Motor Thermal Calculator.xlsx

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2	Motor Thermal Calculator
3	Author:	Aaron M. Johnson	amj1@cmu.edu
4	License:	CC BY 4.0	https://creativecommons.org/licenses/by/4.0/
5	To Use:	Change values in yellow boxes to match motor and operating conditions.
6		Green boxes are calculated automatically.
7
8	Assumptions:	Single heating and cooling cycle, starting at ambient temperature.
9		Using Euler integration, may not be accurate for large dt or fast responses.
10		Thermal characteristics assume rotating motor to spread heat among coils, stall will lead to hot spots.
11		Dynamic characteristics are for ambient temperature (e.g. torque is calculated at ambient).
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14	Motor Characteristics:
15	Dynamic:
16		Torque constant:	k_t	26.1	mNm/A	(Maxon line 12)
17		Gear ratio:	G	80	:1
18	Thermal:
19		Winding resistance (at 25C):	R	0.836	Ohms	(Maxon line 10)
20		Thermal resistance housing-ambient:	R_2	8.96	K/W	(Maxon line 17)
21		Thermal resistance winding-housing:	R_1	0.74	K/W	(Maxon line 18)
22		Thermal time constant winding:	t_1	4.12	s	(Maxon line 19)
23		Thermal time constant motor:	t_2	968	s	(Maxon line 20)
24		Max winding temperature:	T_max	155	C	(Maxon line 22)
25		Thermal resistance coefficient	a_cu	0.0039	Ohm/K	(Maxon p.46)
26	Calculated:
27		Thermal mass winding:	C_1	5.568	J/K	t_1/R_1
28		Thermal mass motor:	C_2	108.036	J/K	t_2/R_2
29
30	Operating Conditions:
31		Ambient temperature	T_amb	25	C
32		Temperature safety margin	T_s	25	C	Safe temp:	130
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36	Temperature Response Curves:
37						Equivalent motor current:
38	(optional)	Output torque	tau	24	Nm	11.494	A
39		Applied current	i	11.494	A
40		On duration	t_on	30	s	Integration dt:
41		Total duration	t_max	60	s	0.25	s
42	Calculated:					(Fixed 240 steps in table)
43		Max temp	T_core	183.180	C
44		Time of T_core>T_safe	t	10.75	s	(Up to dt resolution)
45		Time of T_core>T_max	t	18.25	s
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47
48	Thermal Equations:
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53								Integrating equations:
54								Timestep	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63	64	65	66	67	68	69	70	71	72	73	74	75	76	77	78	79	80	81	82	83	84	85	86	87	88
55		Image source, with discussion of motor selection.						Time	0	0.25	0.50	0.75	1.00	1.25	1.50	1.75	2.00	2.25	2.50	2.75	3.00	3.25	3.50	3.75	4.00	4.25	4.50	4.75	5.00	5.25	5.50	5.75	6.00	6.25	6.50	6.75	7.00	7.25	7.50	7.75	8.00	8.25	8.50	8.75	9.00	9.25	9.50	9.75	10.00	10.25	10.50	10.75	11.00	11.25	11.50	11.75	12.00	12.25	12.50	12.75	13.00	13.25	13.50	13.75	14.00	14.25	14.50	14.75	15.00	15.25	15.50	15.75	16.00	16.25	16.50	16.75	17.00	17.25	17.50	17.75	18.00	18.25	18.50	18.75	19.00	19.25	19.50	19.75	20.00	20.25	20.50	20.75	21.00	21.25	21.50	21.75	22.00
56								R	0.836	0.852	0.868	0.883	0.897	0.910	0.924	0.936	0.948	0.960	0.971	0.982	0.992	1.002	1.011	1.021	1.029	1.038	1.046	1.054	1.061	1.069	1.076	1.083	1.089	1.096	1.102	1.108	1.113	1.119	1.124	1.129	1.134	1.139	1.144	1.148	1.153	1.157	1.161	1.165	1.169	1.173	1.177	1.181	1.184	1.188	1.191	1.194	1.198	1.201	1.204	1.207	1.210	1.213	1.216	1.218	1.221	1.224	1.227	1.229	1.232	1.234	1.237	1.239	1.242	1.244	1.246	1.249	1.251	1.253	1.255	1.258	1.260	1.262	1.264	1.266	1.268	1.270	1.273	1.275	1.277	1.279	1.281	1.283	1.285	1.287	1.289	1.291	1.292
57		Heat generated at 25C	h	110.451	W	i^2 R		h	110.451	112.587	114.635	116.599	118.482	120.289	122.022	123.686	125.283	126.817	128.290	129.705	131.065	132.372	133.630	134.839	136.002	137.122	138.201	139.239	140.240	141.205	142.135	143.033	143.899	144.735	145.543	146.324	147.078	147.808	148.515	149.199	149.861	150.503	151.125	151.729	152.315	152.884	153.437	153.975	154.498	155.007	155.503	155.986	156.456	156.916	157.364	157.802	158.229	158.647	159.057	159.457	159.849	160.233	160.610	160.980	161.343	161.699	162.049	162.394	162.733	163.066	163.394	163.718	164.037	164.351	164.662	164.968	165.271	165.570	165.865	166.158	166.447	166.733	167.017	167.298	167.576	167.852	168.126	168.398	168.667	168.935	169.201	169.465	169.727	169.988	170.248	170.505	170.762
58		Heat flow through R_1		h_1 = (T_core-T_case)/R_1				dT_core/dt		19.838	19.018	18.236	17.489	16.777	16.098	15.449	14.831	14.241	13.678	13.141	12.629	12.140	11.674	11.229	10.805	10.400	10.014	9.646	9.294	8.959	8.639	8.334	8.043	7.766	7.501	7.249	7.008	6.778	6.559	6.350	6.151	5.961	5.780	5.607	5.442	5.285	5.135	4.992	4.856	4.726	4.602	4.484	4.372	4.265	4.162	4.065	3.972	3.883	3.799	3.718	3.642	3.569	3.499	3.433	3.370	3.309	3.252	3.198	3.146	3.096	3.049	3.004	2.961	2.921	2.882	2.845	2.810	2.777	2.745	2.715	2.687	2.659	2.633	2.609	2.585	2.563	2.542	2.522	2.503	2.485	2.468	2.452	2.437	2.422	2.408	2.395	2.383
59		Derivative of T_core		dT_core/dt = (h - h_1 ) / C_1				T_core	25	29.960	34.714	39.273	43.645	47.840	51.864	55.726	59.434	62.994	66.414	69.699	72.856	75.891	78.810	81.617	84.318	86.918	89.422	91.833	94.157	96.397	98.556	100.640	102.651	104.592	106.467	108.280	110.032	111.726	113.366	114.954	116.491	117.982	119.427	120.828	122.189	123.510	124.794	126.042	127.256	128.438	129.588	130.709	131.802	132.868	133.909	134.925	135.918	136.889	137.839	138.768	139.679	140.571	141.446	142.304	143.146	143.974	144.787	145.586	146.372	147.146	147.909	148.660	149.400	150.130	150.851	151.562	152.265	152.959	153.645	154.324	154.996	155.660	156.319	156.971	157.617	158.258	158.894	159.524	160.150	160.771	161.388	162.001	162.611	163.216	163.818	164.417	165.013
60		Heat flow through R_2		h_2 = (T_case-T_amb)/R_2				dT_case/dt		0.000	0.062	0.121	0.178	0.232	0.284	0.333	0.380	0.425	0.468	0.510	0.549	0.587	0.623	0.657	0.690	0.721	0.751	0.780	0.808	0.834	0.859	0.883	0.906	0.928	0.950	0.970	0.989	1.008	1.026	1.043	1.059	1.075	1.090	1.104	1.118	1.131	1.144	1.156	1.168	1.179	1.190	1.200	1.210	1.220	1.229	1.238	1.246	1.254	1.262	1.270	1.277	1.284	1.291	1.298	1.304	1.310	1.316	1.322	1.327	1.333	1.338	1.343	1.348	1.352	1.357	1.361	1.366	1.370	1.374	1.378	1.382	1.385	1.389	1.392	1.396	1.399	1.402	1.406	1.409	1.412	1.415	1.418	1.421	1.423	1.426	1.429	1.432
61		Derivative of T_case		dT_case/dt = (h_1 - h_2) / C_2				T_Case	25	25	25.016	25.046	25.090	25.148	25.219	25.302	25.398	25.504	25.621	25.748	25.886	26.032	26.188	26.352	26.525	26.705	26.893	27.088	27.290	27.498	27.713	27.934	28.161	28.393	28.630	28.873	29.120	29.372	29.628	29.889	30.154	30.422	30.695	30.971	31.250	31.533	31.819	32.108	32.400	32.694	32.992	33.292	33.594	33.899	34.206	34.516	34.827	35.141	35.456	35.774	36.093	36.414	36.737	37.061	37.387	37.715	38.044	38.374	38.706	39.039	39.374	39.709	40.046	40.384	40.724	41.064	41.405	41.748	42.091	42.436	42.781	43.127	43.475	43.823	44.172	44.521	44.872	45.223	45.576	45.929	46.282	46.637	46.992	47.348	47.704	48.062	48.420
62		Resistance temperature correction		R_T = R_25 * (1 + a_cu * (T-25))				T_safe	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130	130
63								T_max	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155	155
64	Steady state calculations (without resistance temperature correction):
65		Steady state core temp	T_core(inf)	1096.37	C	(i^2 R)(R_1+R_2) + T_amb
66		Steady state case temp	T_case(inf)	1014.64	C	(i^2 R)(R_2) + T_amb
67		Steady state relative temp	T_core-T_case	81.73	C	(i^2 R)(R_1)
68		Time to reach full steady state	t_ss	4840	s	~5*t_2
69		Time to reach relative steady state	t_rs	21		~5*t_1
70		(Relative temp, T_core-T_case, will reach steady state much faster, then both temperatures increase slowly)
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