Circuit voltage u= (root number 2) ucos (wt+θ); θ is the voltage phase.
Circuit current i= (root number 2) icos (wt+ψ); ψ is the current phase, and the phase difference is φ = θ-ψ;
Then the actual power p of the circuit is: (u, I is the effective value, multiplied by the root number 2 to indicate the maximum value).
p=ui=2UIcos(wt+θ)cos(wt+ψ)
=UIcos(θ-ψ)+UIcos(2wt+θ+ψ) (this step is the triangular sum-product formula).
=UIcosφ+UIcos(2wt+θ+ψ)
As can be seen from the above formula, the actual power includes two parts: one part is always positive, which means that the circuit is always consuming power, that is, the first half is the actual power consumed by the circuit;
The other part is time-varying, which shows that the circuit is constantly exchanging energy with external circuits, only exchanging without consumption, and this part of power is not the power consumed by the circuit.
So in order to represent the actual power consumed by the circuit, we remember that this part of power is active power P, which can be obtained from the above formula: p = uicos φ;
We also define the apparent power of the circuit S=UI, so obviously COS φ = P/S.
Supplement: Phase is a physical quantity defined in vibration and fluctuation phenomena, which reflects the initial information of vibration or fluctuation. For example, when describing the motion of a simple pendulum, it is inevitable to show its initial position, such as the highest point, the lowest point and any point between these two points ... Because the mathematical description of vibration or fluctuation is a trigonometric function: one period represents 2π, then the phase of an initial position corresponds to [0,2π], which is generally defined as [-π,+π] for convenience of expression.