To analyze circuits using the Mesh Analysis, all sources in the circuit must be Voltage Sources
(If one of the sources is a current source, we first have to convert it to a voltage source - if possible - and then use the Mesh Analysis.)
- We use the Mesh Analysis to find the values of currents.
- When using the Mesh Analysis, we always try to reduce the number of loops so that we simplify the process of analyzing (one of these is to convert the current source to a voltage source)
To analyze circuits using the Mesh Analysis, we follow the steps below:-
Example:- Find the value of the current through the 10Ω resistance using the Mesh Analysis?
Step 1: We identify a number of closed and independent current paths (loops).
Step 2: We make an assumption about the directions of the currents, either clockwise or counterclockwise (and preferably clockwise).
Step 3: We determine the polarity of resistors based on the directions of currents in paths (loops).
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| Steps 1, 2, 3 |
Step 4: We then form the equations of currents. ( The number of equations is equal to the number of paths).
In this example, we have two paths so we are creating two equations
We take the first path through which the first current passes:-
We write the equation for the first path as follows:
1. The first-current signal is positive and the rest of the currents' signals are negative (that’s because the first equation is for the first path through which the first current passes).
2. To find the first-current factor, we add all the resistors in the first path. The second-current factor is the resistance common between the first and second paths. If there is no common resistance between the first and second paths, the second-current factor is then zero.
3. After the equal sign (=), we write the voltage generated in the first path. In the first path, we have two voltage sources that are opposite in direction. Therefore; we subtract the smaller voltage from the larger voltage (if they are in the same direction we are bringing them together). Then, we look at the direction of the larger voltage source, If it is opposite in direction with the first-path current (which we have previously assumed its direction in step 2), the voltage is then negative. If the direction of the larger voltage source is in the same direction as the first- path current, the voltage is then positive.
4. If possible, we simplify equations (by dividing the two sides of the equation by the same number). In this example we will divide the two sides of the equation by 2
We take the second path through which the second current passes:-
We write the equation for the second path as follows:
1. The second-current signal is positive and the rest of the currents' signals are negative (that’s because the second equation is for the second path through which the second-current passes).
2. To find the second-current factor, we add all the resistors in the second path. The first-current factor is the resistance common between the first and second paths, If there is no common resistance between the first and second paths, the first-current factor is then zero.
3. After the equal sign (=), we write the voltage generated in the second path. In the second path, we have two voltage sources that have the same direction. Therefore; we bring them together. We then look at the direction of the total voltage source if it is opposite to the direction of the second-path current (which we have previously assumed its direction in step 2), the voltage is then negative. If the direction of the total voltage source is in the same direction as the second-path current, the voltage is then positive.
4. If possible, we simplify equations (by dividing the two sides of the equation by the same number). In this case we will divide the two sides of the equation by 5
Step 5: We solve these equations using one of the solution methods (Deletion and Compensation – Saros – Determinants).
In this example, I will use the determinants (You could use the Deletion and Compensation method)
Notes:-
· If the resulting current value is positive, our assumption of the current direction in step 2 was correct, but if the resulting current value is negative, the correct direction of the current is the opposite of the assumed direction.
· In this example, the first and second currents both pass through the resistance 10Ω (that’s because it is a common resistance between the first and second paths) Therefore:
§ If the first and second currents have the same direction, the current passing through the resistance 10Ω equals the sum of the two currents and its direction is the same direction of the first and second currents.
§ If the first and second currents are opposite in direction, the current passing through the resistance 10Ω equals the difference between the two currents and it has the direction of the larger current.
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