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):

v(t)=L{di(t) \over dt}

:

v(0)=1=L*(-{\frac {AR}{L}})

:

A=-1/R

If the current in the inductor is initially zero, then:

i(0)=0=A+C

Which implies that:

C=-A=1/R

So the response to a DC voltage source turning on at t=0 to one volt (called the unit response μ) is:

i_{\mu }(t)={\frac {1}{R}}(1-e^{-{\frac {t}{\frac {L}{R}}}})

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

i_{\delta }(t)={\frac {e^{-{\frac {t}{\frac {L}{R}}}}}{L}}

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

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="27">0</number><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="28">t</identifier></children></limboth><infixop role="implicit" annotation="clearspeak:unit" id="58">⁢<content><operator role="multiplication" id="56">⁢</operator><operator role="multiplication" id="57">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="30">f</identifier><fenced role="leftright" id="51"><content><fence role="open" id="31">(</fence><fence role="close" id="35">)</fence></content><children><infixop role="subtraction" id="50">−<content><operator role="subtraction" id="33">−</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="32">t</identifier><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="34">τ</identifier></children></infixop></children></fenced><appl role="simple function" annotation="clearspeak:simple" id="54"><content><punctuation role="application" id="53">⁡</punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="36">g</identifier></content><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="36">g</identifier><fenced role="leftright" id="52"><content><fence role="open" id="37">(</fence><fence role="close" id="39">)</fence></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="38">τ</identifier></children></fenced></children></appl></children></infixop><prefixop role="integral" font="normal" id="55">d<content><identifier role="latinletter" font="italic" id="40">d</identifier></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="41">τ</identifier></children></prefixop></children></integral><subscript role="latinletter" id="45"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="43">C</identifier><number role="integer" font="normal" annotation="clearspeak:simple" id="44">1</number></children></subscript></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -2.338ex; width:53.329ex; height:6.176ex;', 'sanetex' => '{\\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}}', 'speech' => 'i left parenthesis t right parenthesis equals integral Subscript 0 Superscript t Baseline i Subscript delta Baseline left parenthesis t minus tau right parenthesis upper V Subscript s Baseline left parenthesis tau right parenthesis d tau equals integral Subscript 0 Superscript t Baseline f left parenthesis t minus tau right parenthesis g left parenthesis tau right parenthesis d tau plus upper C 1', ), )"): {\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. 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data-semantic-parent="55">τ</mi></mrow></mrow><mo data-semantic-type="operator" data-semantic-role="addition" data-semantic-id="42" data-semantic-parent="70" data-semantic-operator="infixop,+">+</mo><msub data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-id="45" data-semantic-children="43,44" data-semantic-parent="70"><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="43" data-semantic-parent="45">C</mi><mrow class="MJX-TeXAtom-ORD"><mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple" data-semantic-id="44" data-semantic-parent="45">1</mn></mrow></msub></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\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}}</annotation></semantics></math>', 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id="27">0</number><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="28">t</identifier></children></limboth><infixop role="implicit" annotation="clearspeak:unit" id="58">⁢<content><operator role="multiplication" id="56">⁢</operator><operator role="multiplication" id="57">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="30">f</identifier><fenced role="leftright" id="51"><content><fence role="open" id="31">(</fence><fence role="close" id="35">)</fence></content><children><infixop role="subtraction" id="50">−<content><operator role="subtraction" id="33">−</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="32">t</identifier><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="34">τ</identifier></children></infixop></children></fenced><appl role="simple function" annotation="clearspeak:simple" id="54"><content><punctuation role="application" id="53">⁡</punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="36">g</identifier></content><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="36">g</identifier><fenced role="leftright" id="52"><content><fence role="open" id="37">(</fence><fence role="close" id="39">)</fence></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="38">τ</identifier></children></fenced></children></appl></children></infixop><prefixop role="integral" font="normal" id="55">d<content><identifier role="latinletter" font="italic" id="40">d</identifier></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="41">τ</identifier></children></prefixop></children></integral><subscript role="latinletter" id="45"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="43">C</identifier><number role="integer" font="normal" annotation="clearspeak:simple" id="44">1</number></children></subscript></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -2.338ex; width:53.329ex; height:6.176ex;', 'sanetex' => '{\\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}}', 'speech' => 'i left parenthesis t right parenthesis equals integral Subscript 0 Superscript t Baseline i Subscript delta Baseline left parenthesis t minus tau right parenthesis upper V Subscript s Baseline left parenthesis tau right parenthesis d tau equals integral Subscript 0 Superscript t Baseline f left parenthesis t minus tau right parenthesis g left parenthesis tau right parenthesis d tau plus upper C 1', ), )"): {\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:

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data-semantic-parent="55">τ</mi></mrow></mrow><mo data-semantic-type="operator" data-semantic-role="addition" data-semantic-id="42" data-semantic-parent="70" data-semantic-operator="infixop,+">+</mo><msub data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-id="45" data-semantic-children="43,44" data-semantic-parent="70"><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="43" data-semantic-parent="45">C</mi><mrow class="MJX-TeXAtom-ORD"><mn data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple" data-semantic-id="44" data-semantic-parent="45">1</mn></mrow></msub></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\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}}</annotation></semantics></math>', 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id="43">C</identifier><number role="integer" font="normal" annotation="clearspeak:simple" id="44">1</number></children></subscript></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -2.338ex; width:53.329ex; height:6.176ex;', 'sanetex' => '{\\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}}', 'speech' => 'i left parenthesis t right parenthesis equals integral Subscript 0 Superscript t Baseline i Subscript delta Baseline left parenthesis t minus tau right parenthesis upper V Subscript s Baseline left parenthesis tau right parenthesis d tau equals integral Subscript 0 Superscript t Baseline f left parenthesis t minus tau right parenthesis g left parenthesis tau right parenthesis d tau plus upper C 1', ), )"): {\displaystyle i_o(t) = \int_0^t e^{-(t-\tau)}\sin (t-\tau) (1 + \cos \tau) d\tau + C_1}

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id="43">C</identifier><number role="integer" font="normal" annotation="clearspeak:simple" id="44">1</number></children></subscript></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -2.338ex; width:53.329ex; height:6.176ex;', 'sanetex' => '{\\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}}', 'speech' => 'i left parenthesis t right parenthesis equals integral Subscript 0 Superscript t Baseline i Subscript delta Baseline left parenthesis t minus tau right parenthesis upper V Subscript s Baseline left parenthesis tau right parenthesis d tau equals integral Subscript 0 Superscript t Baseline f left parenthesis t minus tau right parenthesis g left parenthesis tau right parenthesis d tau plus upper C 1', ), )"): {\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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id="54"><content><punctuation role="application" id="53">⁡</punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="36">g</identifier></content><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="36">g</identifier><fenced role="leftright" id="52"><content><fence role="open" id="37">(</fence><fence role="close" id="39">)</fence></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="38">τ</identifier></children></fenced></children></appl></children></infixop><prefixop role="integral" font="normal" id="55">d<content><identifier role="latinletter" font="italic" id="40">d</identifier></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="41">τ</identifier></children></prefixop></children></integral><subscript role="latinletter" id="45"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="43">C</identifier><number role="integer" font="normal" annotation="clearspeak:simple" id="44">1</number></children></subscript></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -2.338ex; width:53.329ex; height:6.176ex;', 'sanetex' => '{\\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}}', 'speech' => 'i left parenthesis t right parenthesis equals integral Subscript 0 Superscript t Baseline i Subscript delta Baseline left parenthesis t minus tau right parenthesis upper V Subscript s Baseline left parenthesis tau right parenthesis d tau equals integral Subscript 0 Superscript t Baseline f left parenthesis t minus tau right parenthesis g left parenthesis tau right parenthesis d tau plus upper C 1', ), )"): {\displaystyle i_o(0_+) = 0 = \frac{1}{5} - \frac{7}{10} + \frac{1}{2} + C_1}

This implies:

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'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '18', '$t' => 's', ), ), ), 1 => (object) array( 'type' => 'fenced', 'role' => 'leftright', 'id' => '49', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'greekletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '21', '$t' => 'τ', ), ), 'content' => array ( 0 => (object) array( 'type' => 'fence', 'role' => 'open', 'id' => '20', '$t' => '(', ), 1 => (object) array( 'type' => 'fence', 'role' => 'close', 'id' => '22', '$t' => ')', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'punctuation', 'role' => 'application', 'id' => '60', '$t' => '⁡', ), 1 => (object) array( 'type' => 'identifier', 'role' => 'simple function', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '17', '$t' => 'V', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'multiplication', 'id' => '65', '$t' => '⁢', ), ), ), 2 => (object) array( 'type' => 'prefixop', 'role' => 'integral', 'font' => 'normal', 'id' => '64', '$t' => 'd', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'greekletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '24', '$t' => 'τ', ), ), 'content' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'id' => '23', '$t' => 'd', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'largeop', 'role' => 'integral', 'id' => '5', '$t' => '∫', ), ), ), 2 => (object) array( 'type' => 'infixop', 'role' => 'addition', 'id' => '70', '$t' => '+', 'children' => array ( 0 => (object) array( 'type' => 'integral', 'role' => 'integral', 'id' => '59', 'children' => array ( 0 => (object) array( 'type' => 'limboth', 'role' => 'integral', 'id' => '29', 'children' => array ( 0 => (object) array( 'type' => 'largeop', 'role' => 'integral', 'id' => '26', '$t' => '∫', ), 1 => (object) array( 'type' => 'number', 'role' => 'integer', 'font' => 'normal', 'annotation' => 'clearspeak:simple', 'id' => '27', '$t' => '0', ), 2 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '28', '$t' => 't', ), ), ), 1 => (object) array( 'type' => 'infixop', 'role' => 'implicit', 'annotation' => 'clearspeak:unit', 'id' => '58', '$t' => '⁢', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '30', '$t' => 'f', ), 1 => (object) array( 'type' => 'fenced', 'role' => 'leftright', 'id' => '51', 'children' => array ( 0 => (object) array( 'type' => 'infixop', 'role' => 'subtraction', 'id' => '50', '$t' => '−', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '32', '$t' => 't', ), 1 => (object) array( 'type' => 'identifier', 'role' => 'greekletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '34', '$t' => 'τ', ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'subtraction', 'id' => '33', '$t' => '−', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'fence', 'role' => 'open', 'id' => '31', '$t' => '(', ), 1 => (object) array( 'type' => 'fence', 'role' => 'close', 'id' => '35', '$t' => ')', ), ), ), 2 => (object) array( 'type' => 'appl', 'role' => 'simple function', 'annotation' => 'clearspeak:simple', 'id' => '54', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'simple function', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '36', '$t' => 'g', ), 1 => (object) array( 'type' => 'fenced', 'role' => 'leftright', 'id' => '52', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'greekletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '38', '$t' => 'τ', ), ), 'content' => array ( 0 => (object) array( 'type' => 'fence', 'role' => 'open', 'id' => '37', '$t' => '(', ), 1 => (object) array( 'type' => 'fence', 'role' => 'close', 'id' => '39', '$t' => ')', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'punctuation', 'role' => 'application', 'id' => '53', '$t' => '⁡', ), 1 => (object) array( 'type' => 'identifier', 'role' => 'simple function', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '36', '$t' => 'g', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'multiplication', 'id' => '56', '$t' => '⁢', ), 1 => (object) array( 'type' => 'operator', 'role' => 'multiplication', 'id' => '57', '$t' => '⁢', ), ), ), 2 => (object) array( 'type' => 'prefixop', 'role' => 'integral', 'font' => 'normal', 'id' => '55', '$t' => 'd', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'greekletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '41', '$t' => 'τ', ), ), 'content' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'id' => '40', '$t' => 'd', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'largeop', 'role' => 'integral', 'id' => '26', '$t' => '∫', ), ), ), 1 => (object) array( 'type' => 'subscript', 'role' => 'latinletter', 'id' => '45', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '43', '$t' => 'C', ), 1 => (object) array( 'type' => 'number', 'role' => 'integer', 'font' => 'normal', 'annotation' => 'clearspeak:simple', 'id' => '44', '$t' => '1', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'operator', 'role' => 'addition', 'id' => '42', '$t' => '+', ), ), ), ), 'content' => array ( 0 => (object) array( 'type' => 'relation', 'role' => 'equality', 'id' => '4', '$t' => '=', ), 1 => (object) array( 'type' => 'relation', 'role' => 'equality', 'id' => '25', '$t' => '=', ), ), ), ), 'streeXml' => '<stree><relseq role="equality" id="71">=<content><relation role="equality" id="4">=</relation><relation role="equality" id="25">=</relation></content><children><appl role="simple function" annotation="clearspeak:simple" id="69"><content><punctuation role="application" id="68">⁡</punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="0">i</identifier></content><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="0">i</identifier><fenced role="leftright" id="46"><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><integral role="integral" id="67"><content><largeop role="integral" id="5">∫</largeop></content><children><limboth role="integral" id="8"><children><largeop role="integral" id="5">∫</largeop><number role="integer" font="normal" annotation="clearspeak:simple" id="6">0</number><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="7">t</identifier></children></limboth><infixop role="implicit" annotation="clearspeak:unit" id="66">⁢<content><operator role="multiplication" id="65">⁢</operator></content><children><appl role="simple function" id="63"><content><punctuation role="application" id="62">⁡</punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="9">i</identifier></content><children><subscript role="simple function" id="11"><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="9">i</identifier><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="10">δ</identifier></children></subscript><fenced role="leftright" id="48"><content><fence role="open" id="12">(</fence><fence role="close" id="16">)</fence></content><children><infixop role="subtraction" id="47">−<content><operator role="subtraction" id="14">−</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="13">t</identifier><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="15">τ</identifier></children></infixop></children></fenced></children></appl><appl role="simple function" annotation="clearspeak:simple" id="61"><content><punctuation role="application" id="60">⁡</punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="17">V</identifier></content><children><subscript role="simple function" id="19"><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="17">V</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="18">s</identifier></children></subscript><fenced role="leftright" id="49"><content><fence role="open" id="20">(</fence><fence role="close" id="22">)</fence></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="21">τ</identifier></children></fenced></children></appl></children></infixop><prefixop role="integral" font="normal" id="64">d<content><identifier role="latinletter" font="italic" id="23">d</identifier></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="24">τ</identifier></children></prefixop></children></integral><infixop role="addition" id="70">+<content><operator role="addition" id="42">+</operator></content><children><integral role="integral" id="59"><content><largeop role="integral" id="26">∫</largeop></content><children><limboth role="integral" id="29"><children><largeop role="integral" id="26">∫</largeop><number role="integer" font="normal" annotation="clearspeak:simple" id="27">0</number><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="28">t</identifier></children></limboth><infixop role="implicit" annotation="clearspeak:unit" id="58">⁢<content><operator role="multiplication" id="56">⁢</operator><operator role="multiplication" id="57">⁢</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="30">f</identifier><fenced role="leftright" id="51"><content><fence role="open" id="31">(</fence><fence role="close" id="35">)</fence></content><children><infixop role="subtraction" id="50">−<content><operator role="subtraction" id="33">−</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="32">t</identifier><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="34">τ</identifier></children></infixop></children></fenced><appl role="simple function" annotation="clearspeak:simple" id="54"><content><punctuation role="application" id="53">⁡</punctuation><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="36">g</identifier></content><children><identifier role="simple function" font="italic" annotation="clearspeak:simple" id="36">g</identifier><fenced role="leftright" id="52"><content><fence role="open" id="37">(</fence><fence role="close" id="39">)</fence></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="38">τ</identifier></children></fenced></children></appl></children></infixop><prefixop role="integral" font="normal" id="55">d<content><identifier role="latinletter" font="italic" id="40">d</identifier></content><children><identifier role="greekletter" font="italic" annotation="clearspeak:simple" id="41">τ</identifier></children></prefixop></children></integral><subscript role="latinletter" id="45"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="43">C</identifier><number role="integer" font="normal" annotation="clearspeak:simple" id="44">1</number></children></subscript></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -2.338ex; width:53.329ex; height:6.176ex;', 'sanetex' => '{\\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}}', 'speech' => 'i left parenthesis t right parenthesis equals integral Subscript 0 Superscript t Baseline i Subscript delta Baseline left parenthesis t minus tau right parenthesis upper V Subscript s Baseline left parenthesis tau right parenthesis d tau equals integral Subscript 0 Superscript t Baseline f left parenthesis t minus tau right parenthesis g left parenthesis tau right parenthesis d tau plus upper C 1', ), )"): {\displaystyle C_1 = 0}

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