Lab 4: The transformer

Transcription

1 ab 4: The transformer EEC 305 July 8 05 Read this lab before your lab eriod and answer the questions marked as relaboratory. You must show your re-laboratory answers to the TA rior to starting the lab. It is a long lab and requires the full 3 hours to comlete. Divide into grous of (or more if necessary). Someone must have the following roles: Exerimentalist: Taking measurements off the oscilloscoe Theorist: Doing calculations and organizing exerimental data Hint: If using excel, ensure all angles are in radians for trig functions NOTE: Each grou member must reare his own lab reort. ab reort is to include re-lab question answers. Introduction The objectives of this lab are to investigate: a) The relationshis between current and voltage in the rimary and secondary windings of a transformer. b) Imedance transformation with a transformer. c) How well a widely-used equivalent circuit describes the behavior of a real transformer. d) How real transformers have a limited frequency range of useful oeration. e) A segment of a ower transmission line for home delivery. Pre-lab questions: ) Based on the theory section rovided derive equation (4) starting from equation set (.A) and (.B). ) Using the voltage and current transformation equations, (4) and (6), obtain the imedance transformation (5). 3) Define leakage inductance. 4) Define magnetization inductance. 5) Describe Eddy currents and how they lead to ower loss in a real transformer 6) Describe hysteresis losses and how they lead to ower loss in a transformer. 7) What stes are taken to minimize Eddy current losses and hysteresis losses in real transformers? 8) Draw a schematic of a large scale ower distribution network starting from the megawatt generator and branching out to different consumer loads (residential, industrial, ).

2 . Theory The starting oint for analyzing the transformer is the basic equivalent circuit of figure. In this circuit, the inductance of the secondary, and M the mutual inductance between the two coils. Current and voltages in the transformer are described by ~ ~ v~ di M di (.A) dt dt ~ ~ v~ M di di dt (.B) dt In the following we will use the symbol v ~ to reresent a comlex voltage, and v to reresent the voltage amlitude. M is related to and by: M k () where k is the fraction of the flux in each turn of coil which also threads each turn of coil. Also, if there are N turns in the rimary and N turns in the secondary: N a N (3) N where a is the turns ratio of the transformer. In a well-designed iron core transformer, N k is close to. In an ideal transformer it is assumed that k =. In this case: v N a (4) v N If a load resistance, R is connected across the secondary of the transformer, it can also be shown v that the inut imedance, Z i, seen "looking into" the rimary is given aroximately by in (ideal transformer exression): i Z in R a (5) The transformer can therefore rovide a very valuable imedance transformation function. It should be noted that (5) is an aroximation which is only valid if a R. In other words, for imedance transformation to work it is vital that the magnitude of the reactance associated with the transformer rimary winding be much larger than the transformed load resistance. Under this condition, it can also be shown that the currents in the rimary and secondary are related by:

3 i a (6) i Figure : Basic transformer equivalent circuit. It is often convenient to redraw the equivalent circuit of figure in the form shown in figure below. The ideal transformer in the center of this circuit has the same turns ratio a as the real transformer, but has erfect flux couling and infinite inductance in the rimary and secondary coils. The inductance k is sometimes called the leakage inductance, while k is the magnetizing inductance. You should be able to show that this circuit gives exactly the same relationshi between voltages and currents at the terminals as the circuit of figure. osses in real transformers are often aroximately modeled by adding to resistances to the equivalent circuit of figure, giving the circuit of figure 3. The resistor R reresents the resistance of the rimary winding, and R the resistance of the secondary winding. Resistance R aroximately reresents losses due to hysteresis and Eddy currents in the core. c Figure : Alternative transformer equivalent circuit.

4 Figure 3 Transformer equivalent circuit allowing for losses Figure 3 is still an aroximate descrition of the real transformer. It makes no allowance for the caacitance between the windings in the rimary and the secondary, for the fact that the resistance of the windings is distributed throughout the coil and cannot be reresented by a single lumed resistor, nor for the deendence of core losses and core ermeability on frequency. When the secondary windings are shorted, the equivalent circuit for the rimary reduces to that of figure 4. This circuit shows that by measuring the inut imedance with the secondary shorted it is ossible to determine the winding resistance and the leakage inductance. Figure 3: Equivalent circuit with secondary shorted. Usually we will have Rc and k both much greater than the series combination of R a R k in which case the equivalent circuit reduces to that of figure 5. and

5 Figure 4: Simlified equivalent circuit with secondary shorted. When the secondary is oen circuit, the equivalent circuit becomes that of figure 6.. Equiment and Procedure Figure 5: Equivalent circuit with secondary oen. The transformer considered in this lab is a tyical low cost audio transformer designed to imedance match all 8 seaker to a transistor amlifier outut stage over the frequency range from aroximately 0 Hz to 0 khz. The core of this transformer consists of "soft" iron lates laminated to reduce Eddy current losses. To comlete the lab, it will be necessary to measure the current flowing in the rimary of the transformer. This will be done by connecting a small resistor in series with the rimary, and measuring the voltage dro across this resistor.

6 Figure 7: Exerimental setu. One side of the transformer rimary is connected to the lug marked "Outut" on the function generator. There is a cable on each side of the rimary connected to the oscilloscoe. A wire is used to connect the 0 Ω resistor in series (see figure 7). The outut labeled CH on the board gives v, while the outut labeled CH is the voltage across the resistor, which can be used to find i. In your lab book draw the equivalent circuit. Are the ground sides of the scoe inuts connected together? How to use the Oscilloscoe: To measure the voltage, ress Measure, then ress more otions until you find amlitude Pk to Pk To measure the hase angle, ress Measure, then scroll down to as above and ress Phase (make sure the hase difference is being calculated from CH to CH otherwise your angle will be negative) If the angle is the comlementary angle (for examle 0 ο instead of 60 ο ), then move the curves horizontally) To scale roerly, usually the Auto Scale will be enough but make sure when taking measurements that you see to 4 cycles. (Not too zoomed in or zoomed out)

7 To find the inut imedance of the transformer, we will need to measure the hase angle between v and i. This can be done by dislaying both the voltage and current waveforms on the oscilloscoe and setting the time base to the longest value for which a half-eriod T/ of both waveforms is visible. etting t be the time difference between the zero crossing of the voltage waveform and the zero crossing of the current waveform, we have: t 360 T Traditionally we would measure the time difference, t o, to find the angle, but newer oscilloscoes are able to directly measure the hase angle. This technique is illustrated in figure 8. (7) Figure 8: Measuring hase difference between v and i ; voltage leads current in this case. In an ideal inductor, current lags voltage by 90 o. We will define this to be a ositive hase angle. If current leads voltage (the case in a caacitor), the hase angle is negative. Throughout this lab, it is imortant to note whether current leads voltage or vice versa.

8 Figure 9: Side-by-side examle of maximizing dislay settings to imrove accuracy of results. 3. Measurements and Calculations Carry out the following measurements on the audio transformer. To do this, connect the sync of the generator to CH4 of the oscilloscoe. The sync will hel in getting a stable wave on the oscilloscoe as it will use CH4 as a trigger for synchronizing both machines. Once the desired voltage has been set on the generator, you are then ready to connect the outut to the rimary coil. a) Using a digital multimeter, measure the transformer s rimary and secondary coil resistances, R and R. Make sure the transformer is not connected to anything. Additionally, verify the values of both resistors using the multimeter. ( mark) b) In this exeriment, we measure the turns ratio, a, of the audio transformer, assuming it s ideal. Set the function generator frequency to khz. Set the voltage to be 4 V eak to eak. v To comute the turns ratio, a, use the oscilloscoe to measure the voltage across the v rimary coil ( v ) and across the secondary coil ( v ). Comare your value to the turns ratio of the transformer secified by the manufacturer, which is 7.9 :. ( mark) c) Introduction to this question: In this exeriment you will comute the inut imedance and its comonents (resistance and reactance) with the secondary coil short circuited. Using the set-u shown in figure 7, you will measure the current and voltage in the rimary to calculate the v ratio, which gives the inut imedance. Note that this will be comlex as we have the effect i of a resistance and an inductance connected in series, as shown in figure 5. The resistance is R Ra and the inductance is ( k). Procedure: Connect the transformer as shown in figure 7, short circuit the secondary coil, add a 0 Ω resistor to the rimary coil and measure the current flowing in the rimary. Set f = khz

9 on the function generator. Measure v, i and the hase angle between them. Remember that the outut connected to "CH " gives v, while the outut connected to "CH " gives the voltage across the resistor. See figure 8 for instructions on how to measure the hase angle. Given true: Comute Z in is reresented as a resistance R s in series with an inductance v i R s cos and sin R s and s. From figure 5, we should have s the following is v s i (8) R s Ra R and s ( k). Comute R Ra from the results of arts (a) and (b) and comare with the measured value of R s. Comute ( k). Don t forget to show all of your calculations! (5 marks) d) Introduction to this question: The urose of this exeriment is to measure the Eddy current losses R c and k. For this we oen circuit the secondary and hence, figure 6 comes into effect. As in the revious exeriment, you shall measure v and i. From that you can estimate the effective series resistance and the inductance of the circuit in figure 6, Rs and s as before. From figure 6 you will see that these values deend uon R s, ( k), R c and k. But we already know the values of R s (art a) and ( k) (half the value of the inductance found in (art c)), so to find the remaining unknowns, ( R c and k ), you must use relationshi (0) given below and then use the values of R s, s, R and ( k) to comute R c and k. Procedure: Oen circuit the secondary, and set f = khz and the eak to eak voltage to 4 V. Record v, i and re-calculate R s and s from equation (8). To determine R c and k in the equivalent circuit 9 of figure 6, we need to account for the winding resistance R and leakage inductance ( k). R' s and Rs ' Rs R ' s are defined as and s ' s k (9) s Additionally, R cos and Z ' sin ' c Z s k (0) s where Z s ' R ' ' () s s and ' tan s () R ' s

10 Comute R c and k. Using k determined here, and ( k) found in art c), estimate k. You have equations and unknowns. Don t forget to show all of your calculations! (6 marks) e) Reeat the measurement described in art d) at frequencies of 00 Hz, 500 Hz and 5 khz, using a eak to eak voltage of V. Remember to kee track if the voltage is lagging or leading the current, which tells you if the reactance of the rimary is caacitive or inductive (see the Exeriment and Procedure section). Use the table format given below to record your data (which includes R c and k ). Comment on the ability of the equivalent circuit of figure 3 to accurately reresent the behavior of the transformer over this frequency range. Seculate on why R c and k aear to deend on frequency. (4 marks) f(hz) v (V) i (A) ( ) R c (Ω) k (H) f) Reeat the measurement of v, i and at f = 00 khz. Is the reactance seen looking into the rimary now inductive or caacitive? Suggest an exlanation for your observation. (3 marks) F (Hz) v (V) i (A) ( ) g) Connect a 0 Ω load resistor across the secondary of the transformer. Kee the eak to eak voltage to V and measure v, i and at frequencies of 00 Hz, khz, 0 khz and 00 khz. Reresenting Z in as a resistor R in arallel with an inductance the following is true. v i Construct a table showing would have R v i (3) R cos and sin R and a R (see equation (5)) and as functions of frequency. For an ideal transformer we would be infinite. Note that exeriment (g) (and (h) to follow) is a reetition of the revious exeriments, with changes:. The 0 ohm load on the secondary winding.. The resulting equivalent circuit looking into the source v is a resistance and an inductance in arallel, as oosed to in series, as has been the case in revious exeriments.

11 With v and i, you will measure the resistance and inductance connected in arallel. Use relationshis (3) to measure the values of R and. Comare the behavior of the real transformer at different frequencies with the ideal model described in (5) and seculate why your results do not satisfy the simle model, esecially at low frequencies. The currents and the voltages distort at some value of v as you increase it. Note that oint and exlain why. Hint: What do you remember of inductors, when it comes to DC currents and very high frequency AC currents? (8 marks) f (Hz) v (V) i (A) ( ) R (Ω) (H) h) eave the 0 Ω load resistor in lace across the secondary and set f = 0 Hz. Increase v towards 0 V, until the i waveform starts to distort. Note the aroximate values of v and i at which the distortion begins. Make a sketch of the current waveform. Also make a sketch of the v waveform under these conditions. Give a brief exlanation for the distortion. ( marks) Transmission ine In this art of the exeriment you will make use of the two transformers on the board. The objective is to examine the voltages along the lines (relative to ground) of a small scale ower transmission system. The transformer used in arts a) to h) is to be configures as a ste u transformer. The other transformer has a center ta on one side and will be used as the ste down transformer. The center ta side will be connected to the residential loads made u of two 0 resistors. The transmission line losses are reresented by a 0 resistor connected between the two transformers. The electrical circuit is shown in figure 0. Setu the circuits. Figure 0: Small scale transmission system.

12 Set u the transmission line as shown in the figure below. Set the frequency to khz and the eak to eak voltage to 4 V. a) Measure the inut voltage, as well as the voltages across each of the coils in the transmission line. When measuring the voltage across the coil with 3 oututs, do measurements. Take each measurement from the ground to the other outut. Exlain the change in the voltage throughout the line as well as its hase. (### marks)