Lab3

3A – Simple Regression

Determine a simple linear regression model for $V_{in}$ as a function of $I$ using thermistor data, and compute $R$ and $\Delta R$.

Reference resistance:

$$R_0=20.221\text{k}\Omega$$

Given / Calculated Data:

$V_{in}$ (V)	$I$ (mA)	$I$ (A)	$R$ ($\Omega$)	$\Delta R$ ($\Omega$)
0.5	0.025	0.000025	20000.000	-221.000
2	0.108	0.000108	18518.519	-1702.481
3	0.166	0.000166	18072.289	-2148.711
4	0.232	0.000232	17241.379	-2979.621
6	0.337	0.000337	17804.153	-2416.847
7	0.449	0.000449	15590.201	-4630.799
8	0.524	0.000524	15267.176	-4953.824
10	0.904	0.000904	11061.947	-9159.053

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Fit Results Table (I, ΔR, fit ΔR, residual)

I (mA)	ΔR (Ω)	Fit ΔR (Ω)	Residual (Ω)
0.025	-221.000	-501.124	280.124
0.053	-1353.075	-954.819	-398.256
0.108	-1702.481	-1588.071	-114.410
0.166	-2148.711	-2179.231	30.520
0.203	-2979.621	-2568.839	-410.782
0.232	-2979.621	-2877.117	-102.504
0.303	-3719.350	-3201.298	-518.052
0.337	-2416.847	-3449.968	1033.121
0.449	-3516.101	-3783.982	267.881
0.449	-4630.799	-3783.982	-846.817
0.524	-4953.824	-3949.532	-1004.292
0.624	-5797.923	-4124.251	-1673.672
0.762	-6442.786	-4239.126	-2203.660
0.885	-6661.678	-4271.316	-2390.362
0.904	-9159.053	-3989.535	-5169.518

1. Fit ΔR calculated using cubic model:
   $\Delta R=-121.17-15.7\times 10^{6}I+2.15\times 10^{10}{I}^2-1.54\times 10^{13}{I}^3$
   ($I$ in amperes)
2. Residual = Observed ΔR - Fit ΔR

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Regression Model:

$$V_{in}=1.282810+11015.491419\cdot I$$

Fit Results Table:

Coefficient	Value	SE	T-ratio
$b_0$	1.282810	0.550642	2.329660
$b_1$	11015.491419	1271.386579	8.664156

• Correlation coefficient: $r=0.962283$

Interpretation of $T$-ratios:

• $|T(b_0)|=2.33$: intercept is statistically significant at the $\alpha \approx 0.05$ level.
• $|T(b_1)|=8.66$: slope is highly significant, indicating $I$ has a strong effect on $V_{in}$.

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Plots:

• Scatter plot of $I$ vs $V_{in}$ with regression line.
• Residual plot (horizontal line at 0).

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3B – Multiple Regression

Objective:
Fit a cubic polynomial model for $\Delta R$ as a function of $I$ using combined data from 3A and 3B tables.

Additional Data (from 3B Table):

$V_{in}$ (V)	$I$ (mA)	$I$ (A)	$R$ ($\Omega$)	$\Delta R$ ($\Omega$)
0.5	0.025	0.000025	20000.000	-221.000
1	0.053	0.000053	18867.925	-1353.075
3.5	0.203	0.000203	17241.379	-2979.621
5	0.303	0.000303	16501.650	-3719.350
7.5	0.449	0.000449	16704.899	-3516.101
9	0.624	0.000624	14423.077	-5797.923
10.5	0.762	0.000762	13778.214	-6442.786
12	0.885	0.000885	13559.322	-6661.678

Typical Errors:

• Voltage (V): ±0.01 V (multimeter accuracy)
• Current (I): ±0.01 mA (observed fluctuation range)

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Regression Model:

$$\Delta R=-121.171774-15726202.978733\cdot I+21510442950.493008\cdot I^2-15423209053119.478516\cdot I^3$$

Fit Results Table:

Coefficient	Value	SE	T-ratio
$b_0$	-121.171774	498.397050	-0.243123
$b_1$	-15726202.978733	5499932.944307	-2.859344
$b_2$	21510442950.493008	15064544044.972355	1.427885
$b_3$	-15423209053119.478516	10988650858090.128906	-1.403558

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Interpretation of $T$-ratios:

• $|T(b_1)|=2.86$: slope term is significant at the 95% level.
• $|T(b_0)|,|T(b_2)|,|T(b_3)|$ are below 2: these parameters are not statistically significant, suggesting possible model overfitting or redundancy in higher-order terms.

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Qualitative Performance Analysis

• Thermistor Behavior:
  The thermistor exhibits Negative Temperature Coefficient (NTC) characteristics:
  • As current increases, power dissipation (($P=I^{2}R$)) heats the device
  • Heating reduces resistance exponentially (Arrhenius law)
  • Creates positive feedback:
    Higher current → More heating → Lower resistance → Further current increase
• Nonlinearity Origin:
  The observed nonlinear ($\Delta R$) vs. (I) relationship stems from:
  • Exponential thermal response: Resistance decays as ($R\propto e^{1/T}$)
  • Power-law self-heating: Temperature rise ($\Delta T\propto I^2$)
  • Combined effect: Results in rapid resistance drop at higher currents

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Conclusion

• 3A: Strong positive correlation ($r\approx 0.962$) between current and voltage, with both $b_0$ and $b_1$ statistically significant.
• 3B: Cubic model captures non-linear $\Delta R$ behavior. Only $b_1$ is clearly significant, indicating the higher-order terms contribute less reliably to the fit.

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