$1$ Nodal Analysis

Consader the following circuit. The node voltages are already labelled, please use these labels in your solution.

Anower the following questions:

$($a$).$ Identify any supernodes $(1$ mark$).$ Each supernode should be identified by a comma$-$separabed list of all ode$).$ If ground is part of a supernode it can be identified as is or yroun. If there aren't any supernodes, then leave this question blank for full marks $($these circuits are randomly generated, it is posaible that there aren't aryy in your cincust$).$

$($b$).$ Cakculate the ounductance $\backslash ($ G$\_\{$n$\}\backslash )$ for each resistor $\backslash ($ R$\_\{\backslash$mathrm$\{$n$\}\}\backslash )(1$ mark$).$ You shoald express your answer in units of $\backslash (\backslash$mathrm$\{$mS$\}\backslash$left$(\backslash$mathrm$\{$kr$\}\backslash$mathrm$\{$t$\}^\{-1\}\backslash$right$)\backslash ).$ It is not necessary to wribe the units, unless yous decide to use samething other than the recommerided sunita.

$($c$)$ Write the nodal equations symbolically $(3$ marks$).$ These equations should include only the node volt$-$

Ploase write cverything symbolically $-$ if there is a current source $\backslash ($ J$\_\{$A$\}=8\backslash$mathrm$\{$~m$\}/\backslash ),$ foe coample, lese $"$IA$"$ in the equations rather than $"3$a $".$ The only actual numbers in your equation should come from dependent voltage or current scurces $($if any$).$ Otherwise please use standand math symbols and brackets $($any of $()\backslash \{\backslash \{$

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

These equations are interprefed mathembtically, so it doesi't mather which terms are on the left$-$hand side or right$-$hand side, or how the berms are grouped. toode voltagis should be still writows symbolically

Yen can enter thete matrioes in aryy form you want as longf the each element is separated from the other elements by some focognizable ch$$nacter $($conama$,$ semicolon, white space, tab, etc.$).$

$($e$)$ Solve the matrix equation to odtain the node voltages $(3$ marks$).$ These should he given in units of V $.$ It is

$($f$).$ Cakculate the current $\backslash ($ i$\_\{$y$\}\backslash )$ shown on the circuit diagram $(2$ marks$).$ Your answer should be given in units of units.