(Electrical) Mesh Current Analysis

We looked at how to analyze circuits using node voltage analysis. Now let's explore how to do the same using mesh current analysis.

Review

In a circuit, a mesh is a closed loop, such as $i_1$ and $i_2$ in the following figure. 

The first step of mesh analysis is to identify all the meshes and assign a clockwise current in each mesh where there is no current source. If there is a current source, then the mesh current should be in direction of the current source.

Kirchoff's Voltage Law (KVL) says that the sum of all the voltages across every element in a mesh is 0. For example, looking at the following circuit we can write KVL equations:
$v_s-v_1-v_2=0$, and
$v_2+v_3+v_4=0$

The voltage across an element shared by two meshes is expressed in terms of both mesh currents. For example, $R_2$ is $v_2=(i_1-i_2)R_2$.

KVL for mesh 1 is: $v_s-i_1R_1-(i_1-i_2)R_2=0$
... and for mesh 2 is: $(i_2-i_1)R_2+i_2R_3+i_2R_4=0$

Notice that the polarity of the voltage across $R_2$ for mesh 2 is now flipped. This is to keep consistent with the direction of $i_2$. The voltage across can be written $R_2$ $v_2=(i_2-i_1)R_2$.

Using KVL, we emerge with the following system of equations:
$(R_1+R_2)i_1-R_2i_2=v_s$
$-R_2i_1+(R_2+R_3+R4)i_2=0$

Question

Given DC voltage sources of $V_1=10V$,  $V_2=9V$, and $V_3=1V$ and resistors with resistances of $R_{1}=5\Omega$, $R_2=10\Omega$, $R_3=5\Omega$, and $R_4=5\Omega$, you connect them into the following circuit:

question_figure.png

Find mesh currents $i_1$ and $i_2$
Bonus: Write your KVL equations

 

Use latex to write equations.


The solutions to Tues & Thurs questions are free.
If you'd like full access to ALL our daily questions & solutions (Mon-Fri), sign-up for LEARNerds Premium!

(Physics) A Smarter Way To Get Into Space

Last week you figured out that you can't just launch straight out of Earth. The atmosphere would burn you alive before you made it. So instead you must launch from low Earth orbit.

"SRBsepfromDiscovery07042006" by NASA - photo from video released by NASA into public domain. Licensed under Public domain via Wikimedia Commons

"SRBsepfromDiscovery07042006" by NASA - photo from video released by NASA into public domain. Licensed under Public domain via Wikimedia Commons

In low Earth orbit you are between 160 km and 2000 km above surface of the Earth free falling towards it. Gravity is the centripetal force keeping you in orbit. Being in low Earth orbit means you are farther from the center of the Earth which means less gravity. You know this also means a smaller escape velocity. 


If you are orbiting an altitude of 1000 km, what is your centripetal acceleration? What would your escape velocity be at this height?

(Econ) You Don't Appreciate Me! -- Part 2

A constant question in every business owner's mind is: “How do I pay less taxes?” One small way is by maximizing depreciation. Two weeks ago we introduced the concept of “depreciation,” specifically “Straight-line Depreciation” (CLICK HERE for a review!). But that’s not the only depreciation method. In actuality, the IRS offers businesses the choice between 2 methods: Straight-line v.s. “MACRS” (Modified Accelerated Cost Recover System). This week, we’re focusing on just MACRS.

So say you’re a proud rural Indiana farmer. And it’s time for you to buy some new cattle (damn Mad Cow Disease!). And that's right… you can even depreciate cattle! It’s going to cost about 100K USD to replace your herd. And all you want to know is “if I use the MACRS method, how much in taxes will I save?” You take a quick look online find snapshots of the following MACRS charts. It’s all new. But you do your best to make sense of it… 

(Partial snapshots taken from: http://en.wikipedia.org/wiki/MACRS)

You only plan to keep the farm another 4 years. So if you spend 100K USD on cows today, and you’re in the 35% tax bracket (assume flat tax)… over the next 4 years, how much in taxes will the MACRS Depreciation Method save you?


      21,000 USD
      29,000 USD
      34,000 USD
      39,000 USD

(Electronics) Pizza Party!

Transient

You and your team have been working on a massive reverse engineering project for the past month, and everyone is beyond burnt out. Your team lead decides to motivate everyone with the promise of a pizza party! Everyone is given one last piece of the puzzle to complete. Here is yours:


What are the Thevenin voltage and resistance between terminal A and B?

(Physics) Escape from Earth

You are the lead project designer for Space X's new mission to the moon. The first step is simple enough. Get off Earth.

Transient

Getting off of Earth means escaping its gravitational field. To do so, you need a fast enough escape velocity that allows you to fly away from Earth without using anymore fuel. Once you reach this point, the kinetic energy of this launch will at least match the Earth's gravitational potential energy.


What is the escape velocity required to break free of Earth's gravity? Assume air resistance is negligible.
Stuck? Click here for a hint
Here are some useful constants.

      5.5 km/s
      11.2 km/s
      106 km/s
      121 km/s