage (pu) Figure 2.8 Open Circuit Characteristics
12 33 Using the same procedure of OC test of an alternator, for every value of field current from 0 to 0.31 A, increased in steps, the corresponding q-axis and d-axis adder circuit output voltages are also noted down as shown in Table 2.2. Table 2.2 Output Voltages of d-axis and q-axis Adder Circuits Field current, I f (A) d-axis output voltage, V 1 (V) q-axis output voltage, V 2 (V) Calibration Figure 2.9 shows the experimental set-up of circuit for flux calibration using a real time flux meter whose sensor is kept along the d-axis of Synchronous Machine. Simultaneously, a sensor diode is also placed along the d-axis as shown in Figure 2.9 for measurement of flux along the axis. A 220 V DC supply is used to provide voltage supply to the field pole coil. Variable rheostat is varied to circulate different value of current from 0 A through the field coil. The sensor region of the flux meter is kept near the vicinity of field coil d-axis pole. The flux meter reading and the sensor diode bridge output voltage for every value of field current are tabulated in Table 2.3.
13 34 The slope (k) of voltage to flux relation is determined from the graph plotted as shown in Figure 2.10 with flux meter reading taken along x-axis and bridge output voltage taken along y-axis. This slope (k), a constant value, can be used for converting the d-axis and q-axis adder circuit output voltages (Table 2.2) into d and q-axis flux respectively. Figure 2.9 Experimental set-up for calibration of flux Figure 2.10 Determination of slope of voltage to flux relation (k)
14 35 Table 2.3 Bridge Output Voltage and Flux Meter Reading for the same Field Current S.No Field Current, I f ( A ) Bridge Output Voltage, V out (mv) Flux Meter Reading (micro-tesla) Estimation of flux A simple calculation is carried out for the determination of d-axis and q-axis flux from the amplifier adder output voltages. The input values for the calculation include the slope value of voltage to flux relation (k), gain of the instrumentation amplifier (A i ), gain of the adder circuit (A s ), number of d- axis sensors (n d ), number of q-axis sensors (n q ), adder output voltage (volt) from four d-axis diode bridges (V 1 ) and adder output voltage (volt) from twelve q-axis diode bridges (V 2 ). Using the relations given in Equations (2.2) and (2.3), direct axis flux ( d ) and quadrature axis flux ( q ) are respectively estimated. d = {V 1 / (n d * A s * A i * k)} (2.2) q = {V 2 / (n q * A s * A i * k)} (2.3)
15 36 The d-axis and the q-axis flux are thus determined by solving the above equations and tabulated against the corresponding values of the field current of the synchronous machine as shown in Table 2.4. Table 2.4 Estimated d-axis and q-axis flux Field current, (A) Flux in d-axis (micro-tesla) Flux in q-axis (micro-tesla) Per unit Conversion of d-q axes flux From Table 2.1, it is evident that the rated terminal voltage of 415 V is attained at a field current of 0.25 A. Hence the base value for per unit conversion of field current is taken to be 0.25 A. The base value of flux in d-axis is micro-tesla, corresponding to base value of the field current. Thus using the base values of both the field current and the flux in d-axis, all the field current values (I f ) and d-q axes fluxes (as in Table 2.4) are converted into their equivalent per unit values and tabulated in Table 2.5. In Figure 2.11, the per unit value of d-axis and q-axis flux are plotted against per unit values of field current.
16 37 Table 2.5 Per Unit Values of d-axis and q-axis flux Field current, I f (pu) d-axis flux, d (pu) q-axis flux, q (pu) Estimated Per unit value of flux d-axis and q-axis Flux (pu) Field Current (pu) d-axis flux (pu) q-axis flux (pu) Figure 2.11 Per unit value of flux saturation characteristics
17 EXPERIMENTAL RESULTS AND DISCUSSION The theoretical concept that the q-axis flux saturates much earlier than the d-axis flux has been experimentally proved from the measured d- axis and q-axis magnetic characteristics of the projected pole (stationary field type) synchronous machine. Further the measured per unit d-axis flux characteristics obtained using Germanium diode flux sensors as in Figure 2.11 matched with the per unit d-axis flux characteristics, obtained by OC test, as in Figure 2.8 and thus validated. Use of Germanium diode flux sensor is the first of its kind for the measurement of d-q axes flux. The Germanium diode flux sensor fabricated is one of the simplest instrumentation kits for the measurement of flux. The merits of flux sensors used in this work are discussed below: Cost of the flux sensors devised in this work is comparatively much cheaper than the widely used Hall Effect flux sensors. Construction of the flux sensors is simple compared to other types of sensors. In some of the magnetic flux measuring methods such as search coil method, output exists only at the instant of placing the search coil in the magnetic field or while moving away from it. But in Germanium diode sensor, the output exists as long as the diode is in the magnetic field. Hence, the effect of magnetic field on diodes can be used in the construction of closed loop controls of electrical systems. The germanium diode flux sensor has an excellent transient response i.e., it has a smaller settling time. Compared to the Hall plate flux sensors which are currently in use, the germanium diode flux sensors have better response. Hence, this facilitates use of the proposed sensors for the measurement of pulse fields.
18 SUMMARY In this chapter, an experimental method for the measurement of d- and q-axes magnetic characteristics in a projected pole (stationary field type) synchronous machine by using Germanium diode as flux sensors is presented. The most important findings and conclusions of this method are: 1. The effect of magnetic field on germanium diodes can be used for the measurement of magnetic fields. The effect of the magnetic field on diodes provides low cost magnetic sensors. 2. The use of shielded twisted pair wire, precise resistors and potentiometers increase the accuracy of the measurement. The accuracy of the flux sensor reading can further be increased by selecting more number of germanium diode bridges and precise instruments for the calibration of flux. 3. It has been found that the sensitivity of the bridge at a strong magnetic field is not more than 0.2% per C and hence the error due to the variation in the temperature is negligible. 4. The proposed setup with appropriate modifications can be employed in the measurement of flux in a Rotating Field type Synchronous machine, by employing wireless transfer of electrical data.
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