Is Power factor the same as cos φ?
The formula S * cos φ = P is only valid with sinusoidal values. However, in today’s systems at least the current is far away from being sinusoidal. The “φ” is the phase angle between the fundamental waves of current and voltage.
If there are also some harmonics included, the signal shape in the systems can be expressed by a sum of fundamental waves and integer harmonics. So the calculation becomes simpler and active power can be expressed as:
P = Σ (Un *In*cos φn)
With: n = harmonics,Un = RMS of the n.th harmonic of U, In also Un.
A simple experiment:
Sinusoidal voltage and distorted current as seen here:
It is a real picture of line power. Voltage is sinusoidal, but not current. It is also easy to see that the fundamental waves of voltage and current almost have unity phase.
Because the voltage is sinusoidal,Un = 0 V for n > 1, it means that only on the fundamental components contribute to the RMS value (230 V) and also to the active power.
Result: P = U * I1 *cos φ1. With this the above equation reduces to the fundamental components (the product would be 0 for all n > 1).
Furthermore the (fundamental displacement) reactive power Q1 = 0, because there is no phase shift between U and I.
The apparent power is defined as:
S = Urms * Irms = (Σ (Un²))0,5*(Σ (In²))0,5
Because the current is considerable distortedIn ≠ 0 for n > 1!
For this reason S > P and the power factor λ = P/S ≠ cos φ.
So, the power triangle is a 3-dimensional polyhedron or a universal triangular:
S² = P² + Q1² + Qd²
Qtot = Total reactive power
Q1 = Fundamental displacementreactive power
Qd = Distortion reactive power (often also referred with “D”)
Cos φ results from the ratio of effective power (P) to fundamental apparent power (S1).
Power factor λ= cos Φ results from the ratio of effective power (P) to total apparent power (S) from fundamental and harmonics.