Power Factor and Effective Resistance

Original Forms To Preserve

Steinmetz gives the sine-wave AC power equation:

P_o = ei\cos\theta

He also keeps Joule’s law but warns that in AC the r in:

P = i^2 r

may be effective resistance, not merely true ohmic resistance.

Modern Notation

Using RMS values:

P = VI\cos\phi

R_\mathrm{eff} = \frac{P}{I^2}

Steinmetz’s Distinction

Effective resistance represents total real-power expenditure. It can include conductor heating, magnetic hysteresis, dielectric hysteresis, mutual induction effects, eddy currents, leakage, corona, and other real losses.

True ohmic resistance is a narrower conductor property.

Worked Example

If:

V = 1000\ \mathrm{V},\quad I = 10\ \mathrm{A},\quad \cos\phi = 0.8

then:

P = 1000 \times 10 \times 0.8 = 8000\ \mathrm{W}

and:

R_\mathrm{eff} = \frac{8000}{10^2} = 80\ \Omega

That 80 ohms is an equivalent real-power coefficient. It should not be confused with a DC resistance measurement unless the only loss is conductor heating.

Why Modern Language Can Hide This

Modern phasor analysis often says “real part of impedance.” Steinmetz’s “effective resistance” asks what physical loss processes are being represented by that real part.

Ether-Field Interpretive Reading

Interpretive only: this is a strong bridge to field-centered readings because it lets magnetic lag, dielectric loss, and secondary field motion appear as measurable real-power terms. The source-grounded claim is simply that Steinmetz includes losses outside the conductor under effective resistance.