Boise State University Department of Electrical and Computer Engineering ECE 212L Circuit Analysis and Design Lab

Transcription

1 Bose State Unersty Department of Electrcal and Computer Engneerng ECE Crcut Analyss and Desgn ab Experment #6: Transformers, Mutual Inductance, and Coupled Cols Objectes The objectes of ths laboratory experment are: To obsere the phenomenon of nducte couplng To perform polarty and turns rato tests on a transformer To measure the self and mutual nductances of a transformer To measure the nductance of two coupled seres-adng or seres-opposng cols Theory Consder the two-wndng transformer shown n Fgure (a) and assume that the prmary wndng s energzed wth a current whle the secondary wndng s open-crcuted. The prmary current creates a magnetc flux ϕ n the ron core wth two components: A large porton of ths prmary flux, the mutual flux ϕ m, s channeled through the magnetc core and lnks all turns of the secondary wndng; A small porton of ths prmary flux, the leakage flux ϕ l, lnks all turns of the prmary wndng but none of the turns of the secondary wndng; These two fluxes result n the followng flux lnkages and nductances n each wndng: λ = ϕ = (ϕ l ϕ m ) = () λ = ϕ m = Smlarly, assume that the secondary wndng s energzed wth a current whle the prmary wndng s open-crcuted as shown n Fgure (b). The secondary current creates a magnetc flux ϕ n the ron core wth two components: A large porton of ths secondary flux, the mutual flux ϕ m, s channeled through the magnetc core and lnks all turns of the prmary wndng; A small porton of ths secondary flux, the leakage flux ϕ l, lnks all turns of the secondary wndng but none of the turns of the prmary wndng; These two fluxes result n the followng flux lnkages and nductances n each wndng: λ = ϕ m = (3) λ = ϕ = (ϕ l ϕ m ) = () ()

2 m l (a) m l (b) m m l l M (c) (d) Fgure : Tests for Measurng Transformer Self and Mutual Inductances (a)-(c) and Transformer Symbol (d) If both wndngs are energzed smultaneously as n Fgure (c), the resultng flux lnkages of each col are gen by: λ = (ϕ l ϕ m ϕ m ) = (5) λ = (ϕ m ϕ l ϕ m ) = (6) Accordng to Faraday s law, the oltages nduced n each col are (t) = R dλ (t) = R dλ d = R d = R d where R and R are the resstances of the prmary and secondary wndngs, respectely. d (7) (8)

3 In phasor form, these oltages are gen by Ṽ = (R jω )Ĩ jω Ĩ (9) Ṽ = jω Ĩ (R jω )Ĩ (0) It wll be shown n ths experment that the mutual nductances and between the two cols are equal. From the test n Fgure (a), V = I R (ω ) = I R X () V = ω I = X I () The resstance R can be measured wth an ohmmeter. Gen the rms oltages V and V and the rms current I, both X = ω and X = ω can be determned. Smlarly, from the test n Fgure (b), both X = ω and X = ω can be determned. It can then be erfed that the mutual reactances X = X = X M are equal and so are the mutual nductances = = M. The fnal model of the transformer s shown n Fgure (d). The coeffcent of couplng k between the two wndngs s defned as k = M = M M = ϕ m (ϕ m ϕ l ) ϕ m (ϕ m ϕ l ) From the aboe relatonshp, t s easy to check that k s always less than unty. An alternate method of determnng the mutual nductance of two coupled cols s shown n Fgure (a). Suppose that the two cols are coupled such that they create fluxes that ad each other n the core. The total flux lnkages of the seres-adng combnaton s λ p = λ λ = ( M ) (M ) = ( M) = p p () where p = M the total nductance of ths col and p = = =. ote that p = (R R ) p dλ p = (R R ) p p d p Ṽ p = (R R )Ĩp jω p Ĩ p = V p = I p (R R ) (ω p ) (6) On the other hand, f the cols are connected so that ther fluxes are opposng each other n the core, then the total flux lnkages of the seres-opposng combnaton s λ n = λ λ = ( M ) (M ) = ( M) = n n (7) where n = M the total nductance of ths col and n = = =. ote that n = (R R ) n dλ n = (R R ) n n d n Ṽ n = (R R )Ĩn jω n Ĩ n = V n = I n (R R ) (ω n ) (9) Measurng p and n leads to the followng alue of the mutual nductance M: (3) (5) (8) M = p n (0) 3

4 M M = = p = = n (a) (b) Fgure : (a) Seres-Adng and (b) Seres-Opposng (b) Connecton of two Coupled Cols 3 Equpment Aglent DSO50A Dgtal Storage Osclloscope HP/Aglent 30A Benchtop Multmeter Fluke 0 or True RMS Multmeter.6-V AC Power Supply and two-wndng transformer Procedure. Set up a step-down two-wndng transformer wth the 00-turn col on the prmary sde connected to the.6-v AC power supply and leae the 00-turn col open-crcuted on the secondary sde as shown n Fgure 3(a). Turn the AC power swtch O and apply.6 V to the 00-turn col. Make sure that the labels ndcatng the number of turns (00 and 00 turns, respectely) are showng on the top faces of each col. abel the nput prmary termnals as - wth termnal beng the poste termnal and termnal beng the negate termnal. Assume mentally that termnal s marked wth a dot. Smlarly, label the termnals on the secondary sde as 3-. Assume that termnal 3 s the poste termnal and termnal the negate one. Obsere the oltages (t) and 3 (t) on the osclloscope and record the peak-to-peak ampltudes of both oltages as well as ther phase shft. (Ths phase shft s ether 0 o or 80 o.) Deduce where the second polarty dot should be placed (termnal 3 or termnal ). V,pp (V) V 3,pp (V) θ (deg) Dotted Termnals and. Hook up the two-wndng transformer as shown n Fgure 3(b). Measure and record the DC resstance of col - usng a handheld multmeter. Apply a.6 VAC to the prmary sde and record the prmary rms current and the prmary and secondary rms oltages. The benchtop multmeter s used to read the rms current n the prmary (00-turn) wndng. R (Ω) I (Arms) V (Vrms) V 3 (Vrms)

5 3. Turn off the AC power supply. Unhook the two-wndng transformer. Measure and record the DC resstance of col 3- usng a handheld multmeter. Connect the secondary sde to the AC power supply by applyng 6.3 VAC to the secondary wndng as shown n Fgure 3(c). Record the secondary rms current and the secondary and prmary rms oltages. R (Ω) I (Arms) V 3 (Vrms) V (Vrms). Connect termnals and 3 of the two-wndng transformer as shown n Fgure (a). Apply 6.3 V to termnals and. Record the seres rms current n both cols, the seres rms oltage of both cols, and the nddual oltages across each col. I = I (Arms) V (VAC) V (Vrms) V 3 (Vrms) 5. Connect termnals and of the two-wndng transformer as shown n Fgure (b). Apply.6 V to termnals and 3. Record the seres rms current n both cols, the seres rms oltage of both cols, and the nddual oltages across each col. I = I (Arms) V 3 (Vrms) V (Vrms) V 3 (Vrms) 5

6 V =.6 V 00 t. 00 t. V 3 (a) I A 00 t. 00 t..6 V V V V V (b) I 00 t. 00 t. A V V V V 6.3 V (c) Fgure 3: Polarty and Turns Rato Tests (a) and Self and Mutual Inductance Measurements (b)-(c) 6

7 00 t. 00 t. 3 V = 6.3 V (a) V 3 =.6 V 00 t. 00 t. 3 (b) Fgure : Inductance measurements of (a) seres-opposng and (b) seres-adng cols 5 Data Analyss and Interpretaton. Compute the oltage rato / 3 of the 00-turn and 00-turn cols n Part and compare t to the theoretcal alue correspondng to the turns rato 00/00 =. Explan why t s greater or smaller than the theoretcal alue.. Calculate the self and mutual reactances X and X from the measurements of Part. Deduce the self and mutual nductances and n mh, respectely. 3. Calculate the self and mutual reactances X and X from the measurements of Part 3. Deduce the self and mutual nductances and n mh, respectely.. Verfy that the mutual reactances X = X = X M and the mutual nductances = = M found aboe are equal. 5. Compute the coeffcent of couplng k of the two-wndng transformer from the measured alues of,,, and and check that t s less than unty. 6. Usng the measurements n Parts () and (5), compute the nductances p and n of the seres-adng and seres-opposng col connectons. Then compute = = M = p n and compare ths alue to the preously calculated alues from questons () and (3). 7

8 Bose State Unersty Department of Electrcal and Computer Engneerng ECE Crcut Analyss and Desgn ab Experment #6: Transformers, Mutual Inductance, and Coupled Cols Date: Recorded by: Data Sheet Equpment st Equpment Descrpton BSU Tag umber or Seral umber Aglent DSO50A Dgtal Storage Osclloscope HP/Aglent 30A Benchtop Multmeter Fluke True RMS Multmeter Part : Polarty Test #: V,pp (V) V 3,pp (V) θ (deg) Dotted Termnals and Part : Self and Mutual Inductance Measurement Test #: R (Ω) I (Arms) V (Vrms) V 3 (Vrms) Part 3: Self and Mutual Inductance Measurement Test #: R (Ω) I (Arms) V 3 (Vrms) V (Vrms) Part : Inductance Measurement of Two Seres-Opposng Coupled Cols: I = I (Arms) V (VAC) V (Vrms) V 3 (Vrms) Part 5: Inductance Measurement of Two Seres-Adng Coupled Cols: I = I (Arms) V 3 (Vrms) V (Vrms) V 3 (Vrms)