Notes on DC Circuits

Karen Tennenhouse, modified by Stefan Bracher
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Textbook

University Physics Volume 2: Chapter 10

Theory

The Kirchhoff Loop Rule

(Voltage rule)

We can understand this as being a result of conservation of energy, and of “electrical potential energy” being a meaningful thing (ie the electrostatic force is conservative in the sense we defined in mechanics course.)

The more precise version of this rule is:

The algebraic  total of the changes in potential,  encountered when 
going completely around any loop, must be zero. 

To apply this rule, we assign + and – signs to the potential diff's as follows:

• If we traverse (mentally walk through) a resistor in the direction of the drawn current i, then the change in potential is – iR.
  

If we walk opposite to the current direction, the change is + iR.

• If we traverse (walk through) an emf from its negative pole to positive pole, the change in potential is + ε. If traversed in the opposite direction, the change is – ε .

The Kirchhoff Junction Rule

(Current rule)

“Currents in parallel branches add up to the current going in.”
We can understand it as a result of the conservation of charge:

Charge is not being allowed to stop and pile up somewhere in 
the circuit, and it is not (in net)created or destroyed.

The more precise version of this rule is:

The algebraic sum of currents at any junction is zero. 

To apply this rule, we write the current equation with + and – signs as follows:

• A current approaching (going into) the junction is written as a positive term

• A current leaving the junction gives a negative term.
  

Use of Kirchhoff's Rules

We use the Kirchoff rules in various ways, including:

A) As important facts in themselves.

B) To solve “simple to medium” problems, often with only one source of emf.

C) Kirchoff’s rules can be used to prove two useful shortcut formulas for the equivalent resistance of several connected resistors. “Equivalent” here means that the rest of the circuit would behave the same if the combination were replaced by one resistor Req .

These two shortcut formulas are often used to calculate Req, and sometimes even to re-draw part of the circuit, to simplify a problem.

Specifically:

In series,  resistance adds up: $R_{e}q=R_{1}+R_{2}+....+R_{n}$ 

In parallel,   $1/R_{eq}=1/R_{1}+1/R_{2}+.....+1/R_{n}$ 
 

(You might say that inverses of resistances add up in parallel.)

You should be able to prove these two results, using the simple, intuitive versions of Kirchoff’s laws.
When there are just two resistors in parallel, it’s often faster to use the version

$R_{eq}={\frac {(R_{1}R_{2})}{(R_{1}+R_{2})}}$

Notice that adding more resistors in series causes Req to increase, but adding more resistors in parallel causes Req to decrease.

D) To solve more complicated problems:

Usually these problems involve several emf’s in separate loops, or some complicated arrangement of resistors that cannot be reduced by Req . In this kind of problem, it is not initially obvious which way the currents will really flow, and we need to use a more formal, systematic method.

Formal method of using Kirchhoff's Rules

(In more complicated problems)

(optional step A) If there are several branches containing only resistors, you may like to redraw the circuit, replacing this section with its Req , before starting the main method.

B) Make a large, clear drawing of the cct. Label each emf and resistor.

C) Guess, draw and label a direction for the current in each separate branch.

It does not matter if your guess is correct; that will all come out in the wash when we solve. Correct guesses will just save you a few minuses in step H).

But, if the question gives you current directions, you must use the given ones.

D) Decide which loops you will examine.

[ Some people like to use the interior loops (windows) of the diagram; some people use the outermost loop; sometimes you can see that a certain loop will have a simpler equation. Do what you like.]

For each chosen loop, choose and draw a direction, clockwise OR counterclockwise, that you will traverse (mentally “walk around”) the loop. Again, it does not matter which you choose.

E) Decide which junction (s) to use. ( It’s often very useful to label them with letters.) Apply the Kirchoff current rule, sum of currents at junction = 0, at each chosen junction. Remember, write + terms for currents going IN, and neg terms for currents going OUT.

F) Apply the Kirchoff loop rule (voltage rule) to each chosen loop.

Remember, assign the + and – signs this way:

• If we traverse (mentally walk through) a resistor in the direction of the drawn current i, then the change in potential is – iR.
  

If we walk opposite to the current direction, the change is + iR.

• If we traverse (walk through) an emf from its negative pole to positive pole, the change in potential is + ε. If traversed in the opposite direction, the change is – ε .

By the end of steps E), F) you should have as many independent equations as there are independent unknowns. For example, in a two-loop circuit, we can get three independent equations, and thus can solve for 3 unknowns.

G) Substitute known values or facts, and solve the equations.

H) Interpret and state the final answers. In particular, if a current value comes out negative, it means that the current really flows opposite to the way you drew it. Do NOT change your diagram now.
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How to solve the System of Independent Equations

IT Tools

Gauss-Jordan Elimination Calculator

https://matrix.reshish.com/gauss-jordanElimination.php

C++

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