Use the Phase Angle Formula to Understand Power Delivery

November 5, 2020 Cadence PCB Solutions

Key Takeaways

  • Power delivery in an AC system depends on the phase angle between voltage and current.

  • The phase angle also depends on the impedance of the circuit, which induces a phase change.

  • When there is a phase difference between voltage and current, the real power delivered to a load can be quite low. You can spot when this will occur by looking at graphs for your circuit.

AC system phase angle formula

Keep the power factor high when working with 3-phase power in your AC systems.

Working with power delivery can be dangerous and complex, especially when we consider reactance in practical AC circuits. Ensuring real power delivery to a resistive load depends on keeping the power factor in your circuits high, which then requires keeping the phase angle in your system near zero. Every so often, you’ll need to check the phase angle between voltage and current in a reactive circuit to ensure sufficient power delivery to your load element.

Just by looking at the time difference between delivered voltage and current at your load component, you can determine the phase angle that governs real power delivery. You can then use the phase angle formula for real and apparent power to determine the power factor in your system. You can then determine whether power factor correction will be necessary, which will depend on the input voltage and power in the system.

The Phase Angle Formula

The phase angle for a circuit depends on the phase difference between the voltage and current in the circuit. Assuming we have a simple LTI system with only resistors, capacitors, and inductors, you can determine a simple phase angle relationship between the voltage and current in each circuit element.

Phase differences in various RLC circuits are shown graphically in the image below, where we can see that the voltage and current are displaced from each other in the time domain. Here, the reactive impedance of C and L elements creates a phase difference between voltage and current. This phase difference will become important later when you want to calculate real power delivered to a load component.

Phase angle formula AC circuits

Complex voltage and current in different circuits, as well as their phase differences.

You can determine the phase angle simply by looking at the time difference between the voltage and current waveforms. This is equal to the time difference between adjacent current and voltage peaks, divided by 180 degrees. Alternatively, if you know the impedance in your AC circuit, you can calculate the phase difference you’ll see between voltage and current:

Real power dissipation in AC circuits

Phase angle formula in terms of impedance.

Note that this applies to LTI circuits, where we have the phase being a constant value (i.e., not a function of voltage or current). For a general RLC network, the phase angle may be a function of frequency, even if the circuit is an LTI system. Now that we have a clear definition of phase difference in an AC circuit, we can determine the real power delivered to a load component in an AC circuit.

Power Delivery and Phase Angle

When current and voltage have some phase difference between them, the power in the circuit is represented by a complex number S. The real and imaginary parts of the complex power represent the real delivered power and the reactive power, respectively. The following formula tells you the complex power S, which is written in terms of the complex impedance Z.

Complex impedance and phase angle formula

Complex power in terms of complex impedance.

Note that resistive portions of a circuit always dissipate real power, as we’ll see momentarily, while reactive portions only receive reactive power. Physically, the reactive power represents power that is confined to a reactive element in a circuit (i.e., L and C elements). In contrast, the real power is dissipated as heat in resistive elements.

If you know the phase angle from the impedance or the time difference between current and voltage, then you can calculate the real power delivered to the load. This is defined below in terms of the magnitude of S and the phase angle:

Real power dissipation in AC circuits

Real power delivered to a load component.

To calculate the reactive power, just replace cosine with sine in the above equation. Pay attention to the sign of the phase angle in this calculation, as this will tell you when power is confined to reactive elements versus when it is dissipated in resistive elements. This distinction between apparent power and real power becomes clearer when we examine the current and voltage distribution in a series RLC circuit, as shown in the image below.

Phase angle formula for series RLC circuit

Voltage and current distribution in a series RLC circuit.

In this image, the current and inductor have voltages that are out of phase with each other, so the total reactive power in the LC leg of the circuit is zero. In other words, the C and L elements have opposite reactive behavior at different points in time; one element is generating reactive power while the other is confining it.

Phase Angle from Real and Reactive Power

Another important quantity is the apparent power |S|, which accounts for the real and reactive power together without a phase constant. In another case, you may already know the real power Re[S] and apparent power |S|, and you need to find the phase angle. The ratio of these two quantities is known as the power factor, which is very important in regulated AC-DC conversion systems. The power factor is defined in terms of the phase angle as follows:

Power factor and phase angle formula

Power factor definition in terms of the phase angle.

Ideally, the power factor of a power conversion system should be equal to 1. Because real regulator circuits have a power factor somewhere in the neighborhood of ~0.7, a PFC circuit is normally added at the AC-DC rectification stage to bring the power factor as close to 1 as possible. Calculating the phase angle tells you how you will need to compensate for a low power factor when designing your circuits.

Once you’ve used the phase angle formula and determined what level of power factor correction you need, you can create your board layout with the best PCB layout and design software with a full suite of design tools. Allegro PCB Editor includes the features you need to layout boards for any application, including AC power conversion systems. You can then use Cadence’s analysis tools to simulate and analyze the behavior of your power electronics.



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