Impulse Response

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:

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

Fout bij het parsen (Conversiefout. 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Now the current due to any arbitrary V_S(t) can be found using the convolution integral:

Fout bij het parsen (Conversiefout. 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id="12"><content><punctuation role="application" id="11">⁡</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="10"><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">0</number><infixop role="addition" id="13">+<content><operator role="addition" id="8">+</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="7">A</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="9">C</identifier></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.838ex; width:17.483ex; height:2.843ex;', 'sanetex' => '{\\displaystyle i(0)=0=A+C}', 'speech' => 'i left parenthesis 0 right parenthesis equals 0 equals upper A plus upper C', ), )"): {\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 Fout bij het parsen (Conversiefout. De server ("cli") heeft gemeld: "array ( 'nohash' => array ( ), 'success' => true, '87967e21148b7d7a2a336915259ea167' => (object) array( 'speakText' => 'i left parenthesis 0 right parenthesis equals 0 equals upper A plus upper C', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="i left parenthesis 0 right parenthesis equals 0 equals upper A plus upper C"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-id="14" data-semantic-children="12,5,13" data-semantic-content="4,6"><mrow data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple" data-semantic-id="12" data-semantic-children="0,10" data-semantic-content="11,0" data-semantic-parent="14"><mi data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" 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id="12"><content><punctuation role="application" id="11">⁡</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="10"><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">0</number><infixop role="addition" id="13">+<content><operator role="addition" id="8">+</operator></content><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="7">A</identifier><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="9">C</identifier></children></infixop></children></relseq></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.838ex; width:17.483ex; height:2.843ex;', 'sanetex' => '{\\displaystyle i(0)=0=A+C}', 'speech' => 'i left parenthesis 0 right parenthesis equals 0 equals upper A plus upper C', ), )"): {\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:

Fout bij het parsen (Conversiefout. De server ("cli") heeft gemeld: "array ( 'nohash' => array ( ), 'success' => true, '87967e21148b7d7a2a336915259ea167' => (object) array( 'speakText' => 'i left parenthesis 0 right parenthesis equals 0 equals upper A plus upper C', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="i left parenthesis 0 right parenthesis equals 0 equals upper A plus upper C"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-id="14" data-semantic-children="12,5,13" data-semantic-content="4,6"><mrow data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple" data-semantic-id="12" data-semantic-children="0,10" data-semantic-content="11,0" data-semantic-parent="14"><mi data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" 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=> true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.838ex; width:17.483ex; height:2.843ex;', 'sanetex' => '{\\displaystyle i(0)=0=A+C}', 'speech' => 'i left parenthesis 0 right parenthesis equals 0 equals upper A plus upper C', ), )"): {\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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Fout bij het parsen (Conversiefout. 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=> true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.838ex; width:17.483ex; height:2.843ex;', 'sanetex' => '{\\displaystyle i(0)=0=A+C}', 'speech' => 'i left parenthesis 0 right parenthesis equals 0 equals upper A plus upper C', ), )"): {\displaystyle I_s = 1 + \cos t}

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

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This implies:

Fout bij het parsen (Conversiefout. 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