2 Simulation exercise 2

Transcription

1 2 Simulation exercise 2 2. Basic AC theory This home assignment is intended to improve a students knowledge and computation skills on complex numbers and AC theory. The content of this home assignment bases on previously studied subject(s) on electrical engineering and three aspects are on focus:. A single and a poly- or multiphase system 2. Different types of loads 3. Rotating magnetic field By starting from the rotating magnetic field, the field rotation induces an alternating electric field in a conductor and sets up an alternating voltage in terminals of a coil. By adding and shifting the coils a multiphase voltage system can be established. The loaded system gives rise to currents and depending on the load this can consume active or/and reactive power at fundamental or at a multiplier of the supply frequency. Non-linear loads create harmonic currents in addition to the original AC current at the fundamental frequency.. A single-phase system 2. A two-phase system 3. A three-phase system The n-phase (voltage) system can be described by n equations where phase k=,2 n is expressed at fundamental alternation frequency ω as u k () t = U m ( k ) sin ω t 2π (2.) n Most of the domestic loads use single-phase power and are connected to the single phase system of two wires (excluding PE wire, etc). Form another hand three phase power is a very efficient form of electrical power distribution. All three wires carry the same current and have a constantly balanced power load. A two-phase system representation comes useful when analysing multiphase systems or developing three-phase power control in electrical machines, active filters and so on. Therefore, a load, which is connected to the n-phase system can consume. Active power 2. Reactive power 3. Distorted power Thus the power can be separated whether it occurs in fundamental frequency (linear) or at higher harmonics (non-linear), whether it does useful work or pulsates between the source and load storage. The apparent power can be expressed as S = n n n n n n n U kik = Pk + Qk + Dk (2.2) k = k = k = k = The graphical presentation of the apparent power is shown below. Notice that the S vector do not need to be necessarily only on PQ-plane.

2 Q Reactive Power P Active Power D Distortion Power In AC (circuit) theory the manipulation of sinusoids is simplified by the use of rotating vectors or phasors. Apart from the time phasors there are also space phasors (or vectors). Basically, the time varying currents in three-phase system cause rotating magnetic field in the distributed windings. This can be analysed from the figure below: a positive current in a-phase winding (A+ out from the figure, A- in to the figure) causes field along a-axis, so the positive current in b-phase winding causes field along b-axis. As in consequence the successively changed currents in the distributed windings will cause the change of the field orientation (in the stator). This can be used to accelerate a rotor where the rotor field is along direct x-axis and the stator field in quadrature y-axis. Look at the placement of field B, current I and try to estimate forces F in accordance with Flemming s Left Hand Rule. To calculate the field direction of the resultant one coil and the current it should carry, all we have to do is create three vectors, each in the direction of one coil, and of amplitude equal to the current of each coil. The sum of these vectors will give the direction of the total field and hence of the one coil that will replace the three. In a similar manner of deriving the space field vector, the space vector for the voltage, current and so on can be evaluated. The (space) phase shift between the projection axes is: 2π j 3 a = e (2.3) The resulting (voltage) space vector can be expressed:

3 r u = u α + ju β = K n k= u k a k where K is scaling value that for a three phase system could be 2/3 in order to establish an amplitude invariant expression (i.e. phase vector has the same magnitude as n-phase quantities) or sqrt(2/3) for the power invariant expression. 2.2 Source The first part of the simulation exercise focuses on the voltage source or a supply. Here can you visualise different systems and calculate some quantities of interest. If you execute the simulation file then you can see animation of the voltage u phasors, u(t) function and the voltage space vector. In result you suppose to get the same figure as it is shown below. (2.4) There are two phase voltages defined, study what is different: e.g. are they phase shifted? A fast Fourier transform (fft) is carried out for the voltages (and currents). Next figure shows the time function of voltages and the spectral content of the voltages. Notice that the voltage waveform u(t) at fundamental frequency has amplitude value U m equal to V. This can be taken as normalised value u n*, where the base value is U b =U m and u n* =u(t)/u b. a b.4.2 a b voltage, u(t) [V] 0 - voltage, u [V] angle, θ=ωt [deg] harmonic content, h [-] You are kindly asked to accomplish the following assignments:. Execute the simulation file EltehRP_2.m in Matlab and inform course responsible if this file is not able to create the two latest figures. a. Find the definition/expression for the average voltage and implement it in Matlab, calculate the mean value for ua(t) and ub(t). (3)

4 b. Find the definition/expression for the effective or rms voltage and implement it in Matlab, calculate the rms value for ua(t) and ub(t). (3) 2. Define new voltage u=ua-ub. a. Calculate the mean value for ua(t)-ub(t). (2) b. Calculate the rms value for ua(t)-ub(t). (2) c. What differences can you noticed compared to the previous calculations at point, is it possible to calculate rms value for ua(t)-ub(t) by knowing the spectral content. (2) d. Find the definition/expression for the total harmonic distortion (THD) and implement it in Matlab, calculate the corresponding value. (3) e. Compare voltage space vectors at the points and 2, what are the reasons that they look like this, make copies of figure and 2 (5) 3. Define two phase voltage system so that the phase shift between the phases is 90 degrees and the second phase has u 2 (ωt)=0 when ωt=0. a. What can you say when you are looking the time functions u(t) and the voltage space vector, make copy of figure. (5) 4. Define three phase voltage system so that the phase shift between the phases is 20 degrees and the first phase has u (ωt)=0 when ωt= Loading a. Compare voltage space vectors at the if the supply is symmetric and if one of the phases has 80% lower voltage, what are the reasons that they look like this, make copies of figure at two different cases. (3) b. Define also voltages between the phases and study the phasor diagram, add copy of figure. (2) The second part of the simulation exercise focuses on the loads and a power consumption. Here can you visualise different systems and loads, and calculate some quantities of interest. The tasks to be completed are 5. Define a single phase system where a single-phase supply (Um=V) is loaded either with i(t) u(t) Z active, reactive or/and non-linear loads. a. Include active load, construct instantaneous power waveform p(t) and calculate the consumed power and a power factor PF, make a copy of Matlab figure (5) b. Include reactive load(s), construct instantaneous power waveform p(t) and calculate the consumed power and a power factor PF, make a copy of Matlab figure (5) c. Include non-linear load, construct instantaneous power waveform p(t) and calculate the consumed power and a power factor PF and total harmonic distortion THD, make a copy of Matlab figure (5)

5 6. Define a three phase system where a three-phase supply (Um=V) is loaded either with u (t) i (t) Z u 2 (t) i 2 (t) Z u 3 (t) i 3 (t) Z active, reactive or/and non-linear symmetric loads. a. Include active load, construct instantaneous power waveform p(t) and calculate the consumed power and a power factor PF, make a copy of Matlab figure (5) b. Include reactive load, construct instantaneous power waveform p(t) and calculate the consumed power and a power factor PF, make a copy of Matlab figure (5) c. Include non-linear load, construct instantaneous power waveform p(t) and calculate the consumed power and a power factor PF and total harmonic distortion THD, make a copy of Matlab figure and 3 (5) 7. Study asymmetric load and supply of the three-phase system a. Include a symmetric three phase stator that phase inductance is Ls, resistance Rs and back emf Em, construct instantaneous power waveform p(t) of the machine and calculate the consumed power and a power factor PF, make a copy of Matlab figure (0) b. Include an assymmetric three phase stator where one stator winding is partly short-circuited that phase inductance is Ls/2, resistance Rs/2 and back emf Em/2, construct instantaneous power waveform p(t) of the machine and calculate the consumed power and a power factor PF, make a copy of Matlab figure (0) c. Include a symmetric three phase stator where one of the phases is disconnected from supply, construct instantaneous power waveform p(t) of the machine and calculate the consumed power and a power factor PF, make a copy of Matlab figure (0) 8. Compare two and three-phase systems and figure out how the currents and voltages has to be scaled so that these systems can deliver the same power, motivate! (0) 9. Write the assignment report a. These assignments suppose to indicate engineer s/msc student s respectability and writing skills in clean organized presentation and clear decisions explanation manner. Make your report clear and clean. (0) 2.4 M-file M-file will be made available at the course site in Moodle. Write your student ID number so that you can evaluate the load parameters. n = [ ]; % student identification number

6 2.5 Course information You can upload your results in the course website by using your student identification number (Martikli nr) and passw. (salasöna) that is EltehRP2. The deadline is on Tuesday the 25 th of March points (shown before in parenthesis) has to be completed in order to pass the assignment. Delayed work or obvious plagiarism causes the reduction of the accumulated points respectively by 5 and 30 points. The worst: delayed plagiarism causes reduction of 45 points! Plan your agenda and make your studies enjoyable! Whenever you feel that you are really stacked and run out of good ideas then do not hesitate to contact course supervisor via skype: avo.reinap or by avo.reinap@iea.lth.se. Contact between a student and a teacher is vital! Be creative and cooperative!