Mutual Inductance. L (1) l

Transcription

1 Mutual Inductance Developers Objectives Preparation Background JD Mitchell, AB Overby and K Meehan The objectives of this experient are to design and construct a transforer and deterine its losses as well as odeling it in Pspice. Read section 9.0 of Engineering Electroagnetics, section 3. of Fundaentals of Electric Circuits, and the pages 6 30 of PSpice For Basic Circuit Analysis regarding self inductance and utual inductance. Inductance (L), or self-inductance, is the ratio of the linkage agnetic flux to the current (I) producing the flux (Edinister 69). Flux linkage is the product of the nuber of turns (N), linking each of those turns. Inductance has units of henries (H), and the agnetic flux Wb where one Henry is ual to one Weber-turn per Apere. Eqs. shows a forula for A the inductance of a solenoid, where µ is the pereability of the core, N is the nuber of turns, S is the cross sectional area, and l is the length of the solenoid. N S L () l Note that Eqs. is applicable only to linear edia, in which the flux is proportional to the current (Buck 93). Inductance can also be defined as the property whereby an inductor exhibits opposition to a change in the current flowing through it (Alexander 6). Mutual Inductance When two coils are in close proxiity to each other the flux which results fro current flow, in one coil will link with the other coil inducing current flow in that coil. Mutual inductance (M), easured in henries, is the ability of an inductor to induce current flow across another inductor in close proxiity (Alexander 557). Mutual inductance is illustrated in Figure. M L L I Figure : Flux is generated by current flowing through L. The flux which links to L is while the flux which links to L is. When the agnetic flux of an inductor links to another inductor the two inductors are said to be agnetically coupled. The total agnetic flux, shown in Eqs., is the su of each flux

2 coponent. In Eqs., represents the flux which links to L while represents the flux which links to L. The voltage () induced in L is given by: The voltage () induced in L is given by: The utual inductance is given by or Transforers () d N (N = nuber of turns of L) (3) dt d N (N = nuber of turns of L) (4) dt d M N (5) di NN S (6) l M A transforer is a device coprising of two or ore coils, usually sharing the sae core. Transforers take advantage of utual inductance for various purposes, including stepping up and stepping down voltage, circuit isolation, and filtering. The side of a transforer connected to a voltage source is called the priary winding while the side of a transforer connected to a load is called the secondary winding. The circuit in figure shows an ideal transforer which has no losses. However, all practical transforers have losses which are caused by the resistance of the copper windings, R, the core losses due to hysteresis, R, flux leakages to the air,, and the finite pereability of the core,. An uivalent circuit for the non-ideal transforer is shown in figure. Figure : Non-ideal transforer uivalent circuit. R RM M Ideal Transforer

3 To find the values of R,, R, and two tests are used: the open circuit test and the short circuit test. The open circuit test involves easuring the current passing through the priary winding of the transforer and the power dissipated by the priary winding while the secondary winding is open. The short circuit test involves the sae procedure as the open circuit test except that the secondary winding is shorted. Figures 3 and 4 show the wire diagras for the open circuit test and the short circuit test respectively. Figure 3: Open circuit test wire diagra. Diagra courtesy of Figure 4: Short circuit test wire diagra. Diagra courtesy of irginia Tech ECE departent The results fro the short and open circuit tests provide the expressions for the losses associated with non-ideal transforers. These expressions are given below, where, I,, and I are open circuit voltage, open circuit current, short circuit voltage, and short circuit current respectively. R (7) P (8) I RM P (9) I R I R (0)

4 References Edinister, J., & Edinister, J. (0). Schau's outline of electroagnetics. New York: McGraw-Hill. Charles K. Alecander and Matthew N.O. Sadik, (004). Fundaentals of Electric Circuits, 4 th edittion, McGraw-Hill. Hayt, W. H., & Buck, J. A. (006). Engineering electroagnetics. Boston: McGraw-Hill Higher Education. Tront, J. G. (005). Pspice for basic circuit analysis. Dubuque, IA: McGraw-Hill. Johnson, David W. "Transforer Equivalent Circuit." CLAYMORE. 7 Apr Web. 4 Aug. 0. < Materials. ea. ¼ diaeter iron core. ea. 00 Ω resistor (shunt) 3. ea. k Ω resistor (load) 4. Magnet Wire 5. ea. dual trace oscilloscope 6. ea. ANDY board 7. ea. ellean oscilloscope 8. ¼ heat shrink tubing. Procedure Modeling:. Design a : transforer with between 00 and 00 turns per inductor, sharing the sae ferrite core. Place the heat sink tubing about the ferrite core so that when the core is reoved the inductor keeps its shape.. Design separate air core inductors with the sae diaeter as your ferrite core. 3. Using PSpice, odel and perfor an AC sweep on your transforer with a kω load across the secondary winding. Use a 5 sine wav as the voltage source. 4. Deterine the fruency response of your ideal transforer. 5. Perfor the short circuit and open circuit tests and deterine the values of the loss coponents for a non-ideal transforer. When easuring current use a 00Ω shunt resistor. To calculate power, use the easured current and voltage across the priary side of the transforer. 6. Deterine the transfer function of your non-ideal transforer circuit. 7. Using MATLAB create a bode plot of the transfer function. Measureents: 8. Use the elleon oscilloscope to perfor a fruency sweep of your transforer. How does it copare to the Bode plot? 9. Deterine the cutoff fruencies. PSpice: 0. Model your transforer with its loss coponents and perfor an AC sweep fro 0Hz MHz. How does this plot copare to the others.. Repeat steps to 0 for a transforer with an air core.

5 . Repeat steps to 0 for two inductors with separate air cores in close proxiity to one another. (Try to ake the touch) 3. How do the loss coponents copare for a transforer with a ferrite core, a transforer with an air core, and for two inductors with separate cores copare. What is the cause of these differences? Modeling an Ideal Transforer: The FRM_LINEAR eleent fro the analog library is used to odel an ideal transforer in Pspice. To ake the transforer ideal, open the Property Editor window for the FRM_LINEAR eleent and ake sure that the COUPLING colun is set to. The COUPLING colun sets the coupling coefficient (k) of the transforer. The coupling coefficient is a easure of the agnetic coupling between two coils (Alexander 566). A axiu k of would ean coplete agnetic coupling between coils, while a k of 0 eans no agnetic coupling between coils. The uation for k is shown below. M k () L L The circuit used to odel and ideal transistor is shown in figure 5. OFF = 0 AMPL = 5 FREQ = 60 0 R T R4 G R LOAD k Figure 5: The ideal transforer in PSpice. R and R represent the loss of the windings. R4 is used as a duy resistor to satisfy a PSpice ruireent that there be DC path between any node and ground. Figure 6 shows the Transforer Property Editor. The values of your inductances ust be entered into the Property Editor coluns L_ALUE and L_ALUE. L_ALUE and L_ALUE control the turns ratio of the transforer using the forula Z n, () Z where Z and Z are the ipedances of the priary and secondary respectively and n is the turns ratio of the transforer. So for the transforer in figure 6, n = 3. Figure 7 shows the results of tie doain analysis of the ideal transforer.

6 Figure 7: Transient Analysis of an ideal transforer. Modeling Mutual Inductance Another way to odel the circuit shown in figure 7 is to use the K_Linear eleent found in the analog library. Using the K_Linear eleent, users can specify the aount of agnetic coupling between up to 6 different inductor eleents. To use the K_Linear eleent, the coupling coefficient and inductors to be coupled ust be selected. Figure 8 shows the circuit diagra of a circuit odeled using K_Linear while Figure 9 shows the property editor window for the K_Linear eleent. Just as before R4 is being used as a duy resistor. In figure 8 the coupling coefficient is set to, aking the transforer ideal, and the inductors selected are L and L. R R OFF = 0 AMPL = 5 FREQ = 60 L 0uH L 0uH LOAD k 0 R4 00MEG K K K_Linear COUPLING = Figure 8: Transforer odeled using K_Linear. Figure 9: K_Linear Property Editor Window.

7 Note that in PSpice Figure 0 is uivalent to Figure. Figure 0 Figure The pin nuber assigned in PSpice can be deterined by double clicking on the pin in question.