Mesh analysis in a resistor network
Mesh current method in a resistor network.
The method of mesh analysis in a resistor network is very useful to know the currents in a resistor network (circuit with only resistors). This method, a little more extended, is also applied to circuits where there are resistances and reactances.
Using this method, the resistors current values are calculated and their voltages and powers are also obtained. This method is based on Kirchhoff’s voltage Law.
Mesh current analysis or the mesh current method is a technique used to find the currents circulating around a mesh with in any closed path of a circuit.
Steps to follow for the analysis by the mesh current method:
- Draw the circuit to be analyzed so that there is no one connection crossing over another. (if possible)
- Convert current sources into voltage sources.
- Draw the direction of the currents in the sense of the needles of the clock. The currents are called I1, I2, I3, … .etc. See example at the end.
- Make a table with the equations obtained from the circuit (Kirchhoff’s law of voltages). The number of rows in the table is the same as the number of currents established in step 3.
- Columns with unknown variables are placed on the left side of the equal sign and the column with the constant values on the right side.
Example of a mesh analysis in a resistor network
For the I1 mesh: V1 + V2 + V3 – 14V = 0 (1)
For the I2 mesh: -V2 + 14V + V4 – 12V = 0 (2)
For the I3 mesh: -V3 -12V + V5 = 0 (3)
Voltages in each resistor are:
V1 = I1 x R1
V2 = (I1-I2) x R2
V3 = (I1-I3) x R3
V4 = I2 x R4
V5 = I3 x R5
We replace the previous voltage values in formulas (1), (2) and (3), and we obtain:
For the I1 mesh: I1 x R1 + (I1-I2) x R2 + (I1-I3) x R3 – 14V = 0
For the I2 mesh: – (I1-I2) x R2 + 14V + I2 x R4 – 12V = 0
For the I3 mesh: – (I1-I3) x ??R3 -12V + I3 x R5 = 0
We use the following values of the resistors: R1 = 10 ohms, R2 = 6 ohms, R3 = 4 ohms, R4 = 8 ohms, R5 = 2 ohms.
For the I1 mesh: 10I1 + 6 (I1-I2) + 4 (I1-I3) – 14V = 0
For the I2 mesh: -6 (I1-I2) + 14V + 8I2 – 12V = 0
For the I3 mesh: -4 (I1-I3) – 12V + 2I3 = 0
Then we get:
For the I1 mesh: (10 + 6 + 4) I1 – 6I2 – 4I3 -14V = 0 …. 20I1 – 6I2 – 4I3 = 14
For the I2 mesh: 6I2 + 8I2 – 6I1 + 14V – 12V = 0 …. 6I2 + 8I2 – 6I1 = -2
For the I3 mesh: 4I3 + 2I3 – 4I1 = -12
The final equations are: (as indicated in step 4)
20I1 – 6I2 – 4I3 = 14
– 6I1 + 14I2 + 0I3 = -2
– 4I1 + 0I2 + 6I3 = -12
Since there are three unknowns values (I1, I2, I3), there are three rows in the table. Using the substitution method or the determinants method, the following values are obtained:
I1 = 0.348 amps
I2 = 0.006285 amps
I3 = -1.768 amps. The minus sign indicates that the direction of current I3 is opposite to that assumed at the beginning.