## Introduction: Analyzing Simple Circuits

Before you get started on analyzing a simple circuit, there are few thing you must know. Analyzing simple circuits will require a few things from you:

· Basic algebra skills (You may also want a calculator handy in order to do some calculations quickly).

· Basic knowledge of how circuits work.

· An empty sheet of paper.

· A pencil, or a pen if no pencil is available, although a pencil is preferable in case an error is made.

· About 15-60 minutes depending upon the problem and your experience with circuits.

In addition to these four things there are also a few concepts you need to understand. This Instructable is not meant for beginners, you must understand that there are physical and electromagnetic laws that circuits revolve around. The exact calculus and physics behind each of the laws will not be explained, but you should be able to understand their basic concepts. There are only three laws that you will need to know in order to work with circuits:

· Ohm’s law

· Kirchhoff’s Current Law (KCL)

· Kirchhoff’s Voltage Law (KVL)

The most important law in analyzing circuits is Ohm’s Law. Ohm's Law states the voltage through an element is equal to that of the element’s current times the current flowing through the element, or Voltage=Resistance*Current (V=IR). Voltage (V) is the amount of electric potential measured in volts (symbol V). Resistance (R) is the difficulty it takes to pass an electric current through an element measured in ohms (symbol Ω). Current (i) is the flow of electron charge measured in ampere (symbol A). Ohms law is the bread and butter, and will be where most of the calculations take place during analysis.

Before Kirchhoff’s Laws are explained, there a few definitions you must first understand. There are figures at the top of this page which are numbered to correspond with each of the following definitions, they should help you fully understand each definition and in some cases its respective symbol:

1.) Power Source: an element that gives power to the circuit, usually a battery; if the current flows to the negative side, the power source is positive. This is not the only way to draw a battery.

2.) Resistor: the most common element, and only type you will be dealing with.

3.) Node: a junction were two or more circuit elements are connected together giving a connection point between two or more branches. A node is indicated by a dot.

4.) Branch: a single or group of components which are connected between two nodes.

5.) Loop: a simple closed path in which no circuit element or node is encountered more than once.

6.) Series: elements are said to be in “series”, if two or more elements are connected to one other without any branches.

7.) Parallel: elements are said to be in “Parallel”, if two or more elements are connected to one other with branches.

8.) Wire: a line in which no elements is connected to. Wires are important since voltages do not change within a wire, therefore two nodes connected with a wire will have the same voltage.

9.) Ground: the point on the circuit where voltage is set to 0. If the “ground” location is not specified, it is assumed to be at any node at the bottom of the circuit where there are no elements directly in-between the negative side of the power source and “ground” i.e. there are only wires in-between the negative side of power and “ground”. You can draw as many grounds as you would like as long as they are only connected with wires.

10.) Voltage drop: the voltage in-between two nodes is equal to the voltage at the second node minus the voltage at the first node. If the second node is connected to ground, then the “voltage drop” along that wire is equal to the voltage at the first node, since Vn - 0 = Vn where the 0 represents the 0 volts at ground.

11.) Positive Orientation: when writing the voltage drop across an element, write + _ V -, such that the + is on the side that hits the current arrow. This will make more sense in context of a problem.

Kirchhoff’s Current Law states that all current flowing into a node is equal to zero, i1 + i2 + i3 + … + in = 0. Kirchhoff’s Voltage Law states that all voltages added up around a loop must add up to zero, V1 + V2 + V3 + … + Vn = 0; you will use this to calculate voltages at nodes. Images for KCL and KVL are labeled in pencil with their respective name.

The last thing you need to know before you get started is the idea of an “Equivalent resistance”. Equivalent resistance is the idea that a circuit with multiple resistors can be simplified down to a circuit with only one resistor. It is used to find the starting current flowing out of the power source. This is done by two equations, firstly has to do with resistors in series. In order to add up any number of resistors in series you just add them up normally. For example, resistors R1, R2, and R3 in series are equal to a single resistor R4, such that R4= R1+ R2+ R3. In order to add up any number of resistors in parallel you add up one over the sum of the inverses. For example, resistors R1, R2, and R3 in parallel are equal to a single resistor R4, such that R4 = 1/(1/R1+1/R2+1/R3). Each of these two equations can be used together for circuits that are not only in series or parallel. For example, resistors R1 and R2 in parallel while also in series with R3 are equal to a single resistor R4 so that R4=(1/(1/R1+1/R2))=R3. Image for equivalent resistance is at the top labeled Eq. R.

Now that all the definitions and laws have been explained it is time for you to get started on how to analyze circuits. An example will be worked out using the following steps, at the start of each major step. The Sample problem its self is the last image at the top. Read through all of these steps on how to analyze a circuit before doing any problems on your own. Sometimes learning how do to something is best taught through examples.

## Step 1: Read the Problem and Draw the Circuit.

- The hardest part of any circuit is correctly understanding what the question is asking.
- Underline key words in the problem that were mentioned in the introduction.
Once you have read through and underlined key words. Try to draw the circuit, try to draw fairly largely as you will come back to this circuit and add values to it when you calculate them.

If you have trouble drawing any part of the circuit refer to the images in the introduction.

Note sometimes the circuit will be given to you, in these cases redrawing the circuit is optional. When you calculate values you can just write on the given drawing.

## Step 2: Redraw the "Equivalent Resistance" Circuit

- When doing equivalent resistance, start from the right and work inwards towards the power source First you should look for any resistors in series, then for any in parallel.
- It is critical that you do not jump around in the circuit. Instead you should move smoothly from right to left, inwards towards the power source, or you will not be able to calculate the correct circuit.
- This step should not be redrawn once, but instead you should redraw the circuit for each series or parallel groups you find. (For example the sample problem worked above requires to be redrawn twice. Once for the 5 ohm and 25 ohm resistors in series, then again for the now 30 ohm resistor and the other 30 ohm resistor in parallel).

## Step 3: Calculate the Current That Comes Out of the Power Source.

- Now that you have your equivalent circuit with only one power source, one resistor, and ground. This means that all of the voltage from the power source must be used in the resistor.
- By using Ohm’s Law, V = i*R, i1 can be calculated where V is the power voltage and R is the equivalent resistance. (For example in the sample problem worked above V is 15V and R is 15 ohms, so i0 is equal to 1A).

## Step 4: Add the New Value to the Original Circuit.

- Now that you have i0, on your original circuit write an arrow flowing out of the positive end of the power source, and also write the value you just calculated.

## Step 5: Calculate Up to the First Node.

- Now that you have a current along a wire, you can calculate any voltage for any element (before the first node you encounter).
- Use Ohm’s law, V = i*R, to calculate voltage across any elements up to the first node.
- Using KVL, add up voltages to the first node. Take the power voltage and subtract all of the element voltages up to the first node, the difference is the voltage at the first node.

## Step 6: Add the New Value to the Original Circuit.

- Write the voltage you just calculated above the first node on the original circuit you drew.
- If there were no elements in-between the power source and the first node, write the power voltage above the first node. (This is because Vsource – 0 = Vsource, the 0 represents 0V across elements, sense there are no elements.)

## Step 7: Find the Current Going From the Node to Ground.

- Check to ensure there is a connection from node to ground, i.e. make sure that there is a node that directly connects to ground (for example, in the sample problem the only node has a wire connecting the 15V to the ground symbol; the wire has a 30 ohm resistor but that does not matter).
- Due to voltage drop the voltage from the node to ground is equal to the voltage at the node.
- Using Ohm’s law, V = i*R, calculate the current flowing through the wire where V is the voltage at the node, and R is the resistance in the wire (which can either be a series of elements, in which case you add up the resistances, or one single element).

## Step 8: Add the New Value to the Original Circuit.

- Now that you have this new current i, write an arrow on your original circuit flowing from the node towards ground, also write the calculated value.

## Step 9: Find the Current Leaving the Node.

- By using KCL, current entering a node is equal to current leaving the node, you can calculate current leaving the node. (For example in the sample problem worked above i0 = i1 + i2, where therefore i2 is equal to 0.5A).

## Step 10: Deciding Your Next Step.

- While looking at the node you just solved put your finger on this node and move it right along the wire.
- If your finger follows the wire and gets to ground without hitting any other nodes, regardless of how many elements it hits, skip to step 12. Otherwise, i.e. your finger hit another node before ground, continue onto step 11. (There are no other nodes in the sample problem, so it will skip to step 12).

## Step 11: Calculate the Voltage Drop In-between the Two Nodes.

- Using the current you found in step 9, you can calculate any voltage for any element (before the next node you encounter).
- Use Ohm’s law, V = i*R, to calculate voltage across any element in-between the two nodes.
- Using KVL, take the voltage at the original node and subtract all of the element voltages up to the next node, the difference calculated is the voltage at the next node. Write this value above the next node.
- If there are no elements between the two nodes, write the voltage of the original node above the next node.
- Repeat back to step 7.

## Step 12: Finish the Circuit.

- You now know each current going through each wire; for any elements you do not have the voltage across calculate the voltage using Ohm’s Law, V=i*R.
- Write the calculated voltages next to their respective elements, using positive orientation.

## Step 13: Answer the Problem.

- Congratulations, you have now completely analyzed the entire circuit.
- Look back to the original question, and rewrite the solution to what the problem asked for and box your answers.