// Numbas version: exam_results_page_options
{"name": "Poles", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "functions": {}, "tags": [], "ungrouped_variables": ["a", "b", "c", "d", "x", "y", "mod"], "statement": "<p>A system is said to be stable if all the poles of the transfer function lie within the unit circle. Otherwise the system is unstable.</p>\n<p>The poles of a transfer function&nbsp;are found by setting the denominator equal to zero and solving.</p>\n<p>Determine whether the system with the following transfer function is stable or unstable.</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>poles of a&nbsp;transfer function</p>"}, "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "<p style=\"padding-left: 90px;\">\\(G(z)=\\frac{\\var{d}}{\\var{a}s^2+\\var{b}s+\\var{c}}\\)</p>\n<p>The poles are given by \\(z=\\)[[0]]\\(\\pm j\\)[[1]]</p>\n<p>The modulus of the poles =&nbsp;[[2]]</p>", "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "type": "numberentry", "maxValue": "{x}", "showFeedbackIcon": true, "precision": "2", "minValue": "{x}", "variableReplacements": [], "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "type": "numberentry", "maxValue": "{y}", "showFeedbackIcon": true, "precision": "2", "minValue": "{y}", "variableReplacements": [], "allowFractions": false}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReduced": false, "mustBeReducedPC": 0, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "type": "numberentry", "maxValue": "sqrt({c}/{a})", "showFeedbackIcon": true, "precision": "2", "minValue": "sqrt({c}/{a})", "variableReplacements": [], "allowFractions": false}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "shuffleChoices": false, "choices": ["<p>The system is stable</p>", "<p>The system is unstable</p>"], "showCorrectAnswer": true, "displayColumns": 0, "distractors": ["", ""], "displayType": "radiogroup", "type": "1_n_2", "matrix": ["if({mod}<1,1,0)", "if({mod}<1,0,1)"], "showFeedbackIcon": true, "minMarks": 0, "variableReplacements": [], "scripts": {}}], "advice": "<p>To find the poles of a transfer function you must set the denominator to zero and solve.</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a}s^2+\\var{b}s+\\var{c}=0\\)</p>\n<p>This is a quadratic equation.</p>\n<p style=\"padding-left: 30px;\">\\(s=\\frac{-\\var{b}\\pm \\sqrt{\\var{b}^2-4*\\var{a}*\\var{c}}}{2*\\var{a}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(s=\\frac{-\\var{b}\\pm \\sqrt{\\simplify{{b}^2-4*{a}*{c}}}}{\\simplify{2*{a}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(s=\\frac{-\\var{b}}{\\simplify{2*{a}}}\\pm \\frac{j\\sqrt{\\simplify{4*{a}*{c}-{b}^2}}}{\\simplify{2*{a}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(s=\\var{x}\\pm j\\var{y}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>The modulus of s is thus</p>\n<p style=\"padding-left: 30px;\">\\(|s|=\\sqrt{\\var{x}^2+\\var{y}^2}=\\var{mod}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>The system is stable if \\(|s|&lt;1\\) otherwise it is unstable.</p>", "type": "question", "variables": {"x": {"group": "Ungrouped variables", "description": "", "definition": "-{b}/(2*{a})", "name": "x", "templateType": "anything"}, "b": {"group": "Ungrouped variables", "description": "", "definition": "random(1..10#1)", "name": "b", "templateType": "randrange"}, "c": {"group": "Ungrouped variables", "description": "", "definition": "random(13..21#1)", "name": "c", "templateType": "randrange"}, "mod": {"group": "Ungrouped variables", "description": "", "definition": "sqrt({x}^2+{y}^2)", "name": "mod", "templateType": "anything"}, "y": {"group": "Ungrouped variables", "description": "", "definition": "sqrt(4*{a}*{c}-{b}^2)/(2*{a})", "name": "y", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "description": "", "definition": "random(2..6#1)", "name": "a", "templateType": "randrange"}, "d": {"group": "Ungrouped variables", "description": "", "definition": "random(10..100#1)", "name": "d", "templateType": "randrange"}}, "rulesets": {}, "extensions": [], "name": "Poles", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}