Impedance And Reactance

Source: Theory and Calculation of Alternating Current Phenomena - Location: Chapter V, OCR chapter-05 lines 317-383 candidate OCR; scan verification needed View source record

Original Form To Preserve

The OCR candidate shows Steinmetz writing the impedance relation in complex form:

$$Z = r + jx$$

and the AC form of Ohm’s law:

$$E = ZI$$

Variables

Symbol	Meaning	Modern Equivalent
Z	Complex impedance	Z
r	Resistance component	R
x	Reactance component	X
j	Quadrature operator, imaginary unit	j

Modern Notation

$$Z = R + jX$$ $$|Z| = \sqrt{R^2 + X^2}$$ $$\phi = \tan^{-1}\left(\frac{X}{R}\right)$$

Derivation Sketch

1. Represent current as a complex sine-wave quantity.
2. Resistance voltage is in phase with current.
3. Reactance voltage is 90 degrees displaced from current.
4. Add the two components as rectangular components.
5. The result is the complex impedance relation.

Worked Example

If:

$$R = 6\ \Omega,\quad X = 8\ \Omega$$

then:

$$Z = 6 + j8\ \Omega$$ $$|Z| = \sqrt{6^2 + 8^2} = 10\ \Omega$$ $$\phi = \tan^{-1}\left(\frac{8}{6}\right) \approx 53.1^\circ$$

Physical Meaning

The magnitude tells how much voltage is required for a given current. The angle tells how far voltage and current are displaced. The real part represents power loss; the quadrature part represents field storage and return.

Review State

This equation page is mathematically standard and source-aligned, but the exact typography and edition-specific notation still need scan verification.