$1$ Nodal Analysis

Consider the following circuit. The node voltage phasors are already labelled, please use these labels in your solution. All sources are AC with a frequency $=900\mathrm{.}0$kra$\frac{d}{s}.$

Answer the following questions:

$($a$).$ Identify any supernodes $(1$ mark$).$ Each supernode should be identified by a comma$-$separated list of all node voltages that belong in that supernode $($for example, $"v_1,v_5,v_8",$ if nodes $1,5,$ and $8$ form a supernode$).$ If ground is part of a supernode it can be identified as $v_0$ or $v_{\text{g}\text{r}\text{o}\text{u}\text{n}\text{d}\text{.}}.$ If there aren't any supernodes, then leave this question blank for full marks $($these circuits are randomly generated, it is possible that there aren't any in your circuit$).$

$($b$).$ Calculate the admittance $Y_n$ for each resistor $R_n,$ inductor $L_n,$ and capacitor $C_n(1$ mark$).$ You should express your answer in units of $mS(k{}^{-1}).$ It is not necessary to write the units, unless you decide to use something other than the recommended units.

$($c$).$ Write the nodal equations symbolically $(3$ marks$).$ These equations should include only the node voltage phasors $v_n$ and any appropriate constants $($admittances$,$ impedances, voltages, currents$)$ from the circuit. I recommend writing these equations using $Y_n=Z_n^{-1}$ as the admittance of impedance element $Z_n,$ but if you hate admittance you can write the equations in terms of $Z_n($as $"\frac{1}{Z}{n}^{\text{'}\text{'}}$ or $"Z{n}^{-1}{1}^{?}$ or similar$).$

Please write everything symbolically as phasors $-$ if there is a current source $I_A=3$mAsin$t,$ for example, use $"I{A}^2"$ in the equations rather than $"3$ mA $"$ or $")".$ The only actual numbers in your equation should come from dependent voltage or current sources $($if any$).$ Otherwise please use standard math symbols and brackets $($any of $()\{\}[]$ are accepted$)$ if necessary. Implicit multiplication is accepted $($i$.$e$."$Ya v $1"$ is interpreted as $"$Ya$*$v$1").$

If you need to combine multiplication and division in these equations $($which isn't necessary if you use admittances, but don't let me tell you what to do$),$ please be careful to indicate order of operations with brackets as something like $"$a$/$bc$"$ could mean either $"(\frac{a}{b})c"$ or $"(\frac{a}{b}c)".$

These equations are interpreted mathematically, so it doesn't matter which terms are on the left$-$hand side or right$-$hand side, or how the terms are grouped.

$($d$).$ Obtain the matrix equation $(3$ marks$).$ These matrices should be written numerically, except the unknown node voltage phasors should be still written symbolically. Write complex numbers in Cartesian form $($such as $"-3\mathrm{.}5+5\mathrm{.}1j"$ and similar$).$

You can enter these matrices in any form you want as long as each element is separated from the other elements by some recognizable character $($comma$,$ semicolon, white space, tab, etc.$).$ I am expecting your answer to consist of a $55$ square matrix and two $51$ column matrices.

Note: White space is not recommended as a element separator, unless you are careful to not use any white space in a complex number $($i$.$e$.$ write $"x+yj"$ rather than $"x+yj")$ or place complex numbers in brackets.

$($e$).$ Solve the matrix equation to obtain the node voltage phasors $(3$ marks$).$ These should be expressed as a magnitude and a phase, the magnitude given in units of V $,$ and the phase in units of radians. It is not necessary to write the units, unless you decide to use something other than the recommended units. You can find the angle unicode character if you want $($to write $v_1=5\frac{V}{{?}_{0\mathrm{.}857}}$rad, for example$),$ but it is not necessary. I will interpret any pair of numbers as a magnitude followed by a phase.

Please be sure to express the phase to $3$ or $4$ decimal places to ensure your submission is sufficiently accurate. Phases are usually expressed in the range $-$ to $,$ but it doesn't matter here $-$ I will also accept angles in the range $0$ to $2.$

$($f$).$ Calculate the current $i_u(t)$ shown on the circuit diagram $(1\mathrm{.}5$ marks$).$ This current is of the form: