1 Answer
To find the efficiency in the Maximum Power Transfer Theorem, we first need to understand the basics of the theorem and the concept of efficiency.
Maximum Power Transfer Theorem:
The Maximum Power Transfer Theorem states that maximum power is delivered to the load when the load resistance (R_L) is equal to the source resistance (R_s), i.e., when:
\[
R_L = R_s
\]
Efficiency Calculation:
Efficiency (\(\eta\)) is the ratio of the useful power delivered to the load to the total power supplied by the source.
\[
\eta = \frac{P_{\text{load}}}{P_{\text{total}}} \times 100
\]
Where:
1. Power Supplied by the Source (P_total):
The total power supplied by the source, when the load is connected, can be calculated using the formula:
\[
P_{\text{total}} = \frac{V_{\text{source}}^2}{4 R_s}
\]
Where \(V_{\text{source}}\) is the source voltage and \(R_s\) is the source resistance.
2. Power Delivered to the Load (P_load):
The power delivered to the load when \(R_L = R_s\) is:
\[
P_{\text{load}} = \frac{V_{\text{source}}^2}{4 R_L}
\]
Since at maximum power transfer \(R_L = R_s\), we substitute \(R_L\) with \(R_s\).
3. Efficiency Formula:
Now, substitute the values of \(P_{\text{load}}\) and \(P_{\text{total}}\) into the efficiency formula:
\[
\eta = \frac{\frac{V_{\text{source}}^2}{4 R_s}}{\frac{V_{\text{source}}^2}{4 R_s}} \times 100
\]
This simplifies to:
\[
\eta = \frac{1}{2} \times 100 = 50\%
\]
Conclusion:
In the case of the Maximum Power Transfer Theorem, the efficiency is 50% because half of the power is lost in the source resistance, and the other half is delivered to the load.
Maximum Power Transfer Theorem:
The Maximum Power Transfer Theorem states that maximum power is delivered to the load when the load resistance (R_L) is equal to the source resistance (R_s), i.e., when:\[
R_L = R_s
\]
Efficiency Calculation:
Efficiency (\(\eta\)) is the ratio of the useful power delivered to the load to the total power supplied by the source.\[
\eta = \frac{P_{\text{load}}}{P_{\text{total}}} \times 100
\]
Where:
- \(P_{\text{load}}\) is the power delivered to the load.
- \(P_{\text{total}}\) is the total power supplied by the source.
1. Power Supplied by the Source (P_total):
The total power supplied by the source, when the load is connected, can be calculated using the formula:\[
P_{\text{total}} = \frac{V_{\text{source}}^2}{4 R_s}
\]
Where \(V_{\text{source}}\) is the source voltage and \(R_s\) is the source resistance.
2. Power Delivered to the Load (P_load):
The power delivered to the load when \(R_L = R_s\) is:\[
P_{\text{load}} = \frac{V_{\text{source}}^2}{4 R_L}
\]
Since at maximum power transfer \(R_L = R_s\), we substitute \(R_L\) with \(R_s\).
3. Efficiency Formula:
Now, substitute the values of \(P_{\text{load}}\) and \(P_{\text{total}}\) into the efficiency formula:\[
\eta = \frac{\frac{V_{\text{source}}^2}{4 R_s}}{\frac{V_{\text{source}}^2}{4 R_s}} \times 100
\]
This simplifies to:
\[
\eta = \frac{1}{2} \times 100 = 50\%
\]