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Mesh Circuit Analysis

Recently I had a reader comment with a request for an example of Kirchoff’s Voltage Law (KVL) with Ohm’s Law. KVL and Ohm’s Law are both used in the circuit analysis methodology called Mesh Circuit Analysis. Consequently I thought I would analyze a small circuit and demonstrate the general principles of this technique that can be used on any circuit.

The circuit we will use below has a 10 volt source and 4 resistors configured as seen.

 

STEP 1:

 

The first step is to label the circuit as I have done in red ink. This includes the current flowing clockwise through each loop (or mesh) and the voltage drops across each passive element. (In this case all the passive elements are resistors.) Next we will use the picture to write some equations.

 

STEP 2:

If you look at the diagram at the first loop and start at the top, then travel around the loop going clockwise and add up the voltage drops you get the first equation below. Notice the voltage drop across the source is negative because it is not a drop it is an increase.Also notice the sum must equal zero.

The next 2 lines are simply writing the unknown voltage drops in terms of Ohm’s Law. The voltage drop across the resistor is equal to the current through the resistor times the resistance. Notice for resistor 2 the currents are opposing one another. In other words current 1 runs from the top of the circuit toward the bottom of the circuit through resistor 2. Current 2 runs from the bottom to the top. That is why the current component in the second Ohm’s Law equation is i1 minus i2.

Next these two Ohm’s Law equations are substituted in the original equation which gives the 4th line below.

From the 4th line the rest is just algebra. Then do the exact same process for loop 2 as seen above. This gives 2 equations with 2 unknowns. Of course these can be solved by any linear algebra method you prefer.

STEP 3:

Below I solved it using simple substitution. This solves for the unknown currents.

 

STEP 4:

Finally, substitute these found currents back into the Ohm’s Law equations we developed earlier and you find the unknown voltage drops. The last diagram shows the final solution.

 

I hope this answers my reader’s question and has been helpful. Please let me know if anyone needs more help.

The Principle of Voltage Division

The principle of voltage division is derived from Ohm’s Law and from Kirchoff’s Voltage Law (KVL). KVL says that the sum of the voltage drops around a closed loop is equal to zero. Consider the following circuit.

If we apply KVL we have:

From Ohm’s Law we can say:

The current is the same through each resistor since this is a series circuit. If we combine these expressions we get:

If we solve each expression of Ohm’s Law above for the current (i) and substitute that into the last equation, we can show how the total voltage from the voltage source divides across each resistor in proportion to the magnitude of the resistance.

Likewise for resistors 2 and 3 we have:

Finally, we can summarize this in a general expression where VRn is voltage across any general resistor in the series circuit, and Vs is the magnitude of the voltage source.

Ohm’s Law Verified

Our second electical circuits lab was an exercise in verifying Ohm’s Law. To start, let’s look at what Ohm’s Law is and the theory behind it.

All materials have a general property of resisting the flow of electricity through themselves. This material property is quantitatively called the resistivity. (Interesting enough, resistivity is the reciprocal of conductivity.) The more difficult it is for electricity to flow through a material, the higher the resistivity value. Resistivity is purely a material property and does not depend on any particular configuration or geometry of the material. It can however be affected by temperature changes. The resistance, on the other hand, is a function of this material property, but also of the material shape. This relationship is given by:

Where R is the resistance, ρ is the resistivity, L is the length of the portion of material, and A is the uniform cross-sectional area.

The circuit element that we utilize as a model for resistance is called the resistor. The German physicist Georg Ohm investigated the relationship between the voltage across a resistor and the current through a resistor. Through this study he developed the following relationship known today as Ohm’s Law:

In order to verify this relationship for ourselves, we experimentally varied one element of the voltage, current, and resistance, kept one constant, and took readings and recorded values for the third. Then we determined the error between these recorded values and the expected values given by Ohm’s Law. Below are the tables of calculated values, measured values, and percent error for 3 different configurations.

The equipment we utilized and had set up in the first configuration consisted of a power supply (DC), a potentiometer, and the digital multimeter. The apparatus was configured as follows:

Here is the procedure for the first configuration.

  1. Ensure the power supply is in the off position.
  2. For part A we kept the voltage constant at 10 volts. Therefore, the first step is to connect the power supply to the digital multimeter with 2 leads, set the multimeter to read DC voltage, turn the power supply on, and adjust the voltage on the power supply until the multimeter reads 10 volts.
  3. Turn the power supply back off.
  4. Next connect the lead (we used red) from the positive terminal of the power supply to positive or red terminal of the potentiometer.
  5. Using a lead (we used black), connect the negative or black terminal of the potentiometer to the lower left black terminal of the digital multimeter.
  6. Using a lead (we used two leads in series for length and end plugs) Connect the lower right red terminal (for reading current) of the digital multimeter to the negative or black terminal of the power supply. At this point the circuit is complete.
  7. Set the digital multimeter to read milliamperes of current.
  8. Turn the power supply on.
  9. Adjust the switches on the potentiometer to obtain the required value for the current displayed on the multimeter. (See the data tables for more. The values were 10,9,8,7… and so on.)
  10. Add up and record the resistance supplied by the potentiometer that generated the corresponding current value.
  11. Compare this resistance value to the resistance value predicted by Ohm’s Law. Computer the percent error for each measurement.
  12. Repeat steps 9 through 11 for each required current value.

Here is a graph of our results for the first configuration. It shows the calculated values using Ohm’s Law, the measured values, and a predictive trendline with equation using the calculated values.

This first configuration gives you the idea. I have the procedure and graphs for the other implementations if anyone is interested. I have omitted them here for brevity.

Read on. You can’t resist.

This semester I have the pleasure of engaging in a little electrical engineering in my circuits lab. The first area we touched on in here and the first lab revolved around resistors. (Hence the corny title.) Below I have constructed a diagram of the typical two-terminal resistors that we are using and the typical 4 band color coding scheme that they employ.

 

So take as an example the above resistor diagram. The first step in determining the resistance provided by this particular resistor is to place it in a horizontal orientation as seen above with the metallic-colored band to the right. (In the example, the band is a metallic gold color.) Next you read the bands from left to right like reading this text.

The first band in our example is brown. Using the table, we see brown corresponds to the first digit of the resistance, 1. The second band is black. By the table we see the second digit in the resistance is a zero. The third band which is red tells us what the multiplier is. In this case the multiplier is 100. So with this information we can now calculate the resistance. We multiply the first two digits, 10, times the multiplier, 100, and obtain the resistance in ohms, 1000. Usually this is called 1 kilohm. The final band tells us the tolerance of the resistor. In our example the final band is gold, so the tolerance is +/- 5%. This means that the resistor could have a value from 950 ohms to 1050 ohms.

Besides learning how to determine the resistance we used a digital multimeter to measure the actual value and verify it falls within the range of the tolerance.

Any questions? More to come soon.