500 lines
34 KiB
Python
500 lines
34 KiB
Python
"""
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知识卡片内容 - 包含时域、频域、根轨迹分析的公式和方法
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使用纯HTML格式，无需LaTeX渲染库
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"""
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# 时域分析知识卡片
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TIME_DOMAIN_KNOWLEDGE = """
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<div class='knowledge-card'>
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<h3 style='color: #667eea; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
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<span style='font-size: 1.3em;'>📚</span> 时域分析常用公式
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</h3>
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<details open>
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<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #f3e5f5 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
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📐 二阶系统标准形式
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</summary>
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<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
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<p style='margin: 10px 0;'><strong>标准传递函数：</strong></p>
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<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
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<div style='font-size: 1.3em; display: inline-block;'>
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<i>G</i>(<i>s</i>) =
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<div style='display: inline-block; vertical-align: middle; margin: 0 8px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 10px 5px;'>ω<sub>n</sub><sup>2</sup></div>
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<div style='text-align: center; padding: 5px 10px 0;'><i>s</i><sup>2</sup> + 2ζω<sub>n</sub><i>s</i> + ω<sub>n</sub><sup>2</sup></div>
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</div>
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</div>
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</div>
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<h4 style='color: #667eea; margin-top: 20px; font-size: 1.1em;'>📊 关键参数</h4>
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<table style='width: 100%; border-collapse: collapse; margin: 15px 0; font-size: 0.95em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
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<tr style='background: linear-gradient(135deg, #667eea 0%, #764ba2 100%); color: white;'>
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<th style='padding: 12px; text-align: left; font-weight: 700;'>参数</th>
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<th style='padding: 12px; text-align: left; font-weight: 700;'>含义</th>
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<th style='padding: 12px; text-align: left; font-weight: 700;'>影响</th>
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</tr>
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<tr style='background: white;'>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'><strong>ζ (zeta)</strong></td>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'>阻尼比</td>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'>控制超调量和振荡</td>
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</tr>
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<tr style='background: #f8f9ff;'>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'><strong>ω<sub>n</sub></strong></td>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'>无阻尼自然频率 (rad/s)</td>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'>决定响应速度</td>
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</tr>
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</table>
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<h4 style='color: #764ba2; margin-top: 20px; font-size: 1.1em;'>🎯 阻尼比分类</h4>
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<ul style='line-height: 2; margin-left: 20px;'>
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<li><strong style='color: #f5576c;'>ζ < 0:</strong> 不稳定系统</li>
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<li><strong style='color: #ffa726;'>ζ = 0:</strong> 无阻尼振荡</li>
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<li><strong style='color: #4caf50;'>0 < ζ < 1:</strong> 欠阻尼（有振荡）⭐ 最常见</li>
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<li><strong style='color: #2196f3;'>ζ = 1:</strong> 临界阻尼（无超调）</li>
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<li><strong style='color: #9c27b0;'>ζ > 1:</strong> 过阻尼（响应慢）</li>
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</ul>
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</div>
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</details>
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<details open>
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<summary style='background: linear-gradient(135deg, #fff0f8 0%, #fce4ec 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
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📊 时域性能指标（欠阻尼系统）
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</summary>
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<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
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<div style='background: linear-gradient(135deg, #e3f2fd 0%, #f3e5f5 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #667eea;'>
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<strong style='color: #667eea; font-size: 1.1em;'>1️⃣ 上升时间 (Rise Time, t<sub>r</sub>)</strong>
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<p style='margin: 8px 0;'>响应从 10% 上升到 90% 终值所需时间</p>
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<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0;'>
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<div style='font-size: 1.2em; display: inline-block;'>
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<i>t</i><sub>r</sub> ≈
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>1.8</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ω<sub>n</sub></div>
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</div>
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</div>
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</div>
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</div>
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<div style='background: linear-gradient(135deg, #fff4f0 0%, #ffe0b2 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #ff9800;'>
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<strong style='color: #f57c00; font-size: 1.1em;'>2️⃣ 峰值时间 (Peak Time, t<sub>p</sub>)</strong>
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<p style='margin: 8px 0;'>响应达到第一个峰值的时间</p>
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<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0;'>
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<div style='font-size: 1.2em; display: inline-block;'>
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<i>t</i><sub>p</sub> =
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>π</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ω<sub>n</sub>√(1−ζ²)</div>
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</div>
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=
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>π</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ω<sub>d</sub></div>
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</div>
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</div>
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</div>
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<p style='margin: 8px 0; color: #666; font-size: 0.95em;'>其中 ω<sub>d</sub> = ω<sub>n</sub>√(1−ζ²) 为阻尼振荡频率</p>
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</div>
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<div style='background: linear-gradient(135deg, #f0fff4 0%, #c8e6c9 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #4caf50;'>
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<strong style='color: #388e3c; font-size: 1.1em;'>3️⃣ 超调量 (Overshoot, σ%)</strong>
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<p style='margin: 8px 0;'>响应超过稳态值的最大百分比</p>
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<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0; font-size: 1.2em;'>
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σ% = <i>e</i><sup>−πζ/√(1−ζ²)</sup> × 100%
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</div>
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<p style='margin: 8px 0; color: #388e3c; font-weight: 600;'>💡 仅与 ζ 有关！</p>
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<table style='width: 100%; margin: 10px 0; font-size: 0.9em;'>
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<tr style='background: #e8f5e9;'>
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<td style='padding: 6px; border: 1px solid #c8e6c9;'><strong>ζ = 0.5</strong></td>
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<td style='padding: 6px; border: 1px solid #c8e6c9;'>σ% ≈ 16%</td>
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</tr>
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<tr style='background: white;'>
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<td style='padding: 6px; border: 1px solid #c8e6c9;'><strong>ζ = 0.707</strong></td>
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<td style='padding: 6px; border: 1px solid #c8e6c9;'>σ% ≈ 4.3% ⭐ 最佳</td>
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</tr>
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</table>
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</div>
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<div style='background: linear-gradient(135deg, #fff0f8 0%, #f3e5f5 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #9c27b0;'>
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<strong style='color: #7b1fa2; font-size: 1.1em;'>4️⃣ 调节时间 (Settling Time, t<sub>s</sub>)</strong>
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<p style='margin: 8px 0;'>响应达到并保持在稳态值 ±2%（或±5%）范围内的时间</p>
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<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0;'>
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<div style='font-size: 1.2em;'>
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<i>t</i><sub>s</sub> ≈
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>4</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ζω<sub>n</sub></div>
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</div>
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(2% 误差带)
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</div>
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<div style='font-size: 1.2em; margin-top: 10px;'>
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<i>t</i><sub>s</sub> ≈
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>3</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ζω<sub>n</sub></div>
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</div>
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(5% 误差带)
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</div>
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</div>
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</div>
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</div>
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</details>
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<details>
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<summary style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
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🎯 一阶系统特性
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</summary>
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<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
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<p><strong>标准传递函数：</strong></p>
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<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0;'>
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<div style='font-size: 1.3em; display: inline-block;'>
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<i>G</i>(<i>s</i>) =
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<div style='display: inline-block; vertical-align: middle; margin: 0 8px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 10px 5px;'><i>K</i></div>
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<div style='text-align: center; padding: 5px 10px 0;'>τ<i>s</i> + 1</div>
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</div>
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</div>
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</div>
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<p><strong>阶跃响应：</strong></p>
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<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0; font-size: 1.2em;'>
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<i>y</i>(<i>t</i>) = <i>K</i>(1 − <i>e</i><sup>−<i>t</i>/τ</sup>)
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</div>
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<div style='background: #e3f2fd; padding: 12px; border-radius: 8px; margin: 10px 0;'>
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<p><strong>💡 关键时间点：</strong></p>
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<ul style='line-height: 1.8;'>
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<li><strong><i>t</i> = τ:</strong> 达到稳态值的 63.2%</li>
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<li><strong><i>t</i> = 3τ:</strong> 达到稳态值的 95%</li>
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<li><strong><i>t</i> = 4τ:</strong> 达到稳态值的 98.2%</li>
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<li><strong><i>t</i> = 5τ:</strong> 达到稳态值的 99.3%</li>
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</ul>
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</div>
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</div>
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</details>
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</div>
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"""
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# 频域分析知识卡片
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FREQUENCY_DOMAIN_KNOWLEDGE = """
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<div class='knowledge-card'>
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<h3 style='color: #667eea; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
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<span style='font-size: 1.3em;'>📚</span> 频域分析常用方法
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</h3>
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<details open>
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<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
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🎯 增益裕度 (Gain Margin, GM)
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</summary>
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<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
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<p><strong>定义：</strong>在相角为 -180° 时，系统增益可增加的最大倍数</p>
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<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0;'>
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<div style='font-size: 1.3em;'>
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GM<sub>dB</sub> = −20log<sub>10</sub>|<i>G</i>(<i>j</i>ω<sub>pc</sub>)|
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</div>
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</div>
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<p style='margin: 8px 0;'><strong>ω<sub>pc</sub></strong> - 相角交越频率（相角 = -180°）</p>
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<h4 style='color: #667eea; margin-top: 15px;'>判断准则</h4>
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<table style='width: 100%; border-collapse: collapse; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
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<tr style='background: #e8f5e9;'>
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<td style='padding: 10px; border: 1px solid #c8e6c9; font-weight: 600;'>GM > 0 dB</td>
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<td style='padding: 10px; border: 1px solid #c8e6c9;'>✅ 系统稳定</td>
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</tr>
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<tr style='background: #fff3e0;'>
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<td style='padding: 10px; border: 1px solid #ffe0b2; font-weight: 600;'>GM = 0 dB</td>
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<td style='padding: 10px; border: 1px solid #ffe0b2;'>⚠️ 临界稳定</td>
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</tr>
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<tr style='background: #ffebee;'>
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<td style='padding: 10px; border: 1px solid #ffcdd2; font-weight: 600;'>GM < 0 dB</td>
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<td style='padding: 10px; border: 1px solid #ffcdd2;'>❌ 系统不稳定</td>
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</tr>
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</table>
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<div style='background: linear-gradient(135deg, #e3f2fd 0%, #f3e5f5 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
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<p><strong>💡 工程要求：</strong>通常要求 <strong>GM ≥ 6 dB</strong>（约2倍增益余量）</p>
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</div>
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</div>
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</details>
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<details open>
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<summary style='background: linear-gradient(135deg, #f3e5f5 0%, #e1bee7 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
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🎯 相角裕度 (Phase Margin, PM)
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</summary>
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<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
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<p><strong>定义：</strong>在增益为 1 (0dB) 时，系统相角与 -180° 的差值</p>
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<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0; font-size: 1.3em;'>
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PM = 180° + ∠<i>G</i>(<i>j</i>ω<sub>gc</sub>)
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</div>
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<p style='margin: 8px 0;'><strong>ω<sub>gc</sub></strong> - 增益交越频率（幅值 = 1 或 0dB）</p>
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<h4 style='color: #764ba2; margin-top: 15px;'>判断准则</h4>
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<table style='width: 100%; border-collapse: collapse; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
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<tr style='background: #e8f5e9;'>
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<td style='padding: 10px; border: 1px solid #c8e6c9; font-weight: 600;'>PM > 0°</td>
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<td style='padding: 10px; border: 1px solid #c8e6c9;'>✅ 系统稳定</td>
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</tr>
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<tr style='background: #fff3e0;'>
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<td style='padding: 10px; border: 1px solid #ffe0b2; font-weight: 600;'>PM = 0°</td>
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<td style='padding: 10px; border: 1px solid #ffe0b2;'>⚠️ 临界稳定</td>
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</tr>
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<tr style='background: #ffebee;'>
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<td style='padding: 10px; border: 1px solid #ffcdd2; font-weight: 600;'>PM < 0°</td>
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<td style='padding: 10px; border: 1px solid #ffcdd2;'>❌ 系统不稳定</td>
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</tr>
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</table>
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<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
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<p><strong>💡 工程要求：</strong>通常要求 <strong>PM ∈ [30°, 60°]</strong></p>
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<p style='margin: 8px 0;'>• PM ≈ 45° ~ 60°: 良好阻尼特性</p>
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<p style='margin: 8px 0;'>• PM 越大，超调量越小</p>
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</div>
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<h4 style='color: #f57c00; margin-top: 15px;'>与时域性能的关系</h4>
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<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0; font-size: 1.2em;'>
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ζ ≈ <div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>PM</div>
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<div style='text-align: center; padding: 3px 8px 0;'>100</div>
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</div> (PM 以度为单位)
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</div>
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<table style='width: 100%; margin: 10px 0; font-size: 0.9em; box-shadow: 0 2px 6px rgba(0,0,0,0.06); border-radius: 6px; overflow: hidden;'>
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<tr style='background: #f5f5f5; font-weight: 600;'>
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<td style='padding: 8px; border: 1px solid #ddd;'>PM</td>
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<td style='padding: 8px; border: 1px solid #ddd;'>ζ</td>
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<td style='padding: 8px; border: 1px solid #ddd;'>超调量</td>
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</tr>
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<tr><td style='padding: 8px; border: 1px solid #ddd;'>30°</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 0.3</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 37%</td></tr>
|
||
<tr><td style='padding: 8px; border: 1px solid #ddd;'>45°</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 0.45</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 20%</td></tr>
|
||
<tr><td style='padding: 8px; border: 1px solid #ddd;'>60°</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 0.6</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 10% ⭐</td></tr>
|
||
</table>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #fff0f8 0%, #fce4ec 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📊 Bode 图与 Nyquist 图
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<h4 style='color: #667eea;'>Bode 图（伯德图）</h4>
|
||
<p><strong>组成：</strong>幅频特性图 + 相频特性图</p>
|
||
<ul style='line-height: 1.8;'>
|
||
<li><strong>横轴：</strong>频率 ω (对数刻度)</li>
|
||
<li><strong>纵轴（幅频）：</strong>20log|<i>G</i>(<i>j</i>ω)| (dB)</li>
|
||
<li><strong>纵轴（相频）：</strong>∠<i>G</i>(<i>j</i>ω) (度)</li>
|
||
</ul>
|
||
|
||
<div style='background: #e3f2fd; padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 优点：</strong></p>
|
||
<ul style='line-height: 1.6;'>
|
||
<li>便于绘制（渐近线近似）</li>
|
||
<li>直观读取稳定裕度</li>
|
||
<li>串联系统可图形叠加</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<h4 style='color: #764ba2; margin-top: 20px;'>Nyquist 图（奈奎斯特图）</h4>
|
||
<p><strong>定义：</strong>开环频率特性在复平面上的轨迹</p>
|
||
|
||
<div style='background: #fff3e0; padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>🎯 奈奎斯特稳定判据：</strong></p>
|
||
<p style='text-align: center; font-weight: 600; margin: 10px 0; font-size: 1.2em;'><i>Z</i> = <i>P</i> − <i>N</i></p>
|
||
<p style='font-size: 0.95em;'>• <i>Z</i>: 闭环右半平面极点数<br>
|
||
• <i>P</i>: 开环右半平面极点数<br>
|
||
• <i>N</i>: 曲线逆时针包围 (−1, <i>j</i>0) 的圈数</p>
|
||
<p style='margin-top: 10px; color: #f57c00; font-weight: 600;'>稳定条件：<i>Z</i> = 0</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🔧 典型环节频率特性
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<div style='margin-bottom: 15px; padding: 12px; background: white; border-radius: 8px; border-left: 4px solid #667eea;'>
|
||
<h4 style='margin-top: 0; color: #667eea;'>1️⃣ 比例环节 <i>K</i></h4>
|
||
<p>• 幅频：20log<i>K</i> (dB) - 水平线<br>• 相频：0° - 水平线</p>
|
||
</div>
|
||
|
||
<div style='margin-bottom: 15px; padding: 12px; background: white; border-radius: 8px; border-left: 4px solid #f57c00;'>
|
||
<h4 style='margin-top: 0; color: #f57c00;'>2️⃣ 积分环节 1/<i>s</i></h4>
|
||
<p>• 幅频：-20dB/dec 斜率<br>• 相频：-90° - 水平线</p>
|
||
</div>
|
||
|
||
<div style='margin-bottom: 15px; padding: 12px; background: white; border-radius: 8px; border-left: 4px solid #4caf50;'>
|
||
<h4 style='margin-top: 0; color: #4caf50;'>3️⃣ 惯性环节 1/(<i>Ts</i>+1)</h4>
|
||
<p>• 转折频率：ω = 1/<i>T</i><br>• 幅频：低频 0dB，高频 -20dB/dec<br>• 相频：0° → -90°</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
"""
|
||
|
||
# 根轨迹分析知识卡片
|
||
ROOT_LOCUS_KNOWLEDGE = """
|
||
<div class='knowledge-card'>
|
||
<h3 style='color: #667eea; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
|
||
<span style='font-size: 1.3em;'>📚</span> 根轨迹分析要点
|
||
</h3>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🎯 根轨迹基本概念
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<p><strong>定义：</strong>当开环增益 <i>K</i> 从 0 → ∞ 变化时，闭环特征方程根（极点）在 <i>s</i> 平面上的运动轨迹</p>
|
||
|
||
<h4 style='color: #667eea; margin-top: 15px;'>特征方程</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0; font-size: 1.3em;'>
|
||
1 + <i>KG</i>(<i>s</i>)<i>H</i>(<i>s</i>) = 0
|
||
</div>
|
||
|
||
<h4 style='color: #764ba2; margin-top: 15px;'>主要作用</h4>
|
||
<ul style='line-height: 1.8;'>
|
||
<li>✅ 直观显示增益对极点位置的影响</li>
|
||
<li>✅ 判断系统稳定性</li>
|
||
<li>✅ 选择合适的增益值</li>
|
||
<li>✅ 设计控制器参数</li>
|
||
</ul>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #f3e5f5 0%, #e1bee7 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📐 根轨迹绘制基本条件
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #4caf50;'>
|
||
<h4 style='margin-top: 0; color: #388e3c;'>1️⃣ 幅值条件（充要条件）</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0; font-size: 1.2em;'>
|
||
|<i>KG</i>(<i>s</i><sub>0</sub>)<i>H</i>(<i>s</i><sub>0</sub>)| = 1
|
||
</div>
|
||
<p style='margin: 8px 0;'><strong>💡 用途：</strong>确定增益 <i>K</i> 的值</p>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #ff9800;'>
|
||
<h4 style='margin-top: 0; color: #f57c00;'>2️⃣ 相角条件（充要条件）</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0; font-size: 1.2em;'>
|
||
∠<i>G</i>(<i>s</i><sub>0</sub>)<i>H</i>(<i>s</i><sub>0</sub>) = (2<i>k</i>+1) × 180°
|
||
</div>
|
||
<p style='margin: 8px 0; font-size: 0.95em;'>其中 <i>k</i> = 0, ±1, ±2, ...</p>
|
||
<p style='margin: 8px 0;'><strong>💡 用途：</strong>判断 <i>s</i> 平面某点是否在根轨迹上</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #fff0f8 0%, #fce4ec 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🌟 根轨迹基本性质
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
|
||
<table style='width: 100%; border-collapse: collapse; margin: 15px 0; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #667eea 0%, #764ba2 100%); color: white;'>
|
||
<th style='padding: 12px; text-align: left; font-weight: 700;'>规则</th>
|
||
<th style='padding: 12px; text-align: left; font-weight: 700;'>内容</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>起点</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'><i>K</i>=0，位于<strong>开环极点</strong></td>
|
||
</tr>
|
||
<tr style='background: #f8f9ff;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>终点</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'><i>K</i>→∞，位于<strong>开环零点</strong>或无穷远</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>分支数</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'>max(<i>n</i>, <i>m</i>)，<i>n</i>=极点数，<i>m</i>=零点数</td>
|
||
</tr>
|
||
<tr style='background: #f8f9ff;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>实轴段</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'>右侧零极点总数为<strong>奇数</strong>的区段</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>渐近线数</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'><strong><i>n</i> − <i>m</i></strong> 条</td>
|
||
</tr>
|
||
<tr style='background: #f8f9ff;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>渐近线角度</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'>φ<sub>a</sub> = (2<i>k</i>+1)×180°/(<i>n</i>−<i>m</i>)</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>渐近线交点</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'>σ<sub>a</sub> = (Σ<i>p</i> − Σ<i>z</i>)/(<i>n</i>−<i>m</i>)</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🛡️ <i>s</i> 平面稳定性区域
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: #f5f5f5; font-weight: 600;'>
|
||
<th style='padding: 10px; border: 1px solid #ddd;'>区域</th>
|
||
<th style='padding: 10px; border: 1px solid #ddd;'>条件</th>
|
||
<th style='padding: 10px; border: 1px solid #ddd;'>稳定性</th>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>左半平面</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>Re(<i>s</i>) < 0</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd; background: #c8e6c9;'><strong>✅ 稳定</strong></td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>虚轴</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>Re(<i>s</i>) = 0</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd; background: #fff9c4;'><strong>⚠️ 临界稳定</strong></td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>右半平面</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>Re(<i>s</i>) > 0</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd; background: #ffcdd2;'><strong>❌ 不稳定</strong></td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: #e3f2fd; padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 虚轴（<i>x</i>=0）</strong>即为<strong>稳定性边界</strong></p>
|
||
<p style='margin: 8px 0;'>根轨迹穿越虚轴点对应<strong>临界增益 <i>K</i><sub>c</sub></strong></p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📏 阻尼比等值线
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<p>从原点出发的射线代表恒定阻尼比 ζ 的轨迹：</p>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0; font-size: 1.2em;'>
|
||
θ = arccos(ζ)
|
||
</div>
|
||
|
||
<table style='width: 100%; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.06); border-radius: 6px; overflow: hidden;'>
|
||
<tr style='background: #f5f5f5; font-weight: 600;'>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>阻尼比 ζ</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>角度 θ</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>系统特性</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>0.5</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>60°</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>欠阻尼，较大超调</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'><strong>0.707</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'><strong>45°</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'><strong>⭐ 最佳阻尼</strong></td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 工程应用：</strong>通过根轨迹与阻尼比等值线的交点，选择满足性能要求的增益 <i>K</i></p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
"""
|