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Deriving the Instantaneous Power Equation

Key Takeaways

  • Understand what instantaneous power is.

  • Learn how the instantaneous power equation is derived in DC and AC circuits.

  • Explore the difference between instantaneous power and average power.

What Is Instantaneous Power?

Man testing electric current in a control panel

In physics, power is defined by the amount of energy transferred over a period of time. Meanwhile, instantaneous power refers to the power consumed at a particular point in time. Instantaneous power is an important metric in electronics. It enables the designer to project the capability of the components to handle power consumption. It also enables the design of a power supply module that’s capable of supplying the needed power at any particular moment of usage. 

Power in electronics is often expressed as a product of voltage and current. Therefore, instantaneous power in electronics is a measurement of both parameters at a particular point in time. Deriving the instantaneous power equation is dependent on the type of circuit that you’re working on. There isn’t a universal formula that works for all circuits. 

That being said below you can find the power formula. This will describe things like mechanical power, power factor in electronics, apparent power, and general electrical energy. Your power calculation needs to be as precise as possible in this age of improving electrical power efficiency, and as such, you need to be sure you not only have the right energy equation but also that you are starting with the proper attribution of amps, instantaneous voltage, and potential energy to derive the real power of your device. 

How to Derive the Instantaneous Power Equation

Instantaneous power equation for a DC circuit

Expression of instantaneous power in a DC circuit.

The instantaneous power equation is helpful when analyzing a DC circuit. It provides an accurate representation of how much power is delivered to the load at any point in time. Knowing the instantaneous power enables proper thermal management strategies to be carried out. 

The reason there isn’t a universal equation for instantaneous power is that electronics are either powered by a DC or an AC source. Let’s consider a simple closed circuit that consists of a DC source and a resistor. It will have a stable, flat-line voltage level which results in an equally constant current. 

For a DC circuit, the instantaneous power equation is quite simple and it’s represented by the following equation:

P = V x I. 

The instantaneous power equation for a DC circuit can also be expressed by: 

P = V2/R or I2 x R

Calculating the instantaneous power equation for an AC circuit is, however, not so straightforward. In an AC circuit, the voltage level is not a constant value. Instead, both voltage and current are time-varying sinusoidal waveforms. 

Therefore, the instantaneous power equation for an AC circuit is expressed by:

P = VI cosθ + VI cos (2⍵t+ θ).

The first component (VI cosθ) represents the average power while the second component indicates the time-varying characteristic of the equation. 

Instantaneous Power vs. Average Power

Average power in AC circuit analysis.

Average power is a better representation of power consumption in an AC circuit.

As helpful as it is for DC circuits, the instantaneous power equation is quite meaningless for an AC circuit. That’s because the value is always changing respective to time. Just because you’ve calculated 1 Watt at T= 0.5s doesn’t mean that the power delivered will be the same at T=0.7s. 

In other words, the instantaneous power value will not give you an accurate picture of how much power is produced and consumed on an AC circuit. For practical purposes, you’ll want to have an average power value that’s a better representative of power dissipated in an AC circuit.

The average power is given by the formula:

P = VI cosθ

Both V and I are the root-mean-square (RMS) value of the voltage and current. The θ represents the phase angle between the voltage and current. Not only does average power paint a more accurate picture of the amount of power consumed in a circuit, but it’s also easier to include in calculations.

While electrical circuits aren't really dealing in velocity too frequently, understanding the force at work behind your motor and your PDNs is essential for electric energy design principles. Utilize the proper power formula to find the right energy in your electric power system to ensure voltage is maintained and you're not left in the dark regarding things like voltage drop, active power, displacement, or output power. Your electrical circuit for your products deserve the best simulated chance of achieving their peak power when and how you need it.

Now that you’re aware of how the instantaneous power equation is derived for different types of circuits, you’ll want to build them with sophisticated PCB design and analysis software. Allegro not only has all the basic design tools, but also the analytical capability to ensure a circuit is built for the rated power. You can also use InspectAR to accurately assess and improve PCBs using augmented reality and intuitive interaction. Inspecting, debugging, reworking, and assembling PCBs has never been faster or easier.

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts