1 / 29

Transformer Model

Transformer Model. Voltage Equation of a transformer in matrix form is:. where r = diag [r 1 r 2 ], a diagonal matrix, and.

payton
Download Presentation

Transformer Model

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Transformer Model Voltage Equation of a transformer in matrix form is: where r = diag [r1 r2], a diagonal matrix, and The resistances r1 and r2 and the flux linkages l1 and l2 are related to coils 1 and 2, respectively. Because it is assumed that 1 links the equivalent turns of coil 1 and 2 links the equivalent turns of coil 2, the flux linkages may be written as Where

  2. Linear Magnetic System Reluctance is impossible to measure accurately, could be determined using:

  3. Flux Linkages The coefficients of the 1st two terms on the right–hand side depend upon the turns of coil 1 and the reluctance of the magnetic system; i.e. independent of coil 2. Similar situation exist in equation for 2

  4. Self Inductances • From the previous equations one can define Self-Inductances: Where Ll1 and Ll2 are leakage inductances of coil 1 and 2 respectively. Lm1 and Lm2 are the magnetizing inductances of coils 1 and 2 respectively.

  5. Magnetizing Inductances • The two magnetizing Inductances are related as: Where m the Magnetizing Reluctance being common for both coils. The mutual Inductances are defined:

  6. Mutual Inductances Mutual Reluctance being common for both Circuits; Mutual Inductances are related to Magnetizing Inductances too:

  7. Flux Linkages • Flux Linkages may be written as: = Li Where

  8. Flux Linkages • The Flux Linkage may also be derived based on self and mutual inductances:

  9. Example 1A It is instructive to illustrate the method of deriving an equivalent T circuit from open- and short-circuit measurements. For this purpose let us assume that when coil 2 of the two-winding transformer shown in Fig. is open-circuited, the power input to coil 1 is 12 W with an applied voltage is 100 V (rms) at 60 Hz and the current is 1 A (rms). When coil 2 is short-circuited, the current flowing in coil 1 is 1 A when the applied voltage is 30 V at 60 Hz. The power during this test is 22 W. If we assume Ll1= L’l2, an approximate equivalent T circuit can be determined from these measurements with coil 1 selected as the reference coil.

  10. Equivalent T Circuit of Transformer Where i’2 =(N2/N1)i2

  11. Example 1….. The Power may be expressed: Where are phasor and  is the phase angle between them. Solving for  during the open circuit test, we have:

  12. Example 1…… as a reference phasor and in an inductive circuit of the transformer I phasor would lag behind by the angle of  =83.70  V I Z, the impedance may therefore be determined by: That suggests that Xl1+Xm1 = 109.3, while r1 =12 

  13. Example 1……. For short circuit test i1= -I’2 because transformers are designed so that Xm1>> |r’2+jX’12|. Hence using phase angle equation: In this case input Impedance is (r1+r’2)+j(Xl1+X’l2) and that is determined by: That means r’2 = 10 and Xl1=X’l2 both are 10.2

  14. Example 1…. That leads to conclusion that: Xm1= 109.3 -10.2 =99.1 Hence other parameters are: r1 = 12 Lm1 = 262.9mH r’2 = 10 Ll1 = 27.1mH L’l2 = 27.1mH V1  E1 I1 V’2

  15. Phasor Diagram

  16. T circuit ref. to Primary

  17. Equivalent Circuit ref. to Primary

  18. Active & Reactive Power

  19. Magnetic Laws

  20. Flux Linkage of a Coil Fig. 1 shows a coil of N turns. All these N turns link flux lines of Weber resulting in the N flux linkages. In such a case: • Where • N is number of turns in a coil; • e is emf induced, and •  is flux linking to each coil

  21. Design of Transformer • Let's try to proportion a transformer for 120 V,60 Hz supply, with a full-load current of 10 A. The core material is to be silicon-steel laminations with a maximum operating flux density Bmax = 12,000 gauss. This is comfortably less than the saturation flux density, Bsat. The first requirement is to ensure that we have sufficient ampere-turns to magnetize the core to this level with a permissible magnetizing current I0A.

  22. Design of Transformer…. • Let's choose the magnetizing current to be 1% of the full-load current, or 0.1 A. The exact value is not sacred; this might be thought of as an upper limit. Here, we will assume a simple, uniform magnetic circuit for simplicity. If l is the length of the magnetic circuit, H is 0.4πN(√2I0)/l, and the magnetization curve for the core iron gives the H required for the chosen Bmax. From this, we can find the number of turns, N, required for the primary.

  23. Design of Transformer • We could also estimate the ampere-turns required by using an assumed permeability μ. Experience will furnish a satisfactory value. It is not taken from the magnetization curve, but from the hysteresis loop. Let's take μ = 1000. Then, N = Bmaxl / 0.4π√2 μI0. If we estimate l = 20 cm, the number of primary turns required is N = 1350. The rms voltage induced per turn is determined from Faraday's Law: √2 e = (2πf)BmaxA x 10-8. Now, e must be 120 / 1350 = 0.126 V/turn, f is 60, and Bmax = 12,000 gauss. We know everything but A, the cross-sectional area of the core. We find A = 2.8 cm2.

  24. Design of Transformer • Powdered iron and ferrite cores have low Bsat and permeability values. A type 43 ferrite has Bsat = 2750 gauss, but a maximum permeability of 3000, and is recommended for frequencies from 10 kHz to 1 MHz. Silicon iron is much better magnetically, but cannot be used at these frequencies. The approximate dimensions of an FT-114 ferrite core (of any desired material) are OD 28 mm, ID 19 mm, thickness 7.5 mm. The magnetic dimensions are l = 74.17 mm, A = 37.49 mm2, and volume 2778 mm3. Similar information is available for a wide range of cores. There are tables showing how much wire can be wound on them, and even the inductance as a function of the number of turns.

  25. Design Parameters B = /A Laminated Core A: X-section Area l average length N number of turns

  26. Laminations

  27. Flux Density & MMF Bmax

  28. Hysteresis loop

  29. Hysteresis Loop

More Related