{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "wheatstone-bridge-000",
"metadata": {
"execution": {
"iopub.execute_input": "2026-10-16T00:33:14.961382Z",
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"iopub.status.idle": "2026-10-16T00:33:15.647848Z",
"shell.execute_reply": "2026-10-16T00:33:15.647174Z"
},
"jupyter": {
"source_hidden": true
},
"tags": [
"setup"
]
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from pathlib import Path\n",
"import sys\n",
"ROOT=next(p for p in (Path.cwd(),*Path.cwd().parents) if (p/'pyproject.toml').exists())\n",
"sys.path.insert(0,str(ROOT/'src'))\n",
"import numpy as np\n",
"import pandas as pd\n",
"import matplotlib.pyplot as plt\n",
"from IPython.display import display,Markdown,HTML\n",
"from physicslab.common.theme import EXPORT_CSS\n",
"from physicslab.common.manuscripts import table_display,metrics_display,display_plotly,series_plot\n",
"from physicslab.experiments import wheatstone_bridge as adapter\n",
"EXP=ROOT/'experiments/course-03/wheatstone-bridge'\n",
"result=adapter.analyze(EXP)\n",
"import json\n",
"labels=json.loads((EXP/'authoring.json').read_text(encoding='utf-8')).get('metric_labels',{})\n",
"for key,value in result['metrics'].items():value.setdefault('label',labels.get(key,key))\n",
"display(HTML(EXPORT_CSS))\n"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-001",
"metadata": {},
"source": [
"# 单臂电桥测电阻\n",
"\n",
"淮南师范学院实验报告\n",
"\n",
"学院：电子工程学院 专业：物理学 班级：24 物理 1 班\n",
"\n",
"姓名：缪雨轩 学号：2414050120 实验日期：2025 年 11 月 5 日\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-002",
"metadata": {},
"source": [
"## 一、实验名称\n",
"\n",
"单臂电桥测电阻。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-003",
"metadata": {},
"source": [
"## 二、实验目的\n",
"\n",
"1. 理解惠斯通电桥比较法测量电阻的原理。\n",
"2. 掌握 QJ47 型直流单臂电桥的使用。\n",
"3. 计算平衡电阻并依据适用量程和仪器规格分析结果。\n",
"\n"
]
},
{
"attachments": {
"bridge.svg": {
"image/svg+xml": [
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]
}
},
"cell_type": "markdown",
"id": "wheatstone-bridge-004",
"metadata": {},
"source": [
"## 三、实验原理\n",
"\n",
"四臂为 $R_a,R_b,R_x,R$，电源接 A、C，检流计接 B、D。\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"图 1 惠斯通电桥节点与四臂连接示意\n",
"\n",
"\n",
"\n",
"平衡时 $V_B=V_D$、检流计电流为零，左右支路电流分别为 $I_1,I_2$，有\n",
"\n",
"$$\n",
"I_1R_a=I_2R_x,\\qquad I_1R_b=I_2R.\n",
"$$\n",
"\n",
"消去电流：\n",
"\n",
"$$\n",
"\\frac{R_a}{R_b}=\\frac{R_x}{R},\\qquad M=\\frac{R_a}{R_b},\\qquad R_x=MR.\n",
"$$\n",
"\n",
"不平衡时，在开路检流端、理想电源模型下\n",
"\n",
"$$\n",
"\\frac{V_B-V_D}{E}=\\frac{R_b}{R_a+R_b}-\\frac{R}{R_x+R}.\n",
"$$\n",
"\n",
"该电压可解释调节比较臂寻找零点的过程，实际检流计加载会改变不平衡电压，平衡条件保持不变。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-005",
"metadata": {},
"source": [
"#### 平衡附近的桥电压灵敏度\n",
"\n",
"令比较臂 $R=1$、比率臂 $R_b=1,R_a=M$（归一化模型），扫描 $R_x/(MR)$。默认 $M=1$；不把归一化阻值当成本次实际桥臂内阻。"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "wheatstone-bridge-006",
"metadata": {
"execution": {
"iopub.execute_input": "2026-10-16T00:33:15.649587Z",
"iopub.status.busy": "2026-10-16T00:33:15.649338Z",
"iopub.status.idle": "2026-10-16T00:33:15.914171Z",
"shell.execute_reply": "2026-10-16T00:33:15.913294Z"
},
"jupyter": {
"source_hidden": true
}
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import physicslab.common.manuscripts as _presentation\n",
"_presentation._table_captioned=False\n",
"display_plotly(adapter.explore(\"balance\",result))"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-007",
"metadata": {},
"source": [
"图 2 理想开路桥电压与比率臂灵敏度"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-008",
"metadata": {},
"source": [
"## 四、主要仪器设备\n",
"\n",
"QJ47 型直流单双臂电桥、五个待测电阻（标称 $3,47,200,2000,20000\\,\\Omega$）及导线。仪器总体标称 0.05 级，但说明书按测量区间另有等级条件，不可对所有结果统一应用。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-009",
"metadata": {},
"source": [
"## 五、实验步骤\n",
"\n",
"1. 熟悉比率盘、比较臂十进盘、粗调/细调、电源和指零仪开关。\n",
"2. 指零仪内接并调零，按估计阻值选比率 $M$，尽量使用比较臂高位。\n",
"3. 待测电阻接入，先粗调再细调，从高位到低位寻找平衡。\n",
"4. 松开按钮后读取 $M$ 和比较臂总示值 $R$，计算 $R_x=MR$。\n",
"5. 更换电阻重复，核查说明书有效范围、基准值及环境条件后再评价允许误差。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-010",
"metadata": {},
"source": [
"## 六、原始数据记录\n",
"\n",
"\n",
"\n",
"表 1 比率设定、比较臂与源处理结果\n",
"\n",
"\n",
"\n",
"| 标称 / Ω | 比率设定 | 比较臂 / Ω | 源处理结果 / Ω |\n",
"| --- | --- | --- | --- |\n",
"| 3 | 0.1 | 30.66 | 3.066 |\n",
"| 47 | 0.1 | 465.94 | 46.594 |\n",
"| 200 | 100/100 | 200.2 | 200.2 |\n",
"| 2000 | 10 | 197.97 | 1979.7 |\n",
"| 20000 | 100 | 204.21 | 20421 |\n",
"\n",
"$100/100$ 是比率盘设定标识，实际乘数为 1；没有将其作为两次独立读数。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-011",
"metadata": {},
"source": [
"## 七、数据处理\n",
"\n",
"### 1. 平衡值及标称比较\n",
"\n",
"$$\n",
"D=\\frac{R_x-R_{\\mathrm{nom}}}{R_{\\mathrm{nom}}}\\times100\\%.\n",
"$$\n",
"\n",
"\n",
"\n",
"表 2 平衡电阻复算及量程条件\n",
"\n",
"\n",
"\n",
"| 标称 / Ω | $M$ | 重算电阻 / Ω | 标称偏差 / % | 说明书等级条件 |\n",
"| --- | --- | --- | --- | --- |\n",
"| 3 | 0.1 | 3.066 | 2.2 | <10 Ω，超单桥有效范围 |\n",
"| 47 | 0.1 | 46.594 | -0.86383 | $0.05\\%$（参考条件下） |\n",
"| 200 | 1 | 200.2 | 0.1 | $0.05\\%$（参考条件下） |\n",
"| 2000 | 10 | 1979.7 | -1.015 | $0.05\\%$（参考条件下） |\n",
"| 20000 | 100 | 20421 | 2.105 | $0.1\\%$（参考条件下） |\n",
"\n",
"标称偏差只比较标称与读数，不等于仪器校准误差。\n",
"\n",
"### 2. 允许误差与适用性\n",
"\n",
"说明书单电桥有效范围下限为 $10\\,\\Omega$；$10$–$10^4\\,\\Omega$ 的等级指数为 $0.05\\%$，$10^4$–$10^5\\,\\Omega$ 为 $0.1\\%$，更高区间另有规定，参考温度为 $20\\pm2\\,^{\\circ}\\mathrm C$。\n",
"\n",
"其允许误差表达采用\n",
"\n",
"$$\n",
"E_{\\mathrm{lim}}=\\pm c\\left(\\frac{R_N}{10}+X\\right),\n",
"$$\n",
"\n",
"$c$ 为等级指数的无量纲数值，$R_N$ 为基准，$X$ 在说明书中定义为标度盘示值。源示例将 $c=0.0005,R_N=100,X=3.066$ 代入得 $0.006533\\,\\Omega$，算术可复算，但不能据此确认 $3\\,\\Omega$ 结果适用该单桥规格。\n",
"\n",
"\n",
"\n",
"表 3 源报告给出的限差表达（适用条件待核，非新认定结果）\n",
"\n",
"\n",
"\n",
"| 标称 / Ω | 源报告限差表达（待核适用条件） |\n",
"| --- | --- |\n",
"| 3 | $(3.066\\pm0.007)\\,\\Omega$ |\n",
"| 47 | $(46.59\\pm0.03)\\,\\Omega$ |\n",
"| 200 | $(200.2\\pm0.2)\\,\\Omega$ |\n",
"| 2000 | $(1980\\pm2)\\,\\Omega$ |\n",
"| 20000 | $(20420\\pm30)\\,\\Omega$ |\n",
"\n",
"本次各档基准和温度条件未完整记录，且 $3\\,\\Omega$ 在有效范围以下，故不将这些允许误差极限改称标准不确定度。若未来确认半宽为 $a$ 的均匀误差模型，才可据模型取 $u_B=a/\\sqrt3$ 并与其他分量合成。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-012",
"metadata": {},
"source": [
"#### 平衡结果与规格分开核验\n",
"\n",
"复算全部乘积，检查3Ω的量程边界和20kΩ等级区间。限差条件不足时不输出伪确定的不确定度。"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "wheatstone-bridge-013",
"metadata": {
"execution": {
"iopub.execute_input": "2026-10-16T00:33:15.957473Z",
"iopub.status.busy": "2026-10-16T00:33:15.957246Z",
"iopub.status.idle": "2026-10-16T00:33:15.961685Z",
"shell.execute_reply": "2026-10-16T00:33:15.960901Z"
},
"jupyter": {
"source_hidden": true
}
},
"outputs": [
{
"data": {
"text/markdown": [
"表 计算结果\n",
"\n",
"| 标称 / Ω | $M$ | 重算电阻 / Ω | 标称偏差 / % | 说明书等级条件 |\n",
"| --- | --- | --- | --- | --- |\n",
"| 3 | 0.1 | 3.066 | 2.2 | <10 Ω，超单桥有效范围 |\n",
"| 47 | 0.1 | 46.594 | -0.86383 | $0.05\\%$（参考条件下） |\n",
"| 200 | 1 | 200.2 | 0.1 | $0.05\\%$（参考条件下） |\n",
"| 2000 | 10 | 1979.7 | -1.015 | $0.05\\%$（参考条件下） |\n",
"| 20000 | 100 | 20421 | 2.105 | $0.1\\%$（参考条件下） |"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import physicslab.common.manuscripts as _presentation\n",
"_presentation._table_captioned=False\n",
"table_display(result[\"tables\"][\"results\"])"
]
},
{
"cell_type": "markdown",
"id": "wheatstone-bridge-014",
"metadata": {},
"source": [
"## 八、实验结果与讨论\n",
"\n",
"1. 五个平衡结果为 $3.066,46.594,200.20,1979.7,20421\\,\\Omega$，由实际比率和比较臂示值计算，保留未修约值用于复算。\n",
"2. $3\\,\\Omega$ 低于说明书单桥有效范围下限，比较臂 $30.66\\,\\Omega$ 也未充分使用建议高位，不能按标称 0.05 级保证其准确度；低阻宜选择适用量程或四端测量。\n",
"3. $20\\,\\mathrm{k}\\Omega$ 属于 $0.1\\%$ 等级区间，不可统一套用 $0.05\\%$。各档基准、环境条件及重复性缺失，未给出已经核定的完整不确定度。\n",
"4. 比较法减少了对电源幅度和检流计绝对标定的依赖，但仍受比较臂、比率臂、接触和零点灵敏度影响。按平衡公式计算本身不构成对该公式的独立实验验证。\n"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.14.7"
},
"physicslab": {
"chinese_font": "SimSun",
"experiment_id": "course-03/wheatstone-bridge",
"latin_font": "Times New Roman"
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"nbformat_minor": 5
}