So far circuits have been driven by a DC source, an AC source and an exponential source. If we can find the current of a circuit generated by a Dirac delta function or impulse voltage source δ, then the convolution integral can be used to find the current to any given voltage source!
Example Impulse Response
The current is found by taking the derivative of the current found due to a DC voltage source! Say the goal is to find the δ current of a series LR circuit ... so that in the future the convolution integral can be used to find the current given any arbitrary source.
Choose a DC source of 1 volt (the real Vs then can scale off this). The particular homogeneous solution (steady state) is 0. The homogeneous solution to the non-homogeneous equation has the form:
Assume the current initially in the inductor is zero. The initial voltage is going to be 1 and is going to be across the inductor (since no current is flowing):
- ::
If the current in the inductor is initially zero, then:
- Which implies that:
- So the response to a DC voltage source turning on at t=0 to one volt (called the unit response μ) is:
Taking the derivative of this, get the impulse (δ) current is:
Now the current due to any arbitrary VS(t) can be found using the convolution integral:
- Fout bij het parsen (Conversiefout. 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Don't think iδ as current. It is really Fout bij het parsen (Conversiefout. 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EndFraction EndEndFraction Baseline Over upper L EndFraction', ), )"): {\displaystyle {d \over dt}\frac{current}{1 volt}} . VS(τ) turns into a multiplier.
LRC Example
Find the time domain expression for io given that Is = cos(t + π/2)μ(t) amp.
Earlier the step response for this problem was found:
- Fout bij het parsen (Conversiefout. De server ("cli") heeft gemeld: "array ( 'nohash' => array ( ), 'success' => true, '8fb22a73b7e00b43664385de6794ae13' => (object) array( 'speakText' => 'i Subscript delta Baseline left parenthesis t right parenthesis equals StartFraction e Superscript minus StartStartFraction t OverOver StartFraction upper L Over upper R EndFraction EndEndFraction Baseline Over upper L EndFraction', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="i Subscript delta Baseline left parenthesis t right parenthesis equals StartFraction e Superscript minus StartStartFraction t OverOver StartFraction upper L Over upper R EndFraction EndEndFraction Baseline Over upper L EndFraction"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-id="21" data-semantic-children="20,17" data-semantic-content="6"><mrow data-semantic-type="appl" data-semantic-role="simple function" 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data-semantic-type="superscript" data-semantic-role="latinletter" data-semantic-id="15" data-semantic-children="7,14" data-semantic-parent="17"><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="7" data-semantic-parent="15">e</mi><mrow class="MJX-TeXAtom-ORD" data-semantic-type="prefixop" data-semantic-role="negative" data-semantic-id="14" data-semantic-children="13" data-semantic-content="8" data-semantic-parent="15"><mo data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-id="8" data-semantic-parent="14" data-semantic-operator="prefixop,−">−<!-- − --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac data-semantic-type="fraction" data-semantic-role="division" data-semantic-id="13" data-semantic-children="9,12" data-semantic-parent="14"><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="9" data-semantic-parent="13">t</mi><mfrac data-semantic-type="fraction" data-semantic-role="division" data-semantic-id="12" data-semantic-children="10,11" data-semantic-parent="13"><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="10" data-semantic-parent="12">L</mi><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="11" data-semantic-parent="12">R</mi></mfrac></mfrac></mrow></mrow></msup><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="16" data-semantic-parent="17">L</mi></mfrac></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle i_{\\delta }(t)={\\frac {e^{-{\\frac 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EndFraction EndEndFraction Baseline Over upper L EndFraction', ), )"): {\displaystyle i_{o_\mu} = \frac{1}{2}(1 - e^{-t}(\cos t + \sin t))}
The impulse response is going to be the derivative of this:
- Fout bij het parsen (Conversiefout. 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id="13"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="9">t</identifier><fraction role="division" id="12"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="10">L</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="11">R</identifier></children></fraction></children></fraction></children></prefixop></children></superscript><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="16">L</identifier></children></fraction></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -1.838ex; width:13.639ex; height:8.509ex;', 'sanetex' => '{\\displaystyle i_{\\delta }(t)={\\frac {e^{-{\\frac {t}{\\frac {L}{R}}}}}{L}}}', 'speech' => 'i Subscript delta Baseline left parenthesis t right parenthesis equals StartFraction e Superscript minus StartStartFraction t OverOver StartFraction upper L Over upper R EndFraction EndEndFraction Baseline Over upper L EndFraction', ), )"): {\displaystyle i_{o_\delta} = \frac{1}{2}e^{-t}(\cos t + \sin t + \sin t - \cos t) = e^{-t}\sin t} :Fout bij het parsen (Conversiefout. 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- Fout bij het parsen (Conversiefout. 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- Fout bij het parsen (Conversiefout. 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EndFraction EndEndFraction Baseline Over upper L EndFraction', ), )"): {\displaystyle i_o(t) = \frac{\cos t}{5} + \frac{2 \sin t}{5} - \frac{7 e^{-t}\cos t}{10} - \frac{11 e^{-t}\sin t}{10} + \frac{1}{2} + C_1}
The Mupad code to solve the integral (substituting x for τ) is:
f := exp(-(t-x)) *sin(t-x) *(1 + cos(x));<br>S := int(f,x = 0..t)
Finding the integration constant
- Fout bij het parsen (Conversiefout. 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This implies:
- Fout bij het parsen (Conversiefout. 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'simple function', 'id' => '2', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'simple function', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'i', ), 1 => (object) array( 'type' => 'identifier', 'role' => 'greekletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '1', '$t' => 'δ', ), ), ), 1 => (object) array( 'type' => 'fenced', 'role' => 'leftright', 'id' => '18', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '4', '$t' => 't', ), ), 'content' => array ( 0 => (object) array( 'type' => 'fence', 'role' => 'open', 'id' => '3', '$t' => '(', ), 1 => (object) array( 'type' => 'fence', 'role' => 'close', 'id' => '5', '$t' => ')', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'punctuation', 'role' => 'application', 'id' => '19', '$t' => '', ), 1 => (object) array( 'type' => 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=> 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '10', '$t' => 'L', ), 1 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '11', '$t' => 'R', ), ), ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'subtraction', 'id' => '8', '$t' => '−', ), ), ), ), ), 1 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '16', '$t' => 'L', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'relation', 'role' => 'equality', 'id' => '6', '$t' => '=', ), ), ), ), 'streeXml' => '<stree><relseq role="equality" id="21">=<content><relation role="equality" id="6">=</relation></content><children><appl role="simple function" annotation="clearspeak:simple" id="20"><content><punctuation role="application" id="19"></punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="0">i</identifier></content><children><subscript role="simple function" id="2"><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="0">i</identifier><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="1">δ</identifier></children></subscript><fenced role="leftright" id="18"><content><fence role="open" id="3">(</fence><fence role="close" id="5">)</fence></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="4">t</identifier></children></fenced></children></appl><fraction role="division" id="17"><children><superscript role="latinletter" id="15"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="7">e</identifier><prefixop role="negative" id="14">−<content><operator role="subtraction" id="8">−</operator></content><children><fraction role="division" id="13"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="9">t</identifier><fraction role="division" id="12"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="10">L</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="11">R</identifier></children></fraction></children></fraction></children></prefixop></children></superscript><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="16">L</identifier></children></fraction></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -1.838ex; width:13.639ex; height:8.509ex;', 'sanetex' => '{\\displaystyle i_{\\delta }(t)={\\frac {e^{-{\\frac {t}{\\frac {L}{R}}}}}{L}}}', 'speech' => 'i Subscript delta Baseline left parenthesis t right parenthesis equals StartFraction e Superscript minus StartStartFraction t OverOver StartFraction upper L Over upper R EndFraction EndEndFraction Baseline Over upper L EndFraction', ), )"): {\displaystyle C_1 = 0}
Voeg commentaar toe
