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<h2 class="hd hd-2 unit-title">Using Complex Numbers to Solve AC Circuits</h2>
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<p><center><b>AC Circuits</b></center></p><p>This week we will begin to explore AC Driven circuits. Instead of having a battery with a constant voltage output, \(\varepsilon\), as we have previously used, we will instead drive circuits with a power source that has a voltage, \(V(t) = V_0 sin(\omega t) \). This will produce a current that is also sinusoidal in nature, an alternating current. We will explore this at greater length in the next lessons.</p><p><center><b>Complex Numbers</b></center></p><p>We will find that complex numbers will make solving AC circuits much simpler. We therefore will take this lesson to explore complex numbers so that you will have the tools you need to tackle the problems in the following lessons.</p><p> We will start by representing complex numbers in the simplest form:</p>
\[z = x + iy\]
<p>And then move on to the Euler representation:</p>
\[z = |z|e^{i\theta}=|z|(\cos\theta + i\sin\theta )\]
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<li><a href="https://openlearninglibrary.mit.edu/courses/course-v1:MITx+8.02.3x+1T2019/pdfbook/0/#viewer-frame" target="[object Object]">Appendix: Complex Numbers</a></li>
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<h2 class="hd hd-2 unit-title">L30v1: Introduction to Complex Numbers</h2>
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<h3 class="hd hd-2">L30v1: Introduction to Complex Numbers</h3>
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<h2 class="hd hd-2 unit-title">L30Q1: Complex Numbers</h2>
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Complex Numbers
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<p>Express the following complex numbers in terms of modulus \(|z|\) and phase \(\theta\). Express your phase as a real number between \(-\pi/2 \le \theta \le +\pi/2\) in radians. Use \(R\), \(L\), \(C\), \(V\), pi for \(\pi\) and omega for \(\omega\) as necessary. Use sqrt() and arctan() (for inverse tangent) where necessary.</p>
<p><b>(Part a)</b> \(z = +i\)</p>
<p>
<p style="display:inline">[mathjaxinline]|z| =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline] </p>
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<p><b>(Part b)</b> \(z = -i\)</p>
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<p style="display:inline">[mathjaxinline]|z| =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline] </p>
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<p><b>(Part c)</b> \(z = R + i \omega L\)</p>
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<p style="display:inline">[mathjaxinline]|z| =[/mathjaxinline] </p>
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<p>
<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline] </p>
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<p><b>(Part d)</b> \(z = R - \dfrac{i}{\omega C}\)</p>
<p>
<p style="display:inline">[mathjaxinline]|z| =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L30v2: Inverse of a Complex Number</h2>
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Inverse of a complex number
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Consider the complex number [mathjaxinline]z=x+i y[/mathjaxinline], where [mathjaxinline]x=\text {Re}(z)[/mathjaxinline] and [mathjaxinline]y=\text {Im}(z)[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> What is [mathjaxinline]\overline{z}[/mathjaxinline], the complex conjugate of [mathjaxinline]z[/mathjaxinline]? Express your answer in terms of [mathjaxinline]x[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], and [mathjaxinline]i[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]\overline{z}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Calculate the product [mathjaxinline]z\overline{z}[/mathjaxinline]. Express your answer in terms of [mathjaxinline]x[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], and [mathjaxinline]i[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]z\overline{z}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> Assume that [mathjaxinline]x\neq 0[/mathjaxinline] or [mathjaxinline]y\neq 0[/mathjaxinline], the inverse of the complex number [mathjaxinline]z[/mathjaxinline] can be written as follows: </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000003">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \dfrac {1}{z} = \dfrac {1}{z}\dfrac {\overline{z}}{\overline{z}}[/mathjaxinline]
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Calculate [mathjaxinline]\left|\dfrac {1}{z}\right|[/mathjaxinline]. Express your answer in terms of [mathjaxinline]x[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], and [mathjaxinline]i[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]\left|\dfrac {1}{z}\right| =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part d)</b> Calculate [mathjaxinline]\dfrac {1}{|z|}[/mathjaxinline]. Express your answer in terms of [mathjaxinline]x[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], and [mathjaxinline]i[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]\dfrac {1}{|z|} =[/mathjaxinline] </p>
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<h3 class="hd hd-2">L30v2: Inverse of a Complex Number</h3>
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<h2 class="hd hd-2 unit-title">L30Q2: Complex Conjugate</h2>
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Complex Conjugate
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<p>
A complex number, [mathjaxinline]z[/mathjaxinline], has a complex conjugate [mathjaxinline]\overline{z}[/mathjaxinline] which is shown in the figure above. If [mathjaxinline]z\overline{z} =4[/mathjaxinline], calculate the real and imaginary part of [mathjaxinline]z[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\text {Re}(z)=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\text {Im}(z)=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L30Q3: Inverse of a Complex Number</h2>
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The Inverse of a Complex Number
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Consider the complex number [mathjaxinline]z = R+i\omega L[/mathjaxinline], where [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and [mathjaxinline]\omega[/mathjaxinline] are real numbers. </p>
<p><b class="bfseries">(Part a)</b> Calculate the real part of [mathjaxinline]\dfrac {1}{z}[/mathjaxinline]. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline]. <p style="display:inline">[mathjaxinline]\text {Re}(1/z)=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_11_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">(Part b)</b> Calculate the imaginary part of [mathjaxinline]\dfrac {1}{z}[/mathjaxinline]. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\text {Im}(1/z)=[/mathjaxinline]</p>
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<p><b class="bfseries">(Part c)</b> Calculate the phase of [mathjaxinline]\dfrac {1}{z}[/mathjaxinline]. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline]</p>
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Consider the complex numbers [mathjaxinline]z_1=3+i2[/mathjaxinline] and [mathjaxinline]z_2=x_2+iy_2[/mathjaxinline] such that [mathjaxinline]z_1+z_2 = \overline{z}_1[/mathjaxinline]. The real and imaginary parts of [mathjaxinline]z_2[/mathjaxinline] are: </p>
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<p style="display:inline">[mathjaxinline]x_2=[/mathjaxinline]</p>
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Multiplication of complex numbers
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Consider the complex numbers [mathjaxinline]z_1=4+i5[/mathjaxinline] and [mathjaxinline]z_2=-2+i3[/mathjaxinline]. </p>
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The real and imaginary parts of the product [mathjaxinline]z_1 z_2[/mathjaxinline] are: </p>
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<p style="display:inline">[mathjaxinline]\text {Re}(z_1z_2)=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L30v5: Euler Formula for Complex Numbers</h2>
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<h2 class="hd hd-2 unit-title">L30v6: Modulus and Phase for a Complex Number</h2>
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<p><center><b>Complex Number and the Euler Formula</b></center></p><p/><p><center><img src="/assets/courseware/v1/9b02710b493a532d43dbaf2ac03b2d4e/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_lesson30_01_01-01.svg" width="600"/></center></p><p> A complex number \(z = x+i y\) can be represented using the Euler Formula as:</p>
\[z = |z|e^{i\theta}=|z|(\cos\theta + i\sin\theta )\]
<p>where</p>
\[|z|=\sqrt{x^2+y^2}\ \text{is the modulus}\]
\[\theta = \tan^{-1}\dfrac{y}{x}\ \text{is the argument}\]
<p><b>Note:</b> You will be shown that the multiplication and division of complex numbers is easier when using the Euler representation of the numbers.</p>
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<h2 class="hd hd-2 unit-title">L30Q6: Euler Formula Exercises</h2>
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Euler Formula Exercises
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<p><b class="bfseries">(Part a)</b> Express [mathjaxinline]z_1= 3-i\; 3\sqrt {3}[/mathjaxinline] using the Euler Formula. Your answer may include [mathjaxinline]i[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]z_1=[/mathjaxinline]</p>
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<p><b class="bfseries">(Part b)</b> Find the imaginary part of the complex number [mathjaxinline]z_2 = 2 e^{i \pi /3}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\text {Im}(z_2)=[/mathjaxinline]</p>
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<p><b class="bfseries">(Part c)</b> Given [mathjaxinline]z_1 = 1+i[/mathjaxinline] and [mathjaxinline]z_2 = 5\; e^{-i\pi /4}[/mathjaxinline], find the real part of the product [mathjaxinline]z_1 z_2[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]z_1 z_2=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L30Q7: Inverse using Euler Formula</h2>
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Inverse of a Complex Number
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<p><b class="bfseries">(Part a)</b> Consider the complex number [mathjaxinline]z = |z|e^{i\theta }[/mathjaxinline]. Which of the following representations correctly describes the inverse of [mathjaxinline]z[/mathjaxinline]? </p>
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<text> [mathjaxinline]\frac{1}{z} = |z|e^{-i \theta }[/mathjaxinline]</text>
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<text> [mathjaxinline]\frac{1}{z} = |z|e^{+i \theta }[/mathjaxinline]</text>
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<text> [mathjaxinline]\frac{1}{z} = \frac{1}{|z|}e^{-i \theta }[/mathjaxinline]</text>
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<text> [mathjaxinline]\frac{1}{z} = \frac{1}{|z|}e^{+i \theta }[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> Consider the complex number [mathjaxinline]z = \dfrac {V}{R+i\omega L}[/mathjaxinline], where [mathjaxinline]V[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]\omega[/mathjaxinline], and [mathjaxinline]L[/mathjaxinline] are real numbers. The Euler representation of this complex number is [mathjaxinline]z = |z| e^{i\theta }[/mathjaxinline]. Find the modulus, [mathjaxinline]|z|[/mathjaxinline], and the phase, [mathjaxinline]\theta[/mathjaxinline], in terms of [mathjaxinline]V[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and arctan() for [mathjaxinline]\tan ^{-1}()[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|z|=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L30Q8: An Unknown Number</h2>
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An Unknown Number
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<p><b class="bfseries">(Part a)</b> Consider the complex numbers [mathjaxinline]z_1 = Ae^{i\alpha }[/mathjaxinline] and [mathjaxinline]z_2 = Be^{i\beta }[/mathjaxinline], where [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] are their moduli, and [mathjaxinline]\alpha[/mathjaxinline] and [mathjaxinline]\beta[/mathjaxinline] their phases. Let [mathjaxinline]z_3 = Ce^{i\theta }[/mathjaxinline] be a complex number such that [mathjaxinline]z_1= z_2 z_3[/mathjaxinline]. Find [mathjaxinline]C[/mathjaxinline], the modulus of [mathjaxinline]z_3[/mathjaxinline], and [mathjaxinline]\theta[/mathjaxinline], the phase of [mathjaxinline]z_3[/mathjaxinline], in terms of [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], alpha for [mathjaxinline]\alpha[/mathjaxinline], and beta for [mathjaxinline]\beta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]C=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part b)</b> Write the Euler expression, [mathjaxinline]|z|e^{i\theta }[/mathjaxinline], of the complex number [mathjaxinline]z=-i\dfrac {1}{\omega C}[/mathjaxinline] where [mathjaxinline]\omega[/mathjaxinline] and [mathjaxinline]C[/mathjaxinline] are real numbers. Express your answer in terms of [mathjaxinline]i[/mathjaxinline], [mathjaxinline]C[/mathjaxinline] and omega for [mathjaxinline]\omega[/mathjaxinline]. Make sure [mathjaxinline]|z|[/mathjaxinline] is a positive quantity and write the phase in the range [mathjaxinline]-\pi &lt;\theta &lt;+\pi[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]z=[/mathjaxinline]</p>
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<p><b class="bfseries">(Part c)</b> Consider two complex numbers that are functions of time: the complex number [mathjaxinline]V= V_0e^{i\omega t}[/mathjaxinline], with modulus [mathjaxinline]V_0[/mathjaxinline] and phase [mathjaxinline]\omega t[/mathjaxinline], and the complex number [mathjaxinline]I=I_0e^{i(\omega t - \theta )}[/mathjaxinline] of modulus [mathjaxinline]I_0[/mathjaxinline] and phase [mathjaxinline]\omega t - \theta[/mathjaxinline]. Find [mathjaxinline]I_0[/mathjaxinline] and [mathjaxinline]\theta[/mathjaxinline] such that [mathjaxinline]V = I z[/mathjaxinline], where [mathjaxinline]z=-i\dfrac {1}{\omega C}[/mathjaxinline]. Express your answer in terms of [mathjaxinline]C[/mathjaxinline], V_0 for [mathjaxinline]V_0[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]I_0=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part d)</b> Calculate the real part of [mathjaxinline]I(t)[/mathjaxinline]. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], V_0 for [mathjaxinline]V_0[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]\text {Re}(I)=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L30Q9: Complex Equations-Part 1</h2>
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Complex Equations-Part 1
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Consider the the real number [mathjaxinline]a[/mathjaxinline] and the complex number [mathjaxinline]z[/mathjaxinline], of modulus [mathjaxinline]z_0[/mathjaxinline] and phase [mathjaxinline]\theta[/mathjaxinline], [mathjaxinline]z = z_0\; e^{i\theta }[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]a\text {Re}(z)[/mathjaxinline] and [mathjaxinline]\text {Re}(az)[/mathjaxinline]. Express your answer in terms of z_0 for [mathjaxinline]z_0[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], and [mathjaxinline]a[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]a\text {Re}(z)=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\text {Re}(az)=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]a\text {Im}(z)[/mathjaxinline] and [mathjaxinline]\text {Im}(az)[/mathjaxinline]. Express your answer in terms of z_0 for [mathjaxinline]z_0[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], and [mathjaxinline]a[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]a\text {Im}(z)=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\text {Im}(az)=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> Assume that the phase of [mathjaxinline]z[/mathjaxinline] is a function of time given by: [mathjaxinline]\theta =\omega t[/mathjaxinline], where [mathjaxinline]\omega[/mathjaxinline] is a positive constant: [mathjaxinline]z=z_0 \; e^{i \omega t}[/mathjaxinline]. Find the Euler expression of [mathjaxinline]\dfrac {dz}{dt}[/mathjaxinline], the time derivative of [mathjaxinline]z[/mathjaxinline]. Express your answer in terms of [mathjaxinline]i[/mathjaxinline], [mathjaxinline]t[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline], and z_0 for [mathjaxinline]z_0[/mathjaxinline]. Be careful to express your answer in the form of a correct Euler representation with a real modulus. </p>
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<p style="display:inline">[mathjaxinline]\dfrac {dz}{dt}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part d)</b> Calculate the real and the imaginary part of [mathjaxinline]\dfrac {dz}{dt}[/mathjaxinline]. Express your answer in terms of z_0 for [mathjaxinline]z_0[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline], and [mathjaxinline]t[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\text {Re}\left(\dfrac {dz}{dt}\right)=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\text {Im}\left(\dfrac {dz}{dt}\right)=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part e)</b> Calculate the time derivative of the real and the imaginary part of [mathjaxinline]z[/mathjaxinline]. Express your answer in terms of z_0 for [mathjaxinline]z_0[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline], and [mathjaxinline]t[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\dfrac {d}{dt}(\text {Re}(z))=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\dfrac {d}{dt}(\text {Im}(z))=[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L30Q10: Complex Equations-Part 2</h2>
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Complex Equations-Part 2a
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In the next lesson, we will look at a circuit with a voltage source that is a sinusoidal function of time of the form [mathjaxinline]V(t) = V_0 \cos (\omega t)[/mathjaxinline] or [mathjaxinline]V(t) = V_0 \sin (\omega t)[/mathjaxinline] where [mathjaxinline]\omega[/mathjaxinline] is the angular frequency. As a result, the current in the circuit will also have a sinusoidal form, possibly shifted by [mathjaxinline]\phi[/mathjaxinline] with respect to [mathjaxinline]V(t)[/mathjaxinline], given by [mathjaxinline]I(t) = I_0 \cos (\omega t-\phi )[/mathjaxinline] or [mathjaxinline]I(t) = I_0 \sin (\omega t - \phi )[/mathjaxinline]. The constants [mathjaxinline]V_0[/mathjaxinline] and [mathjaxinline]I_0[/mathjaxinline] represent the amplitudes of these functions. </p>
<p>
In this exercise, you will relate the real functions [mathjaxinline]V(t)[/mathjaxinline] and [mathjaxinline]I(t)[/mathjaxinline] with complex functions given by [mathjaxinline]V_ c=V_0\; e^{i\omega t}[/mathjaxinline] and [mathjaxinline]I_ c=I_0\; e^{i(\omega t - \phi )}[/mathjaxinline], where the subscript [mathjaxinline]c[/mathjaxinline] is used to distinguish between the real function and the complex one. </p>
<p><b class="bfseries">(Part a)</b> Calculate the real part of [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline]. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], V_0 for [mathjaxinline]V_0[/mathjaxinline], I_0 for [mathjaxinline]I_0[/mathjaxinline], phi for [mathjaxinline]\phi[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\text {Re}(V_ c)=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\text {Re}(I_ c)=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Calculate the imaginary part of [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline]. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], V_0 for [mathjaxinline]V_0[/mathjaxinline], I_0 for [mathjaxinline]I_0[/mathjaxinline], phi for [mathjaxinline]\phi[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\text {Im}(V_ c)=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\text {Im}(I_ c)=[/mathjaxinline] </p>
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Complex Equations-Part 2b
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<p><b class="bfseries">Note:</b> The complex functions [mathjaxinline]V_ c=V_0\; e^{i\omega t}[/mathjaxinline] and [mathjaxinline]I_ c=I_0\; e^{i(\omega t - \phi )}[/mathjaxinline] are called the complexified functions of [mathjaxinline]V_0\cos (\omega t)[/mathjaxinline] and [mathjaxinline]I_0\cos (\omega t - \phi )[/mathjaxinline], respectively. </p>
<p><b class="bfseries">(Part c)</b> In a previous problem you showed that if [mathjaxinline]a[/mathjaxinline] is a real number, and [mathjaxinline]z[/mathjaxinline] is the complex number [mathjaxinline]z=z_0 e^{i\theta }[/mathjaxinline]: </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle a\text {Re}(z)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \text {Re}(a\; z)[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<tr id="a0000000004">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle a\text {Im}(z)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \text {Im}(a\; z)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
If the phase is a function of time given by [mathjaxinline]\theta = \omega t[/mathjaxinline]: </p>
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[mathjaxinline]\displaystyle \dfrac {d}{dt}(\text {Re}(z))[/mathjaxinline]
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<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle \text {Re}\left(\dfrac {dz}{dt}\right)[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \dfrac {d}{dt}(\text {Im}(z))[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \text {Im}\left(\dfrac {dz}{dt}\right)[/mathjaxinline]
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<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
Use the results above and parts (a) and (b) to convert the equations relating the real functions [mathjaxinline]V(t)[/mathjaxinline] and [mathjaxinline]I(t)[/mathjaxinline] into equations relating the complex functions [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline]. </p>
<p><b class="bfseries">(i)</b> If [mathjaxinline]R[/mathjaxinline] is a real number and [mathjaxinline]V = R\; I[/mathjaxinline], find an expression for [mathjaxinline]V_ c[/mathjaxinline] in terms of I_c for [mathjaxinline]I_ c[/mathjaxinline] and [mathjaxinline]R[/mathjaxinline]. <p style="display:inline">[mathjaxinline]V_ c=[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_23b_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">(ii)</b> If [mathjaxinline]L[/mathjaxinline] is a real number and [mathjaxinline]\dfrac {dI}{dt}=\dfrac {1}{L} V[/mathjaxinline], find an expression for [mathjaxinline]\dfrac {dI_ c}{dt}[/mathjaxinline] in terms of V_c for [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]L[/mathjaxinline]. <p style="display:inline">[mathjaxinline]\dfrac {dI_ c}{dt}=[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w13_23b_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">iii.</b> If [mathjaxinline]C[/mathjaxinline] is a real number and [mathjaxinline]\dfrac {dV}{dt}=\dfrac {1}{C}I[/mathjaxinline], find an expression for [mathjaxinline]\dfrac {dV_ c}{dt}[/mathjaxinline] in terms of I_c for [mathjaxinline]I_ c[/mathjaxinline] and [mathjaxinline]C[/mathjaxinline]. <p style="display:inline">[mathjaxinline]\dfrac {dV_ c}{dt}=[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w13_23b_4_1" class="text-input-dynamath capa_inputtype inline textline">
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