Maximum Power Transfer Theorem in Circuit Design
From civilian power distribution to circuit boards, maximum power transfer is an important point to consider during design. Whether you are designing a power delivery network or an AC circuit, you’ll need to choose the right source and load impedances to transfer maximum power throughout your circuits.
Anytime someone brings up a theorem, most people let their eyes glaze over as the mathematical derivations start to flow. Thankfully, the maximum power transfer theorem (also known as Jacobi’s law) in circuit analysis is deceptively simple and is easily applied in circuit design.
What is the Maximum Power Transfer Theorem?
When designing a DC circuit, especially a power delivery network, you need to ensure that you are delivering maximum power throughout the circuit. The actual power delivered will be a function of the source voltage, the output resistance from the power supply or driving DC component, and the resistance of the load. With the right combination of load resistance and output resistance, you can ensure that your power receives the maximum intended power.
According to Thevenin’s theorem, the circuitry in a DC driver can be reduced to an equivalent voltage source and a series resistor. This output series resistance, along with the resistance of the load, will determine the output current that reaches the load, as well as the power delivered to the load. The power that reaches the load can be determined through a simple application of Ohm’s law, either by hand or with a circuit simulation. A circuit diagram that contains the Thevenin equivalent and the load resistor is shown in the figure below:
Equivalent circuit and power dissipation across a load resistor
The maximum power transfer theorem states that the power delivered to the load resistor is maximized when the load resistance is equal to the series resistance. This can be calculated by taking the derivative of the power equation with respect to the load resistance and calculating the critical point. If you graph the power equation in the above image as a function of the load resistance, you’ll see that the power delivered to the load is maximized when the load and series resistors are equal.
Power delivered to the load resistor
In this configuration, the Thevenin equivalent and the load resistor form a voltage divider, and the voltage dropped across the load will be half the value of the source voltage in the driver. Increasing the load resistor increases the voltage drop but decreases the total current, while decreasing the load resistor decreases the voltage drop and increases the total current.
Maximum Power Transfer with AC and Digital Signals
While the maximum power transfer theorem is normally discussed in terms of DC circuits, it also applies to circuits carrying AC signals. Instead of working with a load and series resistance, you consider the load and series impedance at a given frequency. When working with an AC driver or a driver that outputs digital signals, you also want to ensure maximum power transfer to the circuit from the source, followed by maximum power delivery to the load.
This requires considering the impedance of the source and load. If the source and load are included, an application of Ohm’s law yields the following equation for the instantaneous power transfer in terms of the resistance and reactance values of the sources and loads:
Instantaneous power in an AC circuit
Here, one finds that the maximum power transfer occurs when the source and load impedances are complex conjugates of each other. In other words, the instantaneous power is maximum when the source and load resistances are equal, and when the source and load reactances are equal with opposite sign.
Note that the reactance of the load will induce a phase shift in the current, which will affect the instantaneous power. You can still calculate the average power over an oscillation period by taking an integral. Note that the load reactance will still induce a phase shift in the current, even if the maximum power transfer condition is met.
The Maximum Power Transfer Theorem and Transmission Lines
With AC signals and with high speed digital signals, there is another element to consider. When the length of the conductor between the source and load is long enough, the propagation delay for signals along a trace can be longer than the signal rise time. A rule of thumb is that transmission line effects need to be considered when the propagation delay along a trace is greater than 50% of the rise/fall time of the DC signal being sent through the trace. With analog signal, the trace will act as a transmission line when the propagation delay is greater than one-quarter of the oscillation period of the AC signal.
When an AC driver is in series with a transmission line and a load, the driver’s source impedance and the load impedance should match the impedance of the transmission line. Normally, transmission lines are only matched at the load end to prevent signal reflection back along the transmission line. This will prevent a step-like response in the voltage across the transmission line under multiple reflections and ringing, which in turn prevents inadvertent switching of a digital IC at the load end. With AC signals, this is important for preventing standing waves from forming on the trace, which will cause the transmission line from acting like a transmitting antenna.
When the load end is impedance matched to the transmission line, matching at the source end is usually ignored. Remember that the source driver and the transmission line form their own voltage divider. If the source impedance is matched to the transmission line impedance, then maximum power will be delivered to the transmission line, but the voltage will be half the value of the driving voltage.
In order for the driving voltage to equal the actual voltage dropped across the transmission line, the output impedance of the driver should be very low. In effect, the series combination of the source impedance and transmission line impedance will form their own Thevenin equivalent circuit, and the load impedance should be matched to this combined impedance value.
Maximum Power Transfer versus Maximum Power Efficiency
When working in, especially, AC power distribution schemas, the goal of high power efficiency is important for a low generator impedance to load impedance ratio. Unfortunately, maximum efficiency is not particularly tied to the maximum power transfer theorem. You will still need to design for efficiency as MPTT does not guarantee maximum efficiency, or even high efficiency.
Furthermore, MPTT does not ensure low noise ratios either. If you’re looking to learn more about noise floors, I’d highly suggest reading about them here.
Applications such as RF amplifiers are looking for low noise, low output impedances, and a moderate speaker load impedance. When looking at the maximum power transfer theorem and applying it to an RF amplifier, you’ll often find a mismatch between amplifier input impedance and the used antenna.
Working with the right PCB layout and design software can help your DC and AC networks to satisfy the maximum power transfer theorem. Allegro PCB Designer and Cadence’s full suite of design tools are designed with the tools you need to ensure power delivery throughout your DC networks and AC circuits.
If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.