Whether we can apply the Maximum power transfer theorem to the AC circuits?

Yes. We can apply the maximum power transfer theorem to the AC circuits just like the way we applied the Maximum power transfer theorem to DC circuits. If the circuit consists of multiple AC sources or the combination of DC and AC sources and multiple impedances, then we can use this theorem to get the maximum power across the load impedance.

What are the limitations of the Maximum power transfer theorem?

We can’t apply the Maximum power transfer theorem for non-linear and / or unilateral networks / circuits. The limitations of the Maximum power transfer theorem are the same as that of Thevenin’s theorem. We can’t use this theorem in the circuits, where we want the efficiency to be more than 50 %.

Whether Maximum power transfer theorem can be used for maximum efficiency?

No. As the name implies, we can use the maximum power transfer theorem only for getting the maximum power across the variable load resistance. By using this theorem, we can get the efficiency is at most 50%. It’s not possible to get the maximum efficiency by using this theorem.

Whether Maximum power transfer theorem can be applied to the fixed load resistance?

No. If the load resistance is not variable, then we can’t apply the Maximum power transfer theorem for finding the maximum power across that fixed load resistance. That time, we must choose the minimum Thevenin’s equivalent resistance. So that maximum current will flow through the fixed load resistance. Hence, maximum power will get across the fixed load resistance.

What are Maximum power transfer theorem applications?

We can use the Maximum power transfer theorem in all the applications, where we want maximum power across the load resistance. To get the maximum power across the load resistance, we must properly choose the value of load resistance in such a way that it will match with the Thevenin’s resistance or the internal resistance of the source.

Maximum Power Transfer Theorem

By BYJU'S Exam Prep

Updated on: September 25th, 2023

The maximum power transfer theorem is one of the important Network theorems. The theorem we use for solving the given electrical network/circuit is known as Network Theorem/Circuit theorem. The maximum power transfer theorem formula will help find the maximum power across the load resistance.

We can use the Maximum power transfer theorem whenever the load resistance is variable. In this article, get an overview of this theorem. Here, first will state the maximum power transfer theorem, then the derivation and important formulae.

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Table of content

1. What is the Maximum Power Transfer Theorem?
2. Maximum Power Transfer Theorem Formula
3. Derivation of Maximum Power Transfer Theorem
4. Efficiency of Maximum Power Transfer Theorem
5. Maximum Power Transfer Theorem Problems

What is the Maximum Power Transfer Theorem?

Using the Maximum power transfer theorem, we will know in what condition we can get the maximum power across the variable load resistance. This theorem is just a continuation of Thevenin’s theorem. Thevenin’s equivalent circuit consists of a Thevenin’s voltage source, V_Th, in series with the Thevenin’s resistance, R_Th.

Statement of Maximum Power Transfer Theorem

The statement of maximum power transfer theorem is that the value of load resistance, R_L, must and should be equal to the value of Thevenin’s resistance, R_Th, to get the maximum power across the load resistance, R_L. This is the statement of the maximum power transfer theorem. We can use this theorem for linear and Bilateral networks/circuits.

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Maximum Power Transfer Theorem Formula

We have now discussed the derivation of the maximum power transfer theorem and the efficiency by using this theorem. All the important formulae we used and/or derived regarding this theorem are listed below.

The current, I_L flowing through the load resistor, R_L is
I_L=V_Th/(R_Th+ R_L)

The power dissipated by the load resistance, R_L is
P_L=I_L²R_L=>P_L=V_Th²[RL/(R_Th+R_L)²]

The condition for maximum power across the load resistance is RL=RTh.
The maximum power across the load resistance is
(P_L)_max=V_Th²/4R_Th

The total power supplied by the voltage source in the given circuit is
P_S=I_L²(R_Th+ R_L)
=>P_S=V_Th²/2R_Th

The maximum efficiency we can achieve is 50 % by using the maximum power transfer theorem.

We can find Maximum power transfer theorem applications in the receivers of communication systems. In all the real-time applications, we will use the condition of this theorem to get the maximum power across the variable load. In this article, we discussed the statement of the maximum power transfer theorem, its derivation, and the important formula of this theorem. Here, we applied this theorem to DC circuits. Similarly, we can apply this theorem to AC circuits as well.

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Derivation of Maximum Power Transfer Theorem

From the derivation of this theorem, we will get to know not only the condition at which we can get the maximum power across the load resistance, R_L but also its value. Now, let’s see the maximum power transfer theorem derivation.

In series, we can add the resistances. Since R_Th and R_L are in series, the equivalent resistance will be R_Th+R_L. By using Ohm’s law, we will get the current I_L flowing through the circuit as
I_L=V_Th/(R_Th+R_L)
The power dissipated by load resistance, RL is
P_L=I_L²R_L
=>P_L=[V_Th/(R_Th+R_L)]²R_L
=>P_L=V_Th²[R_L/(R_Th+R_L)²]
In the given circuit, V_Th and R_Th are constant, whereas R_L is variable. So, we must differentiate P_L concerning R_L and equate it to zero.
dP_L/dR_L=V_Th²[{(R_Th+R_L)²(1)-2R_L(R_Th+R_L)}/{R_Th+R_L}⁴]=0

=>(R_Th+R_L)²-2R_L(R_Th+R_L)=0

=>(R_Th+R_L)(R_Th+R_L-2R_L)=0

=>(R_Th-R_L)=0

=>R_L=R_Th
So, the condition for maximum power across the load resistance is R_L=R_Th.
We will get the maximum power dissipated by load resistance, R_L by substituting R_L=R_Th in the P_L formula.
(P_L)_max=V_Th²/(R_Th+R_Th)²

=>(P_L)_max=V_Th²[R_Th/4R_T_h²]

=>(P_L)_max=V_Th²[4R_Th²]

Efficiency of Maximum Power Transfer Theorem

Efficiency is a critical parameter in any circuit/network/system. It specifies how efficiently has produced the output for a given input. Here, we are dealing with both the output and input in terms of power. So, because of this, we can call it Power efficiency or efficiency of power.

The total power supplied by the voltage source in the given circuit is
P_S=I_L²(R_Th+RL)

=>P_S=I_L²(R_Th+R_Th)

=>P_S=2I_L²R_Th

=>P_S=2(V_Th/R_Th)²R_Th

=>P_S=V_Th²/2R_Th
We will represent Power efficiency mathematically as
η=(P_L)_max/P_S

=>η=(V_Th²/4R_Th)/(V_Th²/2R_Th)

=>η=0.5
If we multiply the Power efficiency by 100, it is known as the percentage of power efficiency.
=> η=0.5×100 %

=> η=50 %
The maximum efficiency we can achieve is 50 % by using the maximum power transfer theorem.

Maximum Power Transfer Theorem Problems

Question 1: How many methods can we calculate the value of the load resistance, RL, to get the maximum power across it for the circuit, which has multiple independent DC sources and resistances?

Answer: 2

Question 2: How many methods can we calculate the value of the load resistance, RL, to get the maximum power across it for the circuit, which has multiple independent & dependent DC sources and resistances?