Maximum Power Transfer Theorem

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.

Maximum Power Transfer Formula

According to the image of resistive circuit shown above, maximum transfer of power takes place when:

R_{s} = R_{L}

When this condition is matched, Pmax will be:

P_{max (deliveredToLoad)} = \frac{V_{s}^2}{4R_{s}}=\frac{V_{th}^2}{4R_{th}}

Maximum Power Transfer Theorem Proof

Let us consider a circuit, where a practical voltage source is connected to a load resistance (R_L). The circuit is given below:

Step 1: Calculating the power delivered to the load

The power delivered to the load R_L is:

P_{L} = i^2_{L}*R_{L}

i_{L} = \frac{v_{s}}{R_{s}+R_{L}}

P_{L} = \frac{v_{s}^2*R_{L}}{(R_{s}+R_{L})^2}

Step 2: Differentiating to find the maximum power

To find the value of R_L that absorbs maximum power from the given practical source, we differentiate with respect to RL and equating it with 0.

\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}

On equating it to zero we will get:

2R_{L}(R_{s} + R_{L}) = (R_{s} + R_{L})^2

-> R_{s}= R_{L} (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 R_L when R_L is equal to a the Thevenin equivalent resistance of the network i.e.,

R_L = R_s = R_th

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 P_max is given as:

P_{max (deliveredToLoad)} = \frac{V_{s}^2}{4R_{s}}=\frac{V_{th}^2}{4R_{th}}

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

η_{max} = \frac{P_{L,max}}{P_{s}}

where,

• P_L,max: Maximum amount of power transferred to load
• Ps: Power generated by the source

P_L,max = \frac{V_{th}^2}{4R_{th}}

Calculating P_s

P_{s} = I^2R_{th}+I^2R_{L}

According to the condition of maximum power transfer: R_L = R_s = R_th

P_{s} = 2I^2R_{th}

And we know I= \frac{V_{th}}{R_{s}+R_{L}} = \frac{V_{th}}{2R_{th}}

P_{s} = 2*(\frac{V_{th}}{2R_{th}})^2R_{th}

P_{s} = (\frac{V_{th}^2}{2R_{th}})

Hence efficiency will be:

η_{max} = \frac{(\frac{V_{th}^2}{4R_{th}})}{(\frac{V_{th}^2}{2R_{th}})}

η_{max}= \frac{1}{2} = 50%

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 Z_S) is connected to a load (Z_L), 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:

Consider an equivalent circuit analogous to Thevenin's. When analyzing this circuit across the load terminals, the current flowing is given by:

I = \frac{V_{th}}{(Z_{th} + Z_{L})}

Where:

• Z_L = R_L + jX_L (Load impedance)
• Z_th = R_th + jX_th (Thevenin impedance)

Therefore,

I = \frac{V_{th}}{(R_{L}+R_{th}) + j(X_{L}+X_{th})]}

Magnitude of current is:

I = \frac{V_{th}}{\sqrt{[(R_{L}+R_{th})^2 + (X_{L}+X_{th})^2]}}

The power delivered to the load (P_L) is given by:

P_{L} = I^2R_{L}

P_{L} = \frac{V_{th}^2 * R_{L}}{(R_{L} + R_{th})^2 + (X_{L} + X_{th})^2} -> (1)

To maximize power transfer, we will differentiate the equation-1 and equate it to zero. After simplification we will find that:

X_L + X_TH = 0

X_L = -X_TH (condition for maximum power transfer)

Substituting the value of X_L into equation (1), we obtain:

P_{L} = \frac{V_{th}^2 * RL}{(R_{L} + R_{th})^2}

For maximum power transfer, we will equate the above equation to zero:

R_L + R_th = 2R_L

R_L = R_th

Hence, in an AC circuit, the highest power transfer occurs when the load resistor (R_L) equals the Thevenin resistance (R_th) and X_L equals the negative of X_th. In other words, the load impedance (Z_L) must be equal to the complex conjugate of the corresponding circuit impedance, i.e.,

if Z_L = R_L + jX_L then Z_th = R_th - jX_L

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 (R_th)

Calculate the Thevenin Resistance (R_th) 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 (V_th)

After calculating the R_th , 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.

Solution

Step 1: Find the Thevenin equivalent of the circuit

To find the R_th, remove R_L and short-circuit the independent sources. The final circuit diagram is shown below:

From the above circuit it is clear that v_π = 0. So the dependent current source will behave as an open circuit.

Hence R_{th} = 1k\Omega

In order to obtain maximum power delivered into the load, R_L should be set to R_{th} = 1k\Omega

Step 2: Find the Thevenin voltage of the circuit

To find the V_th consider the circuit given below:

v_oc = −0.03v_π (1000) = −30v_π

where the voltage v_π may be found from simple voltage division:

v_{\Pi} = 2.5*10^{-3}*sin(440t)*(\frac{3864}{300+3864})

V_th = −69.6 sin(440t) mV

Step 3: Calculate the Maximum Power Transfer

P_{max} = \frac{V_{th}^2}{4R_{th}}

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

1. 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.
   
2. Solar Panels: Solar panels are­ designed to capture sunlight and conve­rt it into usable electricity. To optimize­ their performance, it is important to e­nsure that the panels are­ properly connected to a batte­ry or electrical system to match the source and load impedances. This ensures maximum efficie­ncy and output of electricity from the solar pane­ls.
   
3. Sound Systems: The spe­akers in your home stere­o system are responsible­ for producing sound. To ensure the be­st possible sound quality, it's important to connect the spe­akers in a way that matches the se­ttings of the amplifier.
   
4. Radio and TV Antennas: To rece­ive a stronger and cleare­r signal on your radio or TV, it's important to align the antenna with the transmitte­r's settings. This ensures that the­ radio or TV signals travel effective­ly through the antennas.
   
5. 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.