Impulse response

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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:

$i(t)=Ae^{-{\frac {t}{\frac {L}{R}}}}+C$

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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Taking the derivative of this, get the impulse (δ) current is:

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Now the current due to any arbitrary V_S(t) can be found using the convolution integral:

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EndFraction', ), )"): {\displaystyle i(t) = \int_0^t i_\delta (t-\tau) V_s(\tau) d\tau = \int_0^t f(t-\tau)g(\tau)d\tau + C_1}

Don't think i_δ as current. It is really Failed to parse (Conversion error. 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id="2">t</identifier></children></fenced></children></appl><infixop role="implicit" annotation="clearspeak:unit" id="25">⁢<content><operator role="multiplication" id="24">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="5">L</identifier><fraction role="division" id="20"><children><infixop role="implicit" annotation="clearspeak:unit" id="15">⁢<content><operator role="multiplication" id="14">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="6">d</identifier><appl role="simple function" annotation="clearspeak:simple" id="13"><content><punctuation role="application" id="12">⁡</punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="7">i</identifier></content><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="7">i</identifier><fenced role="leftright" id="11"><content><fence role="open" id="8">(</fence><fence role="close" id="10">)</fence></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="9">t</identifier></children></fenced></children></appl></children></infixop><infixop role="implicit" annotation="clearspeak:simple;clearspeak:unit" id="19">⁢<content><operator role="multiplication" id="18">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="16">d</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="17">t</identifier></children></infixop></children></fraction></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -2.005ex; width:13.961ex; height:5.843ex;', 'sanetex' => '{\\displaystyle v(t)=L{di(t) \\over dt}}', 'speech' => 'v left parenthesis t right parenthesis equals upper L StartFraction d i left parenthesis t right parenthesis Over d t EndFraction', ), )"): {\displaystyle {d \over dt}\frac{current}{1 volt}} . V_S(τ) turns into a multiplier.

LRC Example

Find the time domain expression for i_o given that I_s = cos(t + π/2)μ(t) amp.

Earlier the step response for this problem was found:

Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '7de306fe3a699c2eeefb66e62b8e26cb' => (object) array( 'speakText' => 'v left parenthesis t right parenthesis equals upper L StartFraction d i left parenthesis t right parenthesis Over d t EndFraction', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="v left parenthesis t right parenthesis equals upper L StartFraction d i left parenthesis t right parenthesis Over d t EndFraction"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-id="26" data-semantic-children="23,25" data-semantic-content="4"><mrow data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple" data-semantic-id="23" data-semantic-children="0,21" data-semantic-content="22,0" data-semantic-parent="26"><mi data-semantic-type="identifier" data-semantic-role="simple 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data-semantic-parent="13" data-semantic-operator="appl">i</mi><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-id="12" data-semantic-parent="13" data-semantic-added="true" data-semantic-operator="appl">⁡</mo><mrow data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-id="11" data-semantic-children="9" data-semantic-content="8,10" data-semantic-parent="13"><mo stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-id="8" data-semantic-parent="11" data-semantic-operator="fenced">(</mo><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="11">t</mi><mo stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-id="10" data-semantic-parent="11" data-semantic-operator="fenced">)</mo></mrow></mrow></mrow><mrow data-semantic-type="infixop" 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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:

Failed to parse (Conversion error. 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id="8">(</fence><fence role="close" id="10">)</fence></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="9">t</identifier></children></fenced></children></appl></children></infixop><infixop role="implicit" annotation="clearspeak:simple;clearspeak:unit" id="19">⁢<content><operator role="multiplication" id="18">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="16">d</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="17">t</identifier></children></infixop></children></fraction></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -2.005ex; width:13.961ex; height:5.843ex;', 'sanetex' => '{\\displaystyle v(t)=L{di(t) \\over dt}}', 'speech' => 'v left parenthesis t right parenthesis equals upper L StartFraction d i left parenthesis t right parenthesis Over d t 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

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EndFraction', ), )"): {\displaystyle i_o(0_+) = 0 = \frac{1}{5} - \frac{7}{10} + \frac{1}{2} + C_1}

This implies:

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=> array ( 0 => (object) array( 'type' => 'punctuation', 'role' => 'application', 'id' => '12', '$t' => '⁡', ), 1 => (object) array( 'type' => 'identifier', 'role' => 'simple function', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '7', '$t' => 'i', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'multiplication', 'id' => '14', '$t' => '⁢', ), ), ), 1 => (object) array( 'type' => 'infixop', 'role' => 'implicit', 'annotation' => 'clearspeak:simple;clearspeak:unit', 'id' => '19', '$t' => '⁢', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '16', '$t' => 'd', ), 1 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '17', '$t' => 't', ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'multiplication', 'id' => '18', '$t' => '⁢', ), ), ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'multiplication', 'id' => '24', '$t' => '⁢', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'relation', 'role' => 'equality', 'id' => '4', '$t' => '=', ), ), ), ), 'streeXml' => '<stree><relseq role="equality" id="26">=<content><relation role="equality" id="4">=</relation></content><children><appl role="simple function" annotation="clearspeak:simple" id="23"><content><punctuation role="application" id="22">⁡</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="21"><content><fence role="open" id="1">(</fence><fence role="close" id="3">)</fence></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="2">t</identifier></children></fenced></children></appl><infixop role="implicit" annotation="clearspeak:unit" id="25">⁢<content><operator role="multiplication" id="24">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="5">L</identifier><fraction role="division" id="20"><children><infixop role="implicit" annotation="clearspeak:unit" id="15">⁢<content><operator role="multiplication" id="14">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="6">d</identifier><appl role="simple function" annotation="clearspeak:simple" id="13"><content><punctuation role="application" id="12">⁡</punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="7">i</identifier></content><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="7">i</identifier><fenced role="leftright" id="11"><content><fence role="open" id="8">(</fence><fence role="close" id="10">)</fence></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="9">t</identifier></children></fenced></children></appl></children></infixop><infixop role="implicit" annotation="clearspeak:simple;clearspeak:unit" id="19">⁢<content><operator role="multiplication" id="18">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="16">d</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="17">t</identifier></children></infixop></children></fraction></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -2.005ex; width:13.961ex; height:5.843ex;', 'sanetex' => '{\\displaystyle v(t)=L{di(t) \\over dt}}', 'speech' => 'v left parenthesis t right parenthesis equals upper L StartFraction d i left parenthesis t right parenthesis Over d t EndFraction', ), )"): {\displaystyle C_1 = 0}

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