In this article, we will study about Maximum Power Transfer Theorem. The Maximum Power is transferred in the circuit when the load impedance is matched with the source impedance. This theorem helps in increasing the efficiency and performance of the circuit. It is very helpful in circuit design. In this article, we will learn more about Maximum Power Transfer Theorem, We will see the Maximum Power Transfer Formula, Maximum Power Transfer Theorem Proof, Efficiency of Maximum Power Transfer and at last we Will go through Some Examples.
What is the Maximum Power Transfer Theorem?
The statement of Maximum Power Transfer Theorem is as follows:
It states that the maximum power is developed in a load when the load resistance equals the Thevenin resistance of the source to which it is connected.
Condition for Maximum Power Transfer
To achieve power transfer in a circuit, the resistance or impedance of the load must match with the source impedance. This means the load and source should have properties for efficient energy utilization and maximum power delivery.
Resistive Circuit
Maximum Power Transfer Formula
According to the image of resistive circuit shown above, maximum transfer of power takes place when:
[Tex]R_{s} = R_{L}
[/Tex]
When this condition is matched, Pmax will be:
[Tex]P_{max (deliveredToLoad)} = \frac{V_{s}^2}{4R_{s}}=\frac{V_{th}^2}{4R_{th}}
[/Tex]
Maximum Power Transfer Theorem Proof
Let us consider a circuit, where a practical voltage source is connected to a load resistance (RL). The circuit is given below:
Practical Voltage Source Connected to a Load Resistor
Step 1: Calculating the power delivered to the load
The power delivered to the load RL is:
[Tex]P_{L} = i^2_{L}*R_{L}
[/Tex]
[Tex]i_{L} = \frac{v_{s}}{R_{s}+R_{L}}
[/Tex]
[Tex]P_{L} = \frac{v_{s}^2*R_{L}}{(R_{s}+R_{L})^2}
[/Tex]
Step 2: Differentiating to find the maximum power
To find the value of RL that absorbs maximum power from the given practical source, we differentiate with respect to RL and equating it with 0.
[Tex]\frac{dP_{L}}{dR_{L}} = \frac{(R_{s}+R_{L})^2v_{s}^2 – v_{s}^2R_{L}(2)(R_{s}+R_{L})}{(R_{s}+R_{L})^4}
[/Tex]
On equating it to zero we will get:
[Tex]2R_{L}(R_{s} + R_{L}) = (R_{s} + R_{L})^2
[/Tex]
[Tex]-> R_{s}= R_{L}
[/Tex] (condition for maximum delivery of the power)
Note: An alternative way to look at the maximum power transfer theorem is possible in terms of the Thevenin equivalent resistance of a network. The modified circuit is given below:
A network delivers maximum power to a load resistance RL when RL is equal to a the Thevenin equivalent resistance of the network i.e.,
RL = Rs = Rth
Thevenin Equivalent Circuit
Hence we have proven that according to the Maximum Power Transfer Theorem, when the ‘load resistance’ or ‘Thevenin resistance’ is equal to ‘source resistance’, maximum power is delivered. Hence the formula for Pmax is given as:
[Tex]P_{max (deliveredToLoad)} = \frac{V_{s}^2}{4R_{s}}=\frac{V_{th}^2}{4R_{th}}
[/Tex]
Efficiency of Maximum Power Transfer
The Maximum Power Transfer Theorem ensures efficient power transfer and minimize wastage when applied correctly. It is useful in scenarios where maximizing power usage and minimizing waste is essential, such as audio amplifiers.
Efficiency Calculation
[Tex]η_{max} = \frac{P_{L,max}}{P_{s}}
[/Tex]
where,
- PL,max: Maximum amount of power transferred to load
- Ps: Power generated by the source
PL,max = [Tex]\frac{V_{th}^2}{4R_{th}}
[/Tex]
Calculating Ps
[Tex]P_{s} = I^2R_{th}+I^2R_{L}
[/Tex]
According to the condition of maximum power transfer: RL = Rs = Rth
[Tex]P_{s} = 2I^2R_{th}
[/Tex]
And we know [Tex]I= \frac{V_{th}}{R_{s}+R_{L}} = \frac{V_{th}}{2R_{th}}
[/Tex]
[Tex]P_{s} = 2*(\frac{V_{th}}{2R_{th}})^2R_{th}
[/Tex]
[Tex]P_{s} = (\frac{V_{th}^2}{2R_{th}})
[/Tex]
Hence efficiency will be:
[Tex]η_{max} = \frac{(\frac{V_{th}^2}{4R_{th}})}{(\frac{V_{th}^2}{2R_{th}})}
[/Tex]
[Tex]η_{max}= \frac{1}{2} = 50%
[/Tex]
Therefore, the efficiency of Maximum Power Transfer theorem is 50%
Maximum Power Transfer Theorem for AC Circuits
In AC circuits, the Maximum Power Transfer Theorem determines the conditions for transferring the maximum power from a source to a load. This theorem states that in an active AC circuit, where a source with internal impedance (denoted as ZS) is connected to a load (ZL), the highest power transfer occurs when the impedance of the load matches the complex conjugate of the source impedance.
For a passive setup, maximum power is transferred to the load when the impedance of the load equals the complex conjugate of the corresponding impedance observed from the load’s terminals.
Now let us derive the condition for maximum power transfer in the AC circuits:
AC Circuit
Consider an equivalent circuit analogous to Thevenin’s. When analyzing this circuit across the load terminals, the current flowing is given by:
[Tex]I = \frac{V_{th}}{(Z_{th} + Z_{L})}
[/Tex]
Where:
- ZL = RL + jXL (Load impedance)
- Zth = Rth + jXth (Thevenin impedance)
Therefore,
[Tex]I = \frac{V_{th}}{(R_{L}+R_{th}) + j(X_{L}+X_{th})]}
[/Tex]
Magnitude of current is:
[Tex]I = \frac{V_{th}}{\sqrt{[(R_{L}+R_{th})^2 + (X_{L}+X_{th})^2]}}
[/Tex]
The power delivered to the load (PL) is given by:
[Tex]P_{L} = I^2R_{L}
[/Tex]
[Tex]P_{L} = \frac{V_{th}^2 * R_{L}}{(R_{L} + R_{th})^2 + (X_{L} + X_{th})^2}
[/Tex] -> (1)
To maximize power transfer, we will differentiate the equation-1 and equate it to zero. After simplification we will find that:
XL + XTH = 0
XL = -XTH (condition for maximum power transfer)
Substituting the value of XL into equation (1), we obtain:
[Tex]P_{L} = \frac{V_{th}^2 * RL}{(R_{L} + R_{th})^2}
[/Tex]
For maximum power transfer, we will equate the above equation to zero:
RL + Rth = 2RL
RL = Rth
Hence, in an AC circuit, the highest power transfer occurs when the load resistor (RL) equals the Thevenin resistance (Rth) and XL equals the negative of Xth. In other words, the load impedance (ZL) must be equal to the complex conjugate of the corresponding circuit impedance, i.e.,
if ZL = RL + jXL then Zth = Rth – jXL
How to Solve Network using Maximum Power Transfer Theorem?
Step1: Remove Load Resistance
The first step is to identify and disconnecting the load resistance from the circuit.
Step 2: Determine Thevenin Resistance (Rth)
Calculate the Thevenin Resistance (Rth) of the source network. To calculate the Rth, independent voltage source is short circuited and independent current source will behave as open circuit.
Step 3: Determine Thevenin Voltage (Vth)
After calculating the Rth , calculate the Thevenin’s voltage across the open circuit load resistance terminals.
Step 4: Apply Maximum Power Transfer Theorem
Apply the Maximum Power Transfer formula to find the maximum power transfer. It can be calculated using the above derived formula.
Solved Example on Maximum Power Transfer Theorem
Q.1 The circuit shown in figure is a model for the common-emitter bipolar junction transistor amplifier. Choose a load resistance so that maximum power is transferred to it.
Circuit Diagram
Solution
Step 1: Find the Thevenin equivalent of the circuit
To find the Rth, remove RL and short-circuit the independent sources. The final circuit diagram is shown below:
Thevenin Equivalent circuit
From the above circuit it is clear that vπ = 0. So the dependent current source will behave as an open circuit.
Hence [Tex]R_{th} = 1k\Omega
[/Tex]
In order to obtain maximum power delivered into the load, RL should be set to [Tex]R_{th} = 1k\Omega
[/Tex]
Step 2: Find the Thevenin voltage of the circuit
To find the Vth consider the circuit given below:
Thevenin Voltage
voc = −0.03vπ (1000) = −30vπ
where the voltage vπ may be found from simple voltage division:
[Tex]v_{\Pi} = 2.5*10^{-3}*sin(440t)*(\frac{3864}{300+3864})
[/Tex]
Vth = −69.6 sin(440t) mV
Step 3: Calculate the Maximum Power Transfer
[Tex]P_{max} = \frac{V_{th}^2}{4R_{th}}
[/Tex]
Pmax = 1.211 sin2 (440t) μW
Advantages and Disadvantages of Maximum Power Transfer Theorem
Here, some list of Advantages and Disadvantages of Maximum Power Transfer Theorem given below :
Advantages
- It ensures that the maximum available power from a source is efficiently delivered to the load.
- It helps in designing the circuits that minimize power wastage. it leads in making devices more energy-efficient.
- It prevent the overloading of components by matching load resistance with the source resistance which enhance circuit safety.
- It is very easy to apply which helps in quick estimation.
Disadvantages
- It is not applicable in non-linear and unilateral networks.
- Matching resistances may not always be feasible in real-world applications due to component limitations.
- In cases where load resistance doesn’t match, it can result in power loss, reducing circuit efficiency.
- The maximum efficiency up to which Maximum Power Transfer Theorem can reach is 50% and not is applicable for power systems.
Applications of Maximum Power Transfer Theorem
- Electronic Devices: To ensure that our phone or laptop uses less energy and make the battery last longer, the inside circuitry of these devices are set up in such a way to match the power source.
- Solar Panels: Solar panels are designed to capture sunlight and conveÂrt it into usable electricity. To optimize their performance, it is important to eÂnsure that the panels are properly connected to a batteÂry or electrical system to match the source and load impedances. This ensures maximum efficieÂncy and output of electricity from the solar paneÂls.
- Sound Systems: The speÂakers in your home stereÂo system are responsible for producing sound. To ensure the beÂst possible sound quality, it’s important to connect the speÂakers in a way that matches the seÂttings of the amplifier.
- Radio and TV Antennas: To receÂive a stronger and cleareÂr signal on your radio or TV, it’s important to align the antenna with the transmitteÂr’s settings. This ensures that the radio or TV signals travel effectiveÂly through the antennas.
- Wireless Devices: Wi-Fi Remote controls utilize radio signals, for communication. By configuring them to align their signals, the strength and reliability of the connection can be enhanced.
Conclusion
In the above article, we gave seen that Maximum Power Transfer Theorem maximizes the power transfer at Thevenin’s resistance of the circuit. It is applicable to both AC and DC circuits and the derivation is explained above. It finds its application in various fields like electronic devices, solar panels, wireless devices and many more.
FAQs on Maximum Power Transfer Theorem
1. What happens if the load resistance does not match the source resistance in a DC circuit?
In a DC circuit, if the load reÂsistance doesn’t match the source resistance, it can lead to ineÂfficient power transfer. This mismatch can cause power loss and decrease overall circuit efficiency. This theorem works best wheÂn resistances are propeÂrly matched.
2. Why we need to maximize the power transfer in the electrical circuits?
Maximizing power transfer is important because it ensures that power is transferred efficiently from source to load, thus reducing losses and increasing overall circuit performance.
3. Can the Maximum Power Transfer Theorem be applied to AC circuits?
The maximum poweÂr transfer theorem also applieÂs to AC circuits. In these circuits, the poweÂr transfer is optimized when the load impedance matches the complex conjugate of the source impedance.
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