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):
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If the current in the inductor is initially zero, then:
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- Fout bij het parsen (Conversiefout. 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Taking the derivative of this, get the impulse (δ) current is:
- Fout bij het parsen (Conversiefout. 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id="11">A</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="12">R</identifier></children></infixop><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="15">L</identifier></children></fraction></children></prefixop></children></fenced></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -1.838ex; width:23.197ex; height:5.343ex;', 'sanetex' => '{\\displaystyle v(0)=1=L*(-{\\frac {AR}{L}})}', 'speech' => 'v left parenthesis 0 right parenthesis equals 1 equals upper L asterisk left parenthesis minus StartFraction upper A upper R Over upper L EndFraction right parenthesis', ), )"): {\displaystyle i_\delta (t) = \frac{e^{-\frac{t}{\frac{L}{R}}}}{L}}
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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id="3">)</fence></content><children><number role="integer" font="normal" annotation="clearspeak:simple" id="2">0</number></children></fenced></children></appl><number role="integer" font="normal" annotation="clearspeak:simple" id="5">1</number><infixop role="multiplication" id="23">∗<content><operator role="multiplication" id="8">∗</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="7">L</identifier><fenced role="leftright" id="20"><content><fence role="open" id="9">(</fence><fence role="close" id="17">)</fence></content><children><prefixop role="negative" id="19">−<content><operator role="subtraction" id="10">−</operator></content><children><fraction role="division" id="16"><children><infixop role="implicit" annotation="clearspeak:simple;clearspeak:unit" id="14"><content><operator role="multiplication" id="13"></operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="11">A</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="12">R</identifier></children></infixop><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="15">L</identifier></children></fraction></children></prefixop></children></fenced></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -1.838ex; width:23.197ex; height:5.343ex;', 'sanetex' => '{\\displaystyle v(0)=1=L*(-{\\frac {AR}{L}})}', 'speech' => 'v left parenthesis 0 right parenthesis equals 1 equals upper L asterisk left parenthesis minus StartFraction upper A upper R Over upper L EndFraction right parenthesis', ), )"): {\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. 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id="3">)</fence></content><children><number role="integer" font="normal" annotation="clearspeak:simple" id="2">0</number></children></fenced></children></appl><number role="integer" font="normal" annotation="clearspeak:simple" id="5">1</number><infixop role="multiplication" id="23">∗<content><operator role="multiplication" id="8">∗</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="7">L</identifier><fenced role="leftright" id="20"><content><fence role="open" id="9">(</fence><fence role="close" id="17">)</fence></content><children><prefixop role="negative" id="19">−<content><operator role="subtraction" id="10">−</operator></content><children><fraction role="division" id="16"><children><infixop role="implicit" annotation="clearspeak:simple;clearspeak:unit" id="14"><content><operator role="multiplication" id="13"></operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="11">A</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="12">R</identifier></children></infixop><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="15">L</identifier></children></fraction></children></prefixop></children></fenced></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -1.838ex; width:23.197ex; height:5.343ex;', 'sanetex' => '{\\displaystyle v(0)=1=L*(-{\\frac {AR}{L}})}', 'speech' => 'v left parenthesis 0 right parenthesis equals 1 equals upper L asterisk left parenthesis minus StartFraction upper A upper R Over upper L EndFraction right parenthesis', ), )"): {\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. De server ("cli") heeft gemeld: "array ( 'nohash' => array ( ), 'success' => true, '30377ab23b801c8dc2a997a2efe049f5' => (object) array( 'speakText' => 'v left parenthesis 0 right parenthesis equals 1 equals upper L asterisk left parenthesis minus StartFraction upper A upper R Over upper L EndFraction right parenthesis', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="v left parenthesis 0 right parenthesis equals 1 equals upper L asterisk left parenthesis minus StartFraction upper A upper R Over upper L EndFraction right parenthesis"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-id="24" data-semantic-children="22,5,23" data-semantic-content="4,6"><mrow data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple" data-semantic-id="22" data-semantic-children="0,18" data-semantic-content="21,0" data-semantic-parent="24"><mi data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="0" data-semantic-parent="22" data-semantic-operator="appl">v</mi><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-id="21" data-semantic-parent="22" data-semantic-added="true" data-semantic-operator="appl"></mo><mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="18" data-semantic-children="2" data-semantic-content="1,3" data-semantic-parent="22"><mo stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-id="1" data-semantic-parent="18" data-semantic-operator="fenced">(</mo><mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple" data-semantic-id="2" data-semantic-parent="18">0</mn><mo stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-id="3" data-semantic-parent="18" data-semantic-operator="fenced">)</mo></mrow></mrow><mo data-semantic-type="relation" data-semantic-role="equality" data-semantic-id="4" data-semantic-parent="24" data-semantic-operator="relseq,=">=</mo><mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple" data-semantic-id="5" data-semantic-parent="24">1</mn><mo data-semantic-type="relation" data-semantic-role="equality" data-semantic-id="6" data-semantic-parent="24" data-semantic-operator="relseq,=">=</mo><mrow data-semantic-type="infixop" data-semantic-role="multiplication" data-semantic-id="23" data-semantic-children="7,20" data-semantic-content="8" data-semantic-parent="24"><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="23">L</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-id="8" data-semantic-parent="23" data-semantic-operator="infixop,∗">∗<!-- ∗ --></mo><mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="20" data-semantic-children="19" data-semantic-content="9,17" data-semantic-parent="23"><mo stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-id="9" data-semantic-parent="20" data-semantic-operator="fenced">(</mo><mrow data-semantic-type="prefixop" data-semantic-role="negative" data-semantic-id="19" data-semantic-children="16" data-semantic-content="10" data-semantic-parent="20"><mo data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-id="10" data-semantic-parent="19" data-semantic-operator="prefixop,−">−<!-- − --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac data-semantic-type="fraction" data-semantic-role="division" data-semantic-id="16" data-semantic-children="14,15" data-semantic-parent="19"><mrow data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit" data-semantic-id="14" data-semantic-children="11,12" data-semantic-content="13" data-semantic-parent="16"><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="14">A</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-id="13" data-semantic-parent="14" data-semantic-added="true" data-semantic-operator="infixop,"></mo><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="12" data-semantic-parent="14">R</mi></mrow><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="15" data-semantic-parent="16">L</mi></mfrac></mrow></mrow><mo stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-id="17" data-semantic-parent="20" data-semantic-operator="fenced">)</mo></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle v(0)=1=L*(-{\\frac {AR}{L}})}</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="23.197ex" height="5.343ex" style="vertical-align: -1.838ex;" viewBox="0 -1508.9 9987.6 2300.3" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">v left parenthesis 0 right parenthesis equals 1 equals upper L asterisk left parenthesis minus StartFraction upper A upper R Over upper L EndFraction right parenthesis</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 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id="11">A</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="12">R</identifier></children></infixop><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="15">L</identifier></children></fraction></children></prefixop></children></fenced></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -1.838ex; width:23.197ex; height:5.343ex;', 'sanetex' => '{\\displaystyle v(0)=1=L*(-{\\frac {AR}{L}})}', 'speech' => 'v left parenthesis 0 right parenthesis equals 1 equals upper L asterisk left parenthesis minus StartFraction upper A upper R Over upper L EndFraction right parenthesis', ), )"): {\displaystyle i_{o_\delta} = {d i_{o_\mu} \over dt} = 0 + \frac{1}{2}e^{-t}(\cos t + \sin t) - \frac{1}{2}e^{-t}(-\sin t + \cos t)}
- Fout bij het parsen (Conversiefout. 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data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-id="8" data-semantic-parent="23" data-semantic-operator="infixop,∗">∗<!-- ∗ --></mo><mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="20" data-semantic-children="19" data-semantic-content="9,17" data-semantic-parent="23"><mo stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-id="9" data-semantic-parent="20" data-semantic-operator="fenced">(</mo><mrow data-semantic-type="prefixop" data-semantic-role="negative" data-semantic-id="19" data-semantic-children="16" data-semantic-content="10" data-semantic-parent="20"><mo data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-id="10" data-semantic-parent="19" data-semantic-operator="prefixop,−">−<!-- − --></mo><mrow class="MJX-TeXAtom-ORD"><mfrac data-semantic-type="fraction" data-semantic-role="division" data-semantic-id="16" data-semantic-children="14,15" 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'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'multiplication', 'id' => '8', '$t' => '∗', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'relation', 'role' => 'equality', 'id' => '4', '$t' => '=', ), 1 => (object) array( 'type' => 'relation', 'role' => 'equality', 'id' => '6', '$t' => '=', ), ), ), ), 'streeXml' => '<stree><relseq role="equality" id="24">=<content><relation role="equality" id="4">=</relation><relation role="equality" id="6">=</relation></content><children><appl role="simple function" annotation="clearspeak:simple" id="22"><content><punctuation role="application" id="21"></punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="0">v</identifier></content><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="0">v</identifier><fenced role="leftright" id="18"><content><fence role="open" id="1">(</fence><fence role="close" id="3">)</fence></content><children><number role="integer" font="normal" annotation="clearspeak:simple" id="2">0</number></children></fenced></children></appl><number role="integer" font="normal" annotation="clearspeak:simple" id="5">1</number><infixop role="multiplication" id="23">∗<content><operator role="multiplication" id="8">∗</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="7">L</identifier><fenced role="leftright" id="20"><content><fence role="open" id="9">(</fence><fence role="close" id="17">)</fence></content><children><prefixop role="negative" id="19">−<content><operator role="subtraction" id="10">−</operator></content><children><fraction role="division" id="16"><children><infixop role="implicit" annotation="clearspeak:simple;clearspeak:unit" id="14"><content><operator role="multiplication" id="13"></operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="11">A</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="12">R</identifier></children></infixop><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="15">L</identifier></children></fraction></children></prefixop></children></fenced></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -1.838ex; width:23.197ex; height:5.343ex;', 'sanetex' => '{\\displaystyle v(0)=1=L*(-{\\frac {AR}{L}})}', 'speech' => 'v left parenthesis 0 right parenthesis equals 1 equals upper L asterisk left parenthesis minus StartFraction upper A upper R Over upper L EndFraction right parenthesis', ), )"): {\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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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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'id' => '14', '$t' => '', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '11', '$t' => 'A', ), 1 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '12', '$t' => 'R', ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'multiplication', 'id' => '13', '$t' => '', ), ), ), 1 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '15', '$t' => 'L', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'subtraction', 'id' => '10', '$t' => '−', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'fence', 'role' => 'open', 'id' => '9', '$t' => '(', ), 1 => (object) array( 'type' => 'fence', 'role' => 'close', 'id' => '17', '$t' => ')', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'multiplication', 'id' => '8', '$t' => '∗', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'relation', 'role' => 'equality', 'id' => '4', '$t' => '=', ), 1 => (object) array( 'type' => 'relation', 'role' => 'equality', 'id' => '6', '$t' => '=', ), ), ), ), 'streeXml' => '<stree><relseq role="equality" id="24">=<content><relation role="equality" id="4">=</relation><relation role="equality" id="6">=</relation></content><children><appl role="simple function" annotation="clearspeak:simple" id="22"><content><punctuation role="application" id="21"></punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="0">v</identifier></content><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="0">v</identifier><fenced role="leftright" id="18"><content><fence role="open" id="1">(</fence><fence role="close" id="3">)</fence></content><children><number role="integer" font="normal" annotation="clearspeak:simple" id="2">0</number></children></fenced></children></appl><number role="integer" font="normal" annotation="clearspeak:simple" id="5">1</number><infixop role="multiplication" id="23">∗<content><operator role="multiplication" id="8">∗</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="7">L</identifier><fenced role="leftright" id="20"><content><fence role="open" id="9">(</fence><fence role="close" id="17">)</fence></content><children><prefixop role="negative" id="19">−<content><operator role="subtraction" id="10">−</operator></content><children><fraction role="division" id="16"><children><infixop role="implicit" annotation="clearspeak:simple;clearspeak:unit" id="14"><content><operator role="multiplication" id="13"></operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="11">A</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="12">R</identifier></children></infixop><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="15">L</identifier></children></fraction></children></prefixop></children></fenced></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -1.838ex; width:23.197ex; height:5.343ex;', 'sanetex' => '{\\displaystyle v(0)=1=L*(-{\\frac {AR}{L}})}', 'speech' => 'v left parenthesis 0 right parenthesis equals 1 equals upper L asterisk left parenthesis minus StartFraction upper A upper R Over upper L EndFraction right parenthesis', ), )"): {\displaystyle C_1 = 0}
