// Numbas version: exam_results_page_options
{"name": "Transfer function", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"y": {"definition": "{b}/{c}", "templateType": "anything", "name": "y", "group": "Ungrouped variables", "description": ""}, "omega": {"definition": "random(2..20#1)", "templateType": "randrange", "name": "omega", "group": "Ungrouped variables", "description": ""}, "mod": {"definition": "sqrt({x}^2+{y}^2)", "templateType": "anything", "name": "mod", "group": "Ungrouped variables", "description": ""}, "b": {"definition": "200*{n}^2*{omega}", "templateType": "anything", "name": "b", "group": "Ungrouped variables", "description": ""}, "n": {"definition": "random(2..6#1)", "templateType": "randrange", "name": "n", "group": "Ungrouped variables", "description": ""}, "x": {"definition": "{a}/{c}", "templateType": "anything", "name": "x", "group": "Ungrouped variables", "description": ""}, "ag": {"definition": "arctan(-{y}/{x})", "templateType": "anything", "name": "ag", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "200*{n}*(100-{omega}^2)", "templateType": "anything", "name": "a", "group": "Ungrouped variables", "description": ""}, "c": {"definition": "(100-{omega}^2)^2+({n}*{omega})^2", "templateType": "anything", "name": "c", "group": "Ungrouped variables", "description": ""}}, "functions": {}, "variable_groups": [], "preamble": {"js": "", "css": ""}, "tags": [], "statement": "<p>A control system has a transfer function given by \\(\\frac{\\theta_o}{\\theta_i}=\\frac{\\simplify{200*{n}}}{100+j\\var{n}\\omega+(j\\omega)^2}\\) &nbsp;where \\(\\omega\\) is the angular velocity.</p>\n<p>The gain and phase of the function are given by the modulus and argument of&nbsp;\\(\\frac{\\theta_o}{\\theta_i}\\)&nbsp;respectively.&nbsp;</p>", "metadata": {"description": "<p>Gain &amp; Phase of a transfer function</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["n", "omega", "a", "b", "c", "x", "y", "mod", "ag"], "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Transfer function", "rulesets": {}, "parts": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>Calculate the Gain of the function, when \\(\\omega=\\var{omega} rads/sec\\).&nbsp;</p>\n<p style=\"padding-left: 30px;\">Gain =&nbsp;[[0]]</p>\n<p>Calculate the Phase of the function, when \\(\\omega=\\var{omega} rads/sec\\).&nbsp;</p>\n<p style=\"padding-left: 30px;\">Phase =&nbsp;[[1]]</p>", "showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "marks": 0, "gaps": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": [], "correctAnswerFraction": false, "strictPrecision": false, "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "minValue": "{mod}", "precisionPartialCredit": 0, "allowFractions": false, "showPrecisionHint": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "marks": 1, "maxValue": "{mod}", "type": "numberentry", "precision": "1"}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": [], "correctAnswerFraction": false, "strictPrecision": false, "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "minValue": "if({x}<0,{ag}+pi,{ag})", "precisionPartialCredit": 0, "allowFractions": false, "showPrecisionHint": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "marks": 1, "maxValue": "if({x}<0,{ag}+pi,{ag})", "type": "numberentry", "precision": "2"}], "type": "gapfill"}], "advice": "<p>\\(\\frac{\\theta_o}{\\theta_i}=\\frac{\\simplify{200*{n}}}{100+j\\var{n}\\omega+(j\\omega)^2}\\)</p>\n<p>\\(\\frac{\\theta_o}{\\theta_i}=\\frac{\\simplify{200*{n}}}{100+j\\var{n}(\\var{omega})+(j\\var{omega})^2}\\)</p>\n<p>\\(\\frac{\\theta_o}{\\theta_i}=\\frac{\\simplify{200*{n}}}{100+j\\simplify{{n}*{omega}}+j^2\\var{omega}^2}\\)</p>\n<p>\\(\\frac{\\theta_o}{\\theta_i}=\\frac{\\simplify{200*{n}}}{100-\\var{omega}^2+j\\simplify{{n}*{omega}}}\\)</p>\n<p>\\(\\frac{\\theta_o}{\\theta_i}=\\frac{\\simplify{200*{n}}}{\\simplify{100-{omega}^2}+j\\simplify{{n}*{omega}}}\\)</p>\n<p>\\(\\frac{\\theta_o}{\\theta_i}=\\frac{\\simplify{200*{n}}}{\\simplify{100-{omega}^2}+j\\simplify{{n}*{omega}}}\\frac{\\simplify{100-{omega}^2}-j\\simplify{{n}*{omega}}}{\\simplify{100-{omega}^2}-j\\simplify{{n}*{omega}}}\\)</p>\n<p>\\(\\frac{\\theta_o}{\\theta_i}=\\frac{\\var{a}-j\\var{b}}{\\var{c}}\\)</p>\n<p>\\(\\frac{\\theta_o}{\\theta_i}=\\frac{\\var{a}}{\\var{c}}-j\\frac{\\var{b}}{\\var{c}}\\)</p>\n<p>\\(\\frac{\\theta_o}{\\theta_i}=\\var{x}-j\\var{y}\\)</p>\n<p></p>\n<p>The Gain is the modulus of this number and the Phase is the argument</p>\n<p>Gain = \\(\\sqrt{(\\var{x})^2+(\\var{y})^2}=\\sqrt{\\simplify{{x}^2+{y}^2}}=\\var{mod}\\)</p>\n<p>Phase = \\(tan^{-1}\\left(\\frac{-\\var{y}}{\\var{x}}\\right)\\)</p>\n<p></p>", "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/"}]}