1 Answer
Millman's Theorem is a useful tool in circuit analysis, especially when dealing with multiple voltage sources in parallel with resistors. It's a way to simplify the calculation of the voltage across a load resistor when there are several voltage sources influencing it.
Millman’s Theorem Formula:
The general formula for Millman’s Theorem is:
\[
V_{o} = \frac{\sum \left(\frac{V_i}{R_i}\right)}{\sum \left(\frac{1}{R_i}\right)}
\]
Where:
Steps to Use Millman’s Theorem:
\[
V_o = \frac{\sum \left(\frac{V_i}{R_i}\right)}{\sum \left(\frac{1}{R_i}\right)}
\]
- For the numerator: Add the terms \(\frac{V_i}{R_i}\) for each voltage source.
- For the denominator: Add the reciprocals of the resistances \(\frac{1}{R_i}\).
Example:
Let’s say you have the following setup:
Using Millman’s Theorem, the output voltage \(V_o\) across the load will be:
\[
V_o = \frac{\left(\frac{12}{4} + \frac{6}{6}\right)}{\left(\frac{1}{4} + \frac{1}{6}\right)}
\]
Simplify:
\[
V_o = \frac{(3 + 1)}{\left(0.25 + 0.1667\right)}
\]
\[
V_o = \frac{4}{0.4167} \approx 9.6V
\]
So, the output voltage across the load is approximately 9.6V.
When to Use Millman’s Theorem:
Remember, Millman’s Theorem only works when the voltage sources are in parallel, and the resistances are all in parallel as well. If there are other configurations, you may need to use other techniques.
Millman’s Theorem Formula:
The general formula for Millman’s Theorem is:
\[
V_{o} = \frac{\sum \left(\frac{V_i}{R_i}\right)}{\sum \left(\frac{1}{R_i}\right)}
\]
Where:
- \(V_o\) is the output voltage across the load.
- \(V_i\) is the voltage of each independent voltage source.
- \(R_i\) is the resistance connected in parallel with each voltage source.
Steps to Use Millman’s Theorem:
- Identify all Voltage Sources and Resistors:
- Apply the Formula:
\[
V_o = \frac{\sum \left(\frac{V_i}{R_i}\right)}{\sum \left(\frac{1}{R_i}\right)}
\]
- For the numerator: Add the terms \(\frac{V_i}{R_i}\) for each voltage source.
- For the denominator: Add the reciprocals of the resistances \(\frac{1}{R_i}\).
- Calculate the Result:
Example:
Let’s say you have the following setup:
- Voltage Source 1: \(V_1 = 12V\), with resistance \(R_1 = 4 \, \Omega\)
- Voltage Source 2: \(V_2 = 6V\), with resistance \(R_2 = 6 \, \Omega\)
Using Millman’s Theorem, the output voltage \(V_o\) across the load will be:
\[
V_o = \frac{\left(\frac{12}{4} + \frac{6}{6}\right)}{\left(\frac{1}{4} + \frac{1}{6}\right)}
\]
Simplify:
\[
V_o = \frac{(3 + 1)}{\left(0.25 + 0.1667\right)}
\]
\[
V_o = \frac{4}{0.4167} \approx 9.6V
\]
So, the output voltage across the load is approximately 9.6V.
When to Use Millman’s Theorem:
- Millman’s Theorem is helpful when you have multiple voltage sources in parallel with different resistances.
- It's commonly used in power systems and electrical circuits with multiple voltage sources influencing the same load.
Remember, Millman’s Theorem only works when the voltage sources are in parallel, and the resistances are all in parallel as well. If there are other configurations, you may need to use other techniques.