Design of a Transformer

An ac–dc converter steps down the voltage through a transformer and supplies the load through a bridge rectifier, as shown in Figure 13.20a. Design a 60-Hz power transformer of the specifications: primary voltage $V_1 = 120 V, 60-Hz$ (square-wave), secondary voltage output $V_o = 40 V$ , and secondary output current $I_o = 6.5 A$ . Assume transformer efficiency η= 95% and window factor $K_u = 0.4$ . Use E-core.

Question

Design of a Transformer

An ac–dc converter steps down the voltage through a transformer and supplies the load through a bridge rectifier, as shown in Figure 13.20a. Design a 60-Hz power transformer of the specifications: primary voltage [latex]V_1 = 120 V, 60-Hz[/latex] (square-wave), secondary voltage output [latex]V_o = 40 V[/latex], and secondary output current [latex]I_o = 6.5 A[/latex]. Assume transformer efficiency η= 95% and window factor [latex]K_u = 0.4[/latex]. Use E-core.

Accepted Answer

[latex]K_t = 4.44[/latex] for square wave, and [latex]P_o = 40 ×6.5 = 260 W[/latex]. Using Eq. (13.51),

[latex]P_t=P_i+P_o=\frac{P_o}{\eta }+P_o=\left(1+\frac{1}{\eta } ight)P_o[/latex] (13.51)

[latex]P_t=\left(1+\frac{1}{0.95} ight)260=533.7 W[/latex].

From Table 13.1

Table 13.1 Core Configuration Constants

Core Type	[latex]K_j@25^\circ c[/latex]	[latex]K_j@50^\circ c[/latex]	x, Exponent	Core Losses
Pot core	433	632	-0.17	[latex]P_{cu} = P_{fe}[/latex]
Power core	403	590	-0.12	[latex]P_{cu} \gg P_{fe}[/latex]
E-laminated core	366	534	-0.14	[latex]P_{cu} = P_{fe}[/latex]
C-core	323	468	-0.14	[latex]P_{cu} = P_{fe}[/latex]
Single-coil	395	569	-0.14	[latex]P_{cu} \gg P_{fe}[/latex]
Tape-wound core	250	365	-0.13	[latex]P_{cu} = P_{fe}[/latex]

for E-laminated core, [latex]K_j = 366[/latex] and [latex]x = -0.14[/latex]. Let [latex]B_m = 1.4[/latex]. From Eq. (13.57),

[latex]A_p=\left[\frac{P_t imes 10^4}{K_tfB_mK_uK_j} ight]^{\frac{1}{1+x} } (cm^4)[/latex] (13.57)

[latex]A_p=\left[\frac{533.7 imes 10^4}{0.4 imes 60 imes 1.4 imes 0.4 imes 366} ight]^{\frac{1}{1+0.14} } = 206.1 cm^4[/latex]

Choose the E-core type, core2-138EI (Magnetics, Inc) with [latex]A_p = 223.39 cm^4[/latex], core weight
[latex]W_t = 3.901 kg[/latex], core area [latex]A_c = 24.4 cm^2[/latex], and mean length of a turn [latex]l_{mt} = 27.7 cm[/latex].

Using Eq. (13.50),

[latex]E=\frac{E_m}{\sqrt{2} } =\frac{2\pi fN\phi _m}{\sqrt{2} } =4.44fN\phi _m[/latex] (13.50)

the primary number of turns is [latex]N_p=\frac{V_1 imes 10^4}{K_tfB_mA_c}=132[/latex].

Using Eq. (13.52), the primary number of turns is

[latex]V_1=K_tfN_1\phi _m[/latex] (13.52)

[latex] N_p=\frac{V_1 imes 10^4}{K_tfB_mA_c} [/latex]

[latex]=\frac{120 imes 10^4}{4.44 imes 60 imes 1.4 imes 24.4}=132[/latex] (13.58)

The secondary number of turns is

[latex]N_s=\frac{N_p}{V_1}V_o[/latex]

[latex] =\frac{132 imes 40}{120}=44[/latex] (13.59)

From Eq. (13.56), [latex]J=K_jA_p^x=366 × 206.7^{-0.14}=173.6 A/cm^2[/latex]. The primary current is [latex]I_1 = (P_t - P_o)/V_1 = (533.7 - 260)/120 = 2.28 A[/latex]. The primary bare-wire cross-sectional area is [latex]A_{wp} = I_1/J = 0.28 A/173.6 = 0.013 cm^2[/latex].

From the wire-size Table B.2 (Appendix B), we find the primary AWG number 16 with [latex]\sigma _p=131.8 \mu \Omega /cm[/latex]. The primary winding resistance is [latex]R_p = l_{mt} N_p \sigma_p × 10-6 = 27.7 × /132 × 131.8 = 0.48 Ω[/latex]. The primary copper loss is [latex]P_p = I^2_p R_p = 2.28^2 × 0.48 = 2.5 W[/latex]. The secondary bare-wire cross-sectional area is [latex]A_{ws} = I_o/J = 6.5/173.6 = 0.037 cm^2[/latex].

Table B.2 Wire Size
	Bare Area		Resistance	Heavy Synthetics
AWG	Bare Area		[latex]10^{-6} Ω[/latex]	Area		Diameter		Turns-Per:		Turns-Per:		Weight
Wire Size	[latex]cm^210^{-3*}[/latex]	[latex]Cir-Mil^{**}[/latex]	cm at 20°C	[latex]cm^210^{-3}[/latex]	[latex]Cir-Mil^{**}[/latex]	cm	[latex]in.^{2**}[/latex]	cm	[latex]in.^{2**}[/latex]	[latex]cm^2[/latex]	[latex]in.^2[/latex]	gm/cm
10	52.61	10384	32.7	55.9	11046	0.267	0.1051	3.87	9.5	10.73	69.2	0.468
11	41.68	8226	41.37	44.5	8798	0.238	0.0938	4.36	10.7	13.48	89.95	0.375
12	33.08	6529	52.09	35.64	7022	0.213	0.0838	4.85	11.9	16.81	108.4	0.2977
13	26.26	5184	65.64	28.36	5610	0.19	0.0749	5.47	13.4	21.15	136.4	0.2367
14	20.82	4109	82.8	22.95	4556	0.171	0.0675	6.04	14.8	26.14	168.6	0.1879
15	16.51	3260	104.3	18.37	3624	0.153	0.0602	6.77	16.6	32.66	210.6	0.1492
16	13.07	2581	131.8	14.73	2905	0.137	0.0539	7.32	18.6	40.73	262.7	0.1184
17	10.39	2052	165.8	11.68	2323	0.122	0.0482	8.18	20.8	51.36	331.2	0.0943
18	8.228	1624	209.5	9.326	1857	0.109	0.0431	9.13	23.2	64.33	414.9	0.07472
19	6.531	1289	263.9	7.539	1490	0.098	0.0386	10.19	25.9	79.85	515·0	0.0594
20	5.188	1024	332.3	6.065	1197	0.0879	0.0346	11.37	28.9	98.93	638.1	0.04726
21	4.116	812.3	418.9	4.837	954.8	0.0785	0.0309	12.75	32.4	124	799.8	0.03757
22	3.243	640.1	531.4	3.857	761.7	0.0701	0.0276	14.25	36.2	155.5	1003	0.02965
23	2.588	510.8	666	3.135	620·0	0.0632	0.0249	15.82	40.2	191.3	1234	0.02372
24	2.047	404	842.1	2.514	497.3	0.0566	0.0223	17.63	44.8	238.6	1539	0.01884
25	1.623	320.4	1062·0	2.002	396·0	0.0505	0.0199	19.8	50.3	299.7	1933	0.01498
26	1.28	252.8	1345·0	1.603	316.8	0.0452	0.0178	22.12	56.2	374.2	2414	0.01185
27	1.021	201.6	1687.6	1.313	259.2	0.0409	0.0161	24.44	62.1	456.9	2947	0.00945
28	0.8046	158.8	2142.7	1.0515	207.3	0.0366	0.0144	27.32	69.4	570.6	3680	0.00747
29	0.647	127.7	2664.3	0.8548	169·0	0.033	0.013	30.27	76.9	701.9	4527	0.00602
30	0.5067	100·0	3402.2	0.6785	134.5	0.0294	0.0116	33.93	86.2	884.3	5703	0.00472
31	0.4013	79.21	4294.6	0.5596	110.2	0.0267	0.0105	37.48	95.2	1072	6914	0.00372
32	0.3242	64·00	5314.9	0.4559	90.25	0.0241	0.0095	41.45	105.3	1316	8488	0.00305
33	0.2554	50.41	6748.6	0.3662	72.25	0.0216	0.0085	46.33	117.7	1638	10565	0.00241
34	0.2011	39.69	8572.8	0.2863	56.25	0.0191	0.0075	52.48	133.3	2095	13512	0.00189
35	0.1589	31.36	10849	0.2268	44.89	0.017	0.0067	58.77	149.3	2645	17060	0.0015
36	0.1266	25·00	13608	0.1813	36·0	0.0152	0.006	65.62	166.7	3309	21343	0.00119
37	0.1026	20.25	16801	0.1538	30.25	0.014	0.0055	71.57	181.8	3901	25161	0.000977
38	0.08107	16·00	21266	0.1207	24.01	0.0124	0.0049	80.35	204.1	4971	32062	0.000773
39	0.06207	12.25	27775	0.0932	18.49	0.0109	0.0043	91.57	232.6	6437	41518	0.000593
40	0.04869	9.61	35400	0.0723	14.44	0.0096	0.0038	103.6	263.2	8298	53522	0.000464
41	0.03972	7.84	43405	0.0584	11.56	0.00863	0.0034	115.7	294.1	10273	66260	0.000379
42	0.03166	6.25	54429	0.04558	9·00	0.00762	0.003	131.2	333.3	13163	84901	0.000299
43	0.02452	4.84	70308	0.03683	7.29	0.00685	0.0027	145.8	370.4	16291	105076	0.000233
44	0.0202	4·00	85072	0.03165	6.25	0.00635	0.0025	157.4	400·0	18957	122272	0.000195
	A	B	C	D	E	F	G	H	I	J	K	L
This notation means the entry in the column must be multiplied by [latex]10^{-3}[/latex]. *This data from REA Magnetic Wire Datalator. Source: Arnold Engineering Company, Magnetics Technology Center, Marengo, IL. www.grouparnold.com/mtc/index.htm

From the wire-size Table B.2 (Appendix B), we find the secondary AWG number 16 with [latex]\sigma _s=41.37 \mu \Omega /cm[/latex]. The secondary winding resistance is [latex]R_s = l_{mt} N_s\sigma _s × 10^{-6} = 27.7 × /44 × 41.37 = 0.05 Ω[/latex]. The secondary copper loss is [latex]P_s = I^2_o R_s = 6.5^2 × 0.05 = 2.1 W[/latex].
Using Figure B.1 (Appendix B), the transformer core loss is [latex]P_{fe} = 0.557 × 10^{-3} × f ^{1.68} × B_m^{1.86} = 0.557 × 10^{-3} × 60^{1.68} × 1.4^{1.86} = 3.95 W[/latex]. The transformer efficiency is [latex]η= P_o /(P_o + P_p + P_s + P_{fe}) = 260/(260 + 2.5 + 2.1 + 3.9)= 96\%[/latex].

Question 13.6: Design of a Transformer An ac–dc converter steps down the vo...

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