电桥实验

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这个实验不需要提交线上实验报告所以只写了数据处理，图一乐

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简易电桥测电阻

原始数据

供桥电压/{V}	电流计内阻/ \Omega	N	{R_0} / \Omega	{R^{'}_0} / \Omega	{R_x} / \Omega	\Delta d/格	\Delta R_0 = {\|}R_{01} - R_{02}{\|}/\Omega
4.0	2500	1.0	493.0	499.0		2	28
4.0	100	1.0	501.0	495.0		2	6
8.0	2500	1.0	500.0	494.0		2	14
8.0	100	1.0	495.9	502.0		2	3.5

计算R_x

R_{x} = \sqrt{{R_0} * {R_0^{'}}}

R_{x1} = \sqrt{{R_0} * {R_0^{'}}} = \sqrt{493.0 * 499.0} = 496.0 \Omega

R_{x2} = \sqrt{{R_0} * {R_0^{'}}} = \sqrt{501.0 * 495.0} = 498.0 \Omega

R_{x3} = \sqrt{{R_0} * {R_0^{'}}} = \sqrt{500.0 * 494.0} = 497.0 \Omega

R_{x4} = \sqrt{{R_0} * {R_0^{'}}} = \sqrt{495.9 * 502.0} = 498.9 \Omega

待测电阻的不确定度

原理

1. 电桥灵敏度

S = \frac{\Delta d}{|R_{01}-R_{02}|}R_0 \ (1)

2. 电桥灵敏度带来的误差

\Delta R_0^{*} = R_0\frac{0.2}{S} \ (2)

3. 电阻箱R_0的仪器误差为

\Delta_仪 = \Delta_{R_{0仪}} = 0.1\%R_0 + 0.005(K+1) \ (3)

4. 联立(1)和(2)可得

\Delta R_0^* = R0 \frac{0.2}{\frac{\Delta d}{|R_{01}-R_{02}|}R_0} = \frac{0.2|R_{01}-R_{02}|}{\Delta d}

计算

1. 电桥灵敏度误差

\Delta d = 0.2时
\begin{aligned} \\ &\Delta R_{0}^{*} = \frac{0.2|R_{01}-R_{02}|}{\Delta d} = |R_{01}-R_{02}| \ / \ 10 \\ &\Delta R_{0_1}^{*} = \frac{0.2|R_{01}-R_{02}|}{\Delta d} = |R_{01}-R_{02}| \ / \ 10 = 2.8 \\ &\Delta R_{0_2}^{*} = \frac{0.2|R_{01}-R_{02}|}{\Delta d} = |R_{01}-R_{02}| \ / \ 10 = 0.6 \\ &\Delta R_{0_3}^{*} = \frac{0.2|R_{01}-R_{02}|}{\Delta d} = |R_{01}-R_{02}| \ / \ 10 = 1.4 \\ &\Delta R_{0_4}^{*} = \frac{0.2|R_{01}-R_{02}|}{\Delta d} = |R_{01}-R_{02}| \ / \ 10 = 0.35 \\ \end{aligned}

2. 电阻箱R_0的仪器误差

\begin{aligned} \Delta_{仪_1} = \Delta_{R_{0仪}} &= 0.1\%R_0 + 0.005(K+1) \\ &= 0.1\%*493.0+0.005*(4+1) \\ &= 0.518 \\ \Delta_{仪_1^{'}} = \Delta_{R_{0仪}} &= 0.1\%R_0^{'} + 0.005(K+1) \\ &= 0.1\%*499.0+0.005*(4+1) \\ &= 0.524 \\ \end{aligned}

\begin{aligned} \\ \Delta_{仪_2} = \Delta_{R_{0仪}} &= 0.1\%R_0 + 0.005(K+1) \\ &= 0.1\%*501.0+0.005*(4+1) \\ &= 0.526 \\ \Delta_{仪_2^{'}} = \Delta_{R_{0仪}} &= 0.1\%R_0^{'} + 0.005(K+1) \\ &= 0.1\%*495.0+0.005*(4+1) \\ &= 0.52 \\ \end{aligned}

\begin{aligned} \\ \Delta_{仪_3} = \Delta_{R_{0仪}} &= 0.1\%R_0 + 0.005(K+1) \\ &= 0.1\%*500.0+0.005*(4+1) \\ &= 0.525 \\ \Delta_{仪_3^{'}} = \Delta_{R_{0仪}} &= 0.1\%R_0^{'} + 0.005(K+1) \\ &= 0.1\%*494.0+0.005*(4+1) \\ &= 0.519 \\ \end{aligned}

\begin{aligned} \\ \Delta_{仪_4} = \Delta_{R_{0仪}} &= 0.1\%R_0 + 0.005(K+1) \\ &= 0.1\%*495.9+0.005*(4+1) \\ &= 0.5209 \\ \Delta_{仪_4^{'}} = \Delta_{R_{0仪}} &= 0.1\%R_0^{'} + 0.005(K+1) \\ &= 0.1\%*502.0+0.005*(4+1) \\ &= 0.527 \\ \end{aligned}

3. 不确定度U_{R_0}

\begin{aligned} U_{R_0}&=\sqrt{(\Delta_{R_{0仪}})^2+(\Delta_{R_{0}^*})^2}\\ U_{R_{01}}&=\sqrt{(\Delta_{R_{0仪}})^2+(\Delta_{R_{0}^*})^2}\\ &=\sqrt{0.518^2+2.8^2}\\ &=2.85\\ U_{R_{01}^{'}}&=\sqrt{(\Delta_{R_{0仪}})^2+(\Delta_{R_{0}^*})^2}\\ &=\sqrt{0.524^2+2.8^2}\\ &=2.85\\ \\ U_{R_{02}}&=\sqrt{(\Delta_{R_{0仪}})^2+(\Delta_{R_{0}^*})^2}\\ &=\sqrt{0.526^2+0.6^2}\\ &=0.80\\ U_{R_{02}^{'}}&=\sqrt{(\Delta_{R_{0仪}})^2+(\Delta_{R_{0}^*})^2}\\ &=\sqrt{0.52^2+0.6^2}\\ &=0.79\\ \\ U_{R_{03}}&=\sqrt{(\Delta_{R_{0仪}})^2+(\Delta_{R_{0}^*})^2}\\ &=\sqrt{0.525^2+1.4^2}\\ &=1.5\\ U_{R_{03}^{'}}&=\sqrt{(\Delta_{R_{0仪}})^2+(\Delta_{R_{0}^*})^2}\\ &=\sqrt{0.519^2+1.4^2}\\ &=1.49\\ \\ U_{R_{04}}&=\sqrt{(\Delta_{R_{0仪}})^2+(\Delta_{R_{0}^*})^2}\\ &=\sqrt{0.5209^2+0.35^2}\\ &=0.63\\ U_{R_{04}^{'}}&=\sqrt{(\Delta_{R_{0仪}})^2+(\Delta_{R_{0}^*})^2}\\ &=\sqrt{0.527^2+0.35^2}\\ &=0.63\\ \end{aligned}

4. 不确定度U_{R_x}

\begin{aligned} U_{R_x} &\approx \frac{\sqrt{2}}{2}\frac{U_{R_0}}{R_0}R_{x}\\ U_{R_{x1}} &\approx \frac{\sqrt{2}}{2}\frac{U_{R_0}}{R_0}R_{x1} = 2.02\\ U_{R_{x1}^{'}} &\approx \frac{\sqrt{2}}{2}\frac{U_{R_0}}{R_0}R_{x1} = 2.00\\ \\ U_{R_{x2}} &\approx \frac{\sqrt{2}}{2}\frac{U_{R_0}}{R_0}R_{x2} = 0.56\\ U_{R_{x2}^{'}} &\approx \frac{\sqrt{2}}{2}\frac{U_{R_0}}{R_0}R_{x2} = 0.56\\ \\ U_{R_{x3}} &\approx \frac{\sqrt{2}}{2}\frac{U_{R_0}}{R_0}R_{x3} = 1.05\\ U_{R_{x3}^{'}} &\approx \frac{\sqrt{2}}{2}\frac{U_{R_0}}{R_0}R_{x3} = 1.06\\ \\ U_{R_{x4}} &\approx \frac{\sqrt{2}}{2}\frac{U_{R_0}}{R_0}R_{x4} = 0.45\\ U_{R_{x4}^{'}} &\approx \frac{\sqrt{2}}{2}\frac{U_{R_0}}{R_0}R_{x4} = 0.44\\ \end{aligned}

最终取平均值

测量结果

供桥电压/{V}	电流计内阻/ \Omega	N	{R_0} / \Omega	{R^{'}_0} / \Omega	{R_x} / \Omega	\Delta d/格	\Delta R_0 = {\|}R_{01} - R_{02}{\|}/\Omega	不确定度U_{R_x}
4.0	2500	1.0	493.0	499.0	496.0\plusmn2.01	2	28	2.01
4.0	100	1.0	501.0	495.0	498.0\plusmn0.56	2	6	0.56
8.0	2500	1.0	500.0	494.0	497.0\plusmn1.055	2	14	1.055
8.0	100	1.0	495.9	502.0	498.9\plusmn0.445	2	3.5	0.445

箱式电桥测电阻

原始数据

被测电阻	N	R_0 / \Omega	R_x=NR_0/\Omega	\Delta d/格	\Delta R_0 = {\|}R_{01} - R_{02}{\|}/\Omega
R_{x1}	100	507.08		2	0.08
R_{x2}	1000	505.04		2	0.1

不确定度

箱式电桥灵敏度的误差

原理

\Delta R_x^* = NR_0\frac{0.2}{S}

S=\frac{\Delta d}{|R_{01}-R_{02}|}R_0

计算

\Delta R_x^* = NR_0\frac{0.2}{\frac{\Delta d}{|R_{01}-R_{02}|}R_0} = N\frac{0.2|R_{01}-R_{02}|}{\Delta d}

\begin{aligned} \Delta R_{x1}^* &= NR_0\frac{0.2}{\frac{\Delta d}{|R_{01}-R_{02}|}R_0} = N\frac{0.2|R_{01}-R_{02}|}{\Delta d} \\ &=100*0.1*0.08=0.8\\ \Delta R_{x2}^* &= NR_0\frac{0.2}{\frac{\Delta d}{|R_{01}-R_{02}|}R_0} = N\frac{0.2|R_{01}-R_{02}|}{\Delta d}\\ &=1000*0.1*0.1=1.0\\ \end{aligned}

极限误差

E_{lim1}=\plusmn(0.2\%NR_0+100) = 201.416
E_{lim2}=\plusmn(0.2\%NR_0+100) = 6050.4

总不确定度

U_{R_x}=\sqrt{E_{lim}^2+(\Delta R_x^*)^2}
U_{R_{x1}}=\sqrt{E_{lim}^2+(\Delta R_x^*)^2} = 201.42
U_{R_{x2}}=\sqrt{E_{lim}^2+(\Delta R_x^*)^2} = 6050.40

结果

被测电阻	N	R_0 / \Omega	R_x=NR_0/\Omega	\Delta d/格	\Delta R_0 = {\|}R_{01} - R_{02}{\|}/\Omega	不确定度
R_{x1}	100	507.08	50708\plusmn201.42	2	0.08	201.42
R_{x2}	1000	505.04	505040\plusmn6050.40	2	0.1	6050.40