A circuit achieves maximum power transfer when the resistance of the load (the device receiving power) is precisely equal to the internal resistance of the voltage source. This internal resistance is often fixed and expressed in Ohms and is referred to as impedance if the circuit contains components such as coils (inductors) or capacitors.
The impedance of a load (a resistance known as RL) can vary greatly when it is connected to the power source’s output. It can be anything from a total short circuit (unrestricted current flow) to a totally open circuit (no current flow). The amount of power that the load absorbs from the source is impacted by this change in impedance. The load’s impedance must be “matched” to the power source’s impedance in order to get the maximum amount of power from it. Maximum Power Transfer is based on this matching concept itself.
The Maximum Power Transfer Theorem is another useful tool for circuit analysis. This theorem facilitates in optimizing power transfer from a power source to the load, which is resistance denoted by RL. It’s important to match the resistance of the load to the power source’s resistance. The amount of power that enters the load depends on how the power source’s internal resistance and the load’s impedance interact. To see this in action, let’s look at the circuit below.
Thevenin’s Equivalent Circuit
In our Thevenin equivalent circuit, the Maximum Power Transfer Theorem facilitates in delivering the maximum amount of power to the load resistance (RL). According to this theorem, in order to do this, the value of RL must precisely equal the circuit’s Thevenin source resistance (RS).
Stated differently, for optimal power transfer:
RL = RS
The highest amount of power that may be given to the load will be reduced if the resistance of the load is either greater or smaller than the resistance of the Thevenin source.
Let’s now determine what value of RL will allow the circuit to transfer the maximum power.
Solving a Maximum Power Transfer Problem
After that, using the Ohm’s Law formulas shown below:
- I = VS / (RS + RL)
- and P = I2 * RL
Now that the following table has been completed, we can calculate the circuit’s power and current for a range of load resistance values.
Table Comparing Power and Current
| RL (Ω) | I (amps) | P (watts) |
|---|---|---|
| 0 | 4.0 | 0 |
| 5 | 3.3 | 55 |
| 10 | 2.8 | 78 |
| 15 | 2.5 | 93 |
| 20 | 2.2 | 97 |
| 25 | 2.0 | 100 |
| 30 | 1.8 | 97 |
| 40 | 1.5 | 94 |
| 60 | 1.2 | 83 |
| 100 | 0.8 | 64 |
We can make a graph using the information in the table to demonstrate how the load resistance (RL) influences the power (P) provided to the circuit. Keep in mind that at both extremes, the power is zero:
When there is no voltage drop across the load and all current passes through it, there is no load resistance (0 ohms), which indicates a short circuit and zero power.
An open circuit occurs when the load resistance is extremely high (nearing infinity), meaning that no current is flowing through the load and all of the voltage is across it, again producing zero power.
Graph Showing Power Versus Load Resistance
Based on the table and graph, we can observe that when the source resistance (RS) and load resistance (RL) are exactly equal, the maximum power transfer (MPT) occurs. In our scenario, this occurs when:
RS = RL = 25Ω
We refer to this as a “matched condition.” In general, when a device’s impedance and power source (such as a battery) are exactly matched, the maximum amount of power is delivered to the device (such as a speaker).
The relationship between a loudspeaker and an audio amplifier is an excellent example of impedance matching. The nominal input impedance (ZIN) of the speaker may be as low as 8Ω, but the output impedance (ZOUT) of the amplifier may range from 4Ω to 8Ω.
Now, the amplifier will “see” the 8Ω speaker as an 8Ω load if you connect it to the output; this works well because the speaker falls within the amplifier’s recommended output range of 4Ω to 8Ω.
The intriguing bit comes next: an amplifier powering a single 4Ω speaker will behave electrically the same as connecting two 8Ω speakers in parallel. Both setups are within the amplifier’s safe working parameters.
Impedance Matching in Audio Amplifiers for Maximum Power Transfer
To maximize the power output of amplifier circuits, impedance matching is essential. Transformers are employed in the output stages to match the loudspeaker’s impedance, which might be greater or lower than the amplifier’s output, in order to maximize the output of sound power.
As seen in the illustration below, these transformers, suitably called “matching transformers,” serve as a bridge between the amplifier’s output and the speaker.
Impedance Matching using Transformer
Maximum power transmission may still be possible even if the speaker’s impedance (ZLOAD) and the amplifier’s output impedance (Zout) are not exactly match. This is accomplished by utilizing a unique transformer with a precisely selected “turns ratio.” The number of turns in the transformer’s coils is adjusted to provide this ratio, which connects the speaker’s impedance to the amplifier’s output impedance.
In simple terms, the transformer functions as a bridge, converting the resistance on one side (either the speaker or the amplifier) to an alternate value on the other.
Here’s the thing: when the output impedance (ZOUT) of the amplifier and the speaker are both fully resistive (i.e., they just include resistance and no reactive components), there is an equation that can be used to determine the optimal turns ratio for maximal power transmission.
Zout = (NP / NS)2 * Zload
The number of turns in the main coil of the transformer is denoted by NP in this equation, whereas the number of turns in the secondary coil is indicated by NS. We may essentially “transform” the output impedance perceived by the speaker to match the source impedance (the amplifier’s output) by carefully altering this ratio of turns (NP/NS). Maximum power transfer is possible because of this change.
Let’s look at an example to understand how this functions.
Solving Another Maximum Power Transfer Problem
We want to connect an amplifier with a significantly larger output impedance of 1000 Ω to an 8 Ω speaker. We must employ a matching transformer in order to transmit the audio signal’s power to its greatest potential.
In this instance:
- Z1 is the source impedance (1000Ω) of the amplifier.
- Z2 is the load impedance (8Ω) of the speaker.
- N stands for the matching transformer’s turns ratio, which is unknown and requires calculation.
- To get the necessary value of N for this particular situation, we’ll make use of the turns ratio notion and the impedance connection.
- Z1 = N2 * Z2
- ∴ N = √(Z1 / Z2)
- ∴ √(Z1 / Z2)
- = √(1000 / 8)
- = 11.2 : 1
To make calculations easier, we frequently assume tiny, high-frequency audio transformers used in low-power amplifier circuits as perfect. This implies we may overlook any energy losses that occur within the transformer.
In the very next tutorial, we’ll look at DC circuit theory and a method called Star-Delta Transformation. This approach converts balanced circuits having a star (wye) configuration to corresponding delta configurations and vice versa. This is an effective tool for studying and simplifying complicated DC circuits.
References: how to calculate the maximum power transfer? (specific exercise example included)
Maximum power transfer theorem
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