Learn Kirchoff's Rules: Solve Circuit With Series and Parallel Combinations

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In summary, Kirchoff's rules are a set of guidelines for analyzing complex electrical circuits. They involve identifying series or parallel combinations, defining current variables and choosing a positive sense for each variable. Symmetry can be used to simplify the problem and reduce the number of independent variables. Junction equations are written at points where branches meet, while loop equations are written by following a path along the circuit and setting the potential change to zero. These rules can help solve for unknowns and are useful in analyzing circuits with multiple components.
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Kirchoff's Rules

I can not find a good explanation anywhere for Kirchoff's rules. Googling onlygives me links that don't explain it well, and my textbook's explanation is lacking as well. I'll go over my textbook's description, so hopefully someone here can clarify this for me.

1. Identify any series or parallel combinations and find the simplest equivalent circuit. (easy to do)

2. Define a current variable and choose a positive sense for each variable. The direction you label need not be the actual direction of the current.
WHY? If I get the direction wrong, I'll be adding where I should be subtracting, and that'll give me the wrong answer. Every link says this too, but none of them explain it.

3. Use symmetry if possible to reduce the number of independent variables.
What does this mean? What is an independent variable anyway? What would be an example of a dependent variable?

4. Write junction equations until each current appears in at least one equation.
What is a junction equation? I hate my textbook for using terms it doesn't define. Do they mean I1+I2=I3?

5. Write loop equations until each arm of the circuit occurs in at least one of the loops.
What is a loop equation? Do they mean I=V/R

6. Solve for unknowns
Unknown is how to use Kirchoff's rules

So, for example, how would I do this problem:
kir.GIF

 
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  • #2
Hi tony873004,

#2. This is one of the strengths of the Kirchoff approach: you don't need to spend effore ahead of time trying to figure out which way the currents are really going. For example, in the circuit in your diagram we could label the current in all four branches as going upwards. (Of course it's not possible for the currents to really all flow upwards, but that does not matter for the labeling.) Then when we solve the simulataneous equations for the currents, if we get a positive answer then the real current is in the same direction as our labeled current, and a negative answer means the real current is in the opposite direction as our label.

So when I label the current in the leftmost branch as i1 and going upward, after I solve the problem I get i1=(-1/3), which means the real current is flowing downwards (as you might expect).

3. If we change your circuit slightly you can show one way that symmetry can simplify your problem. When we label the currents, we need a current i1 for the leftmost branch (containing R1) and a current i4 for the rightmost branch (containing R3). In your circuit these currents will be different.

However, if R1 had been equal to R3, then you would know that their currents would be the same (since they have the same resistance and are connected to the same points). You could then say that i4=i1 and immediately remove one of the variables so you would have one less to solve for.

4. Yes; you write a junction equation at places where branches come together. Since charge cannot build up at a junction you need

(sum of currents coming into junction)=(sum of currents going out of junction)

where 'coming in' or 'going out' refers to their labelled directions.

5. For the loop equation, you follow a path along the circuit so that you begin and end at the same point, writing down the potential change as you follow the path. Since your final point is the same as your beginning point, then the change in potential must be zero.

So if you start at the bottom of the leftmost loop and go clockwise around that loop, you have:

(change in potential across R1) + (change in potential across V1) = V_f - V_i =0


The potential difference across R1 is i1*R1, and is negative if the direction that you are following the loop is the same direction as the labeled current. The potential difference acroos V1 is given, and is negative if the direction that you are following the loop takes you from the positive terminal to the negative terminal of the battery.

So if i1 is labelled upwards, and we follow the leftmost loop clockwise, we get

- i1*R1 - V1 =0

for the loop equation.
 
  • #3


In a series-parallel circuit, there are three resistors connected in series and two resistors connected in parallel. The total resistance of the circuit is 20 ohms.

To solve this problem using Kirchoff's rules, we would first identify the series and parallel combinations. In this case, we have three resistors connected in series (R1, R2, and R3) and two resistors connected in parallel (R4 and R5). We can simplify the circuit by finding the equivalent resistance of the series combination (R1+R2+R3) and the parallel combination (1/R4+1/R5). Let's say the equivalent resistance for the series combination is Rs and the parallel combination is Rp.

Next, we would define a current variable (let's call it I) and choose a positive sense for each variable. The direction we label does not have to be the actual direction of the current, but it should be consistent throughout the circuit. For example, we could choose the positive direction to be from the positive terminal of the battery to the negative terminal.

Using symmetry, we can reduce the number of independent variables. In this case, we can see that the current through Rs and R4 must be the same, and the current through R5 and R3 must also be the same. This means we only need to solve for two independent variables: the current through Rs and the current through Rp.

A junction equation is an equation that represents the conservation of charge at a junction point in the circuit. In this case, we can write the junction equation as I = I1 + I2, where I is the current flowing into the junction and I1 and I2 are the currents flowing out of the junction.

A loop equation is an equation that represents the conservation of energy around a closed loop in the circuit. In this case, we can write the loop equation as V = IRs + IRp, where V is the voltage of the battery and Rs and Rp are the equivalent resistances we found earlier.

Finally, we can solve for the unknowns (the currents through Rs and Rp) by using the junction and loop equations we wrote and solving for the variables.

In summary, Kirchoff's rules provide a systematic approach to solving complex circuits by breaking them down into simpler components and using conservation laws to solve for the unknown variables. It takes practice and understanding of the concepts to effectively use Kirchoff's rules, but with
 

What are Kirchoff's Rules?

Kirchoff's Rules, also known as Kirchoff's Laws, are two fundamental principles in electrical circuit analysis that are used to determine the voltage and current in a circuit. These rules are based on the principle of conservation of energy and conservation of charge.

What is a series combination in a circuit?

A series combination in a circuit is when two or more components, such as resistors, are connected in a single path. In this configuration, the same current flows through each component, and the total resistance is equal to the sum of the individual resistances.

What is a parallel combination in a circuit?

A parallel combination in a circuit is when two or more components are connected in such a way that the current is divided among them. In this configuration, the voltage across each component is the same, and the total resistance is less than the smallest individual resistance.

How do you solve a circuit with series and parallel combinations using Kirchoff's Rules?

The first step is to identify the series and parallel combinations in the circuit. Then, use Kirchoff's Rules to write equations for conservation of energy and conservation of charge for each loop and junction in the circuit. Finally, solve the resulting system of equations to find the voltage and current in each component.

What are some tips for solving circuits with series and parallel combinations?

Some tips for solving circuits with series and parallel combinations are to draw a clear diagram of the circuit, label all components and their values, and use a systematic approach to applying Kirchoff's Rules. It can also be helpful to simplify the circuit by combining series or parallel components before applying the rules.

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