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get prove that, the circuit efficiency during maximum power transfer from source to load is only 50%. from screen.
[Solved] When the power transferred to the load is maximum, the effic
Concept: When the load impedance matches the Thevenin equivalent resistance of the given circuit, maximum power is transferred to it. i.e. &n
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When the power transferred to the load is maximum, the efficiency of power transfer is
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When the load impedance matches the Thevenin equivalent resistance of the given circuit, maximum power is transferred to it. i.e.
If RL = Rth, Maximum power is transferred to the Load.
Power transfer efficiency =
IL=VthRth+RL PL=Loadpower=IL2RL PL=(VthRth+RL)2×RL --(1)
Pi=Vth(VthRth+RL) --(2) For RL = Rth,
Similarly, Equation-(2) becomes:
Power Transfer Efficiency will be:
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Maximum Power Transfer Theorem (MPTT)
A simple tutorial on Maximum Power Transfer Theorem. You will learn its statement, proof, maximum power vs maximum efficiency, example.
What is Maximum Power Transfer Theorem (MPTT)?
April 5, 2021 By Ravi Teja
In this tutorial, we will learn about Maximum Power Transfer Theorem (MPTT). It is one of the basic yet important laws that states the necessary condition for maximum power transfer (not to be confused with maximum efficiency).
In any electric circuit, the electrical energy from the power supply is delivered to the load where it is converted into a useful work. Practically, the entire supplied power will not be present at the load due to the heating effect and other constraints in the network. Therefore, there exists a certain difference between drawing and delivering powers.
The size of the load always affects the amount of power transferred from the supply source, i.e., any change in the load resistance results a change in the power transferred to the load. Thus, the Maximum Power Transfer Theorem ensures the ideal condition to transfer the maximum power to the load. Let us see ‘how’.
Maximum Power Transfer Theorem Statement
The Maximum Power Transfer Theorem states that in a linear, bilateral DC network, Maximum Power is delivered to the load when the load resistance is equal to the internal resistance of the source.
If it is an independent voltage source, then its series resistance (internal resistance RS) or if it is independent current source, then its parallel resistance (internal resistance RS) must equal to the load resistance RL to deliver maximum power to the load.
Proof of Maximum Power Transfer Theorem
The Maximum Power Transfer Theorem ensures the value of the load resistance, at which the maximum power is transferred to the load.
Consider the below DC two terminal network (left side circuit). The condition for maximum power is determined by obtaining the expression of power absorbed by load using mesh or nodal current methods and then deriving the resulting expression with respect to load resistance RL.
This is quite a complex procedure. But in the previous tutorials, we have seen that the complex part of the network can be replaced with a Thevenin’s equivalent as shown below.
The original two terminal circuit is replaced with a Thevenin’s equivalent circuit across the variable load resistance. The current through the load for any value of load resistance is
Form the above expression, the power delivered depends on the values of RTH and RL. However, as the Thevenin’s equivalent is a constant, the power delivered from this equivalent source to the load entirely depends on the load resistance RL. To find the exact value of RL, we apply differentiation to PL with respect to RL and equating it to zero as shown below:
Therefore, this is the condition of matching the load where the maximum power transfer occurs when the load resistance is equal to the Thevenin’s resistance of the circuit. By substituting the RTH = RL in the previous equation, we get:
The maximum power delivered to the load is,
Total power transferred from source is:
PT = IL2 * (RTH + RL)
PT = 2 * IL2 RL …………….(2)
Hence, the maximum power transfer theorem expresses the state at which maximum power is delivered to the load i.e., when the load resistance is equal to the Thevenin’s equivalent resistance of the circuit. Below figure shows a curve of power delivered to the load with respect to the load resistance.
Note that the power delivered is zero when the load resistance is zero as there is no voltage drop across the load during this condition. Also, the power will be maximum, when the load resistance is equal to the internal resistance of the circuit (or Thevenin’s equivalent resistance). Again, the power is zero as the load resistance reaches to infinity as there is no current flow through the load.
Power Transfer Efficiency
We must remember that this theorem states only maximum power transfer but not for maximum efficiency. If the load resistance is smaller than source resistance, power dissipated at the load is reduced while most of the power is dissipated at the source, then the efficiency becomes lower.
Consider the total power delivered from source equation (equation 2), in which the power is dissipated in the equivalent Thevenin’s resistance RTH by the voltage source VTH.
Therefore, the efficiency under the condition of maximum power transfer is:
Efficiency = Output / Input × 100
= IL2 RL / 2 IL2 RL × 100
= 50 %
Hence, at the condition of maximum power transfer, the efficiency is 50%, that means only half of the generated power is delivered to the load and at other conditions, a small percentage of power is delivered to the load, as indicated in efficiency verses maximum power transfer the curves below.
For some applications, it is desirable to transfer maximum power to the load than achieving high efficiency such as in amplifiers and communication circuits.
Maximum Power Transfer Theorem
Maximum power transfer theorem(MPTT) states the condition of max. power transfer Rth=RL..proof of maximum power transfer theorem ..Efficiency.
Putting the value of I from equation (1) in equation(2),we get:
The maximum power delivery is possible if;
Equation(3) can be solved as follows.
Thus, the condition for Maximum Power Transfer is the source delivers maximum power to load, if the source resistance is equal to the load resistance. Now , we will calculate the system efficiency under maximum power transfer condition.
Efficiency of Maximum Power Transfer
To calculate the efficiency, we first calculate the maximum power transfer when the load resistance is equal to the source resistance.Maximum Power Delivered to the Load
Total power transferred from source is:
We can calculate the maximum efficiency by Dividing equation(4) and equation(5)
Hence, at the condition of maximum power transfer the efficiency is 50%. At maximum power transfer condition, the source delivers 50 % of the generated power to the load, and at other conditions the source drivers small percentage of power to the load.
Load Resistance Value for Maximum Power Transfer
Application of Maximum Power Transfer Theorem
The efficiency of the electrical system is 50 % when the load resistance is equal to the source resistance. It means source deliver 50 % power to the load and remaining 50% power lost in the circuit. The MPTT is not applicable where large power transfer is taking place.In view of this MPTT is not applicable for power transmission. In communication circuit, the magnitude of power transfer is very small and low efficiency is not a problem in communication circuit. The maximum power transfer theorem has tremendous applications in communication circuit for impedance matching.Related postsWhat is Millman’s Theorem? Millman’s Theorem FormulaKirchhoff’s Current Law, KCL and Junction Rule,Solved Problems