// 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": [{"functions": {}, "rulesets": {}, "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "variables": {"a": {"group": "Ungrouped variables", "definition": "random(2..6#1)", "templateType": "randrange", "name": "a", "description": ""}, "mod": {"group": "Ungrouped variables", "definition": "sqrt({x}^2+{y}^2)", "templateType": "anything", "name": "mod", "description": ""}, "c": {"group": "Ungrouped variables", "definition": "random(13..21#1)", "templateType": "randrange", "name": "c", "description": ""}, "d": {"group": "Ungrouped variables", "definition": "random(10..100#1)", "templateType": "randrange", "name": "d", "description": ""}, "b": {"group": "Ungrouped variables", "definition": "random(1..10#1)", "templateType": "randrange", "name": "b", "description": ""}, "y": {"group": "Ungrouped variables", "definition": "sqrt(4*{a}*{c}-{b}^2)/(2*{a})", "templateType": "anything", "name": "y", "description": ""}, "x": {"group": "Ungrouped variables", "definition": "-{b}/(2*{a})", "templateType": "anything", "name": "x", "description": ""}}, "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>", "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>", "metadata": {"description": "<p>poles of a&nbsp;transfer function</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "parts": [{"marks": 0, "showFeedbackIcon": true, "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>", "gaps": [{"marks": 1, "maxValue": "{x}", "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerFraction": false, "precisionType": "dp", "showCorrectAnswer": true, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "minValue": "{x}", "variableReplacements": [], "type": "numberentry", "allowFractions": false, "precision": "2"}, {"marks": 1, "maxValue": "{y}", "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerFraction": false, "precisionType": "dp", "showCorrectAnswer": true, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "minValue": "{y}", "variableReplacements": [], "type": "numberentry", "allowFractions": false, "precision": "2"}, {"marks": 1, "maxValue": "sqrt({c}/{a})", "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerFraction": false, "precisionType": "dp", "showCorrectAnswer": true, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "minValue": "sqrt({c}/{a})", "variableReplacements": [], "type": "numberentry", "allowFractions": false, "precision": "2"}], "scripts": {}, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true}, {"minMarks": 0, "displayType": "radiogroup", "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "choices": ["<p>The system is stable</p>", "<p>The system is unstable</p>"], "maxMarks": 0, "showCorrectAnswer": true, "matrix": ["if({mod}<1,1,0)", "if({mod}<1,0,1)"], "marks": 0, "showFeedbackIcon": true, "displayColumns": 0, "scripts": {}, "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false}], "name": "Poles", "extensions": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}