Equation Verification Queue

The records below are the first canonical equation candidates with their current OCR anchors. They remain candidates until the exact printed equation, variables, signs, primes, subscripts, and surrounding definitions are checked against the scan.

Velocity, Frequency, and Wave Length

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Field	Value
Equation ID	`rli-velocity-frequency-wavelength`
Source	Radiation, Light and Illumination
Chapter	Nature And Different Forms Of Radiation
Source location	Lecture I, cleaned OCR lines 248-259 and 697-703
Original form	`f = S / lambda; equivalently S = f lambda`
Modern form	`v = f lambda`
Candidate status	source-located candidate
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Nature And Different Forms Of Radiation, lines 248-259

246: NATURE AND DIFFERENT FORMS OF RADIATION. 1
247:
248: The frequency of radiation follows from the velocity of light,
249: and the wave length.
250:
251: The average wave length of visible radiation, or light, is about
252: lw = 60 microcentimeters,* that is, 60 X 10~8 cm. (or about
253: ^<y^<5-<y in.) and since the speed is S = 3 X 1010 cm. the frequency
254:
255: a
256:
257: is / = r- = 500 X 1012, or 500 millions of millions of cycles per
258:
259: LW
260:
261: second, that is, inconceivably high compared with the frequencies

OCR context for Nature And Different Forms Of Radiation, lines 697-703

695:
696: Zero point chosen at c = 128 cycles per second.
697: Speed of radiation S = 3 X lu10 cm.
698:
699:
700: Cycles.
701:
702: Wave Length in Air
703: (or Vacuum).
704:
705: Octave: Q^/

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Rectangular Complex Quantity

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Field	Value
Equation ID	`ac-symbolic-rectangular-form`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Symbolic Method
Source location	Chapter V, Symbolic Method, cleaned OCR lines 285-290
Original form	`I = i + ji'`
Modern form	`I = I_real + j I_quadrature`
Candidate status	source-located candidate
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Symbolic Method, lines 285-290

283: reduced to the elementary algebra of complex quantities.
284:
285: 31. If / = I + ji' is a sine wave of alternating current, and
286: r is the resistance, the voltage consumed by the resistance is in
287: phase with the current, and equal to the product of the current
288: and resistance. Or
289:
290: rl = ri -\- jri'.
291:
292: If L is the inductance, and x = 2x/L the inductive react-

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The Operator j

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Field	Value
Equation ID	`ac-symbolic-operator-j`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Symbolic Method
Source location	Chapter V, Symbolic Method, cleaned OCR lines 317-333 and promoted Fig. 24 crop
Original form	`j^2 = -1`
Modern form	`j = sqrt(-1), used as a 90 degree rotation operator in phasor analysis`
Candidate status	source-located candidate
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Symbolic Method, lines 317-333

315: jxl = jxi — xi' .
316:
317: Hence, the voltage required to overcome the resistance, r, and
318: the reactance, x, is
319:
320: {r -\- jx)I;
321: that is,
322:
323: Z = r •\- jx is the expression of the impedance of t he circuit
324: in complex quantities.
325:
326: Hence, if / = ^ + ji' is the current, the voltage required to
327: overcome the impedance, Z = r -\- jx, is
328:
329: E ^ ZI = {r+ jx) {i + ji')
330: = {ri + j^xi') -\- j{ri' + xi) ;
331: hence, since j^ = — 1
332:
333: E = (ri — xi') + j(ri' + xi) ;
334: or, ii E = e -\- je' is the impressed voltage and Z = r -\- jx the
335: impedance, the current through the circuit is

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Inductive Reactance

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Field	Value
Equation ID	`ac-inductive-reactance`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Symbolic Method
Source location	Chapter V, Symbolic Method, cleaned OCR lines 292-315 and 373-376
Original form	`x = 2 pi f L`
Modern form	`X_L = omega L = 2 pi f L`
Candidate status	source-located candidate; OCR symbol defects present
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Symbolic Method, lines 292-315

290: rl = ri -\- jri'.
291:
292: If L is the inductance, and x = 2x/L the inductive react-
293: ance, the e.m.f. produced by the reactance, or the counter e.m.f.
294:
295: 1 In this representation of the sine wave by the exponential expression of
296: the complex quantity, the angle 0 necessarily must be expressed in radians,
297: and not in degrees, that is, with one complete revolution or cycle as 2 tt. or
298:
299: 180
300: with — = 57.3° as unit.
301:
302:
303: SYMBOLIC METHOD 35
304:
305: of self-induction, is the product of the current and reactance,
306: and lags in phase 90° behind the current; it is, therefore, repre-
307: sented by the expression
308:
309: — jxl = — jxi -\- xi'.
310:
311: The voltage required to overcome the reactance is consequently
312: 90° ahead of the current (or, as usually expressed, the current
313: lags 90° behind the e.m.f.), and represented by the expression
314:
315: jxl = jxi — xi' .
316:
317: Hence, the voltage required to overcome the resistance, r, and

OCR context for Symbolic Method, lines 373-376

371: both may be combined in the name reactance.
372:
373: We therefore have the conclusion that
374: If r = resistance and L — inductance,
375:
376: thus X = 2 TcJL = inductive reactance.
377:
378: If C = capacity, Xi = ^ — 77-, = condensive reactance,

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Condensive Reactance

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Field	Value
Equation ID	`ac-condensive-reactance`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Symbolic Method
Source location	Chapter V, Symbolic Method, cleaned OCR lines 351-380
Original form	`x_1 = 1 / (2 pi f C)`
Modern form	`X_C = 1 / (omega C) = 1 / (2 pi f C)`
Candidate status	source-located candidate; OCR symbol defects present
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Symbolic Method, lines 351-380

349: I i + ji' i^ -F i'^ i^ -\- i"- "^ ^ i- + i'^ '
350:
351: 32. If C is the capacity of a condenser in series in a circuit
352: in which exists a current I = i + ji' , the voltage impressed upon
353:
354: the terminals of the condenser is E = ^ .^, 90° behind the cur-
355:
356:
357: 36 ALTERNATING-CURRENT PHENOMENA
358:
359: ji
360:
361: rent; and may be represented by — o'— 779 or — jxj, where
362:
363: Ztt/u
364:
365: ^1 ~ o — Tr* i^ ^^6 condensive reactance or condensance of the
366: Z irjL
367:
368: condenser.
369:
370: Condensive reactance is of opposite sign to inductive reactance;
371: both may be combined in the name reactance.
372:
373: We therefore have the conclusion that
374: If r = resistance and L — inductance,
375:
376: thus X = 2 TcJL = inductive reactance.
377:
378: If C = capacity, Xi = ^ — 77-, = condensive reactance,
379:
380: Z — r -{- j(x — Xi) is the impedance of the circuit.
381: Ohm's law is then re-established as follows:
382:

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Complex Impedance

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Field	Value
Equation ID	`ac-impedance-complex-form`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Symbolic Method
Source location	Chapter V, Symbolic Method, cleaned OCR lines 317-333
Original form	`Z = r + jx`
Modern form	`Z = R + jX`
Candidate status	source-located candidate
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Symbolic Method, lines 317-333

315: jxl = jxi — xi' .
316:
317: Hence, the voltage required to overcome the resistance, r, and
318: the reactance, x, is
319:
320: {r -\- jx)I;
321: that is,
322:
323: Z = r •\- jx is the expression of the impedance of t he circuit
324: in complex quantities.
325:
326: Hence, if / = ^ + ji' is the current, the voltage required to
327: overcome the impedance, Z = r -\- jx, is
328:
329: E ^ ZI = {r+ jx) {i + ji')
330: = {ri + j^xi') -\- j{ri' + xi) ;
331: hence, since j^ = — 1
332:
333: E = (ri — xi') + j(ri' + xi) ;
334: or, ii E = e -\- je' is the impressed voltage and Z = r -\- jx the
335: impedance, the current through the circuit is

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Symbolic Ohm’s Law

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Field	Value
Equation ID	`ac-symbolic-ohms-law`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Symbolic Method
Source location	Chapter V, Symbolic Method, cleaned OCR lines 326-383
Original form	`E = ZI, I = E / Z, Z = E / I`
Modern form	`V = ZI`
Candidate status	source-located candidate
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Symbolic Method, lines 326-383

324: in complex quantities.
325:
326: Hence, if / = ^ + ji' is the current, the voltage required to
327: overcome the impedance, Z = r -\- jx, is
328:
329: E ^ ZI = {r+ jx) {i + ji')
330: = {ri + j^xi') -\- j{ri' + xi) ;
331: hence, since j^ = — 1
332:
333: E = (ri — xi') + j(ri' + xi) ;
334: or, ii E = e -\- je' is the impressed voltage and Z = r -\- jx the
335: impedance, the current through the circuit is
336:
337: I _^ _e^je'_
338: Z r + jx'
339:
340: or, multiplying numerator and denominator by (r — jx) to
341: eliminate the imaginary from the denominator, we have
342:
343: Y _ (e -\- je') (r — jx) _er -\- e'x . e'r — ex ^
344:
345: or, if £" = e + je' is the impressed voltage and 7 = t + ji' the
346: current in the circuit, its impedance is
347:
348: jE ^ e + je' ^ (e + je') (i - ji') ^ ei + e'i' . e'i - ei'
349: I i + ji' i^ -F i'^ i^ -\- i"- "^ ^ i- + i'^ '
350:
351: 32. If C is the capacity of a condenser in series in a circuit
352: in which exists a current I = i + ji' , the voltage impressed upon
353:
354: the terminals of the condenser is E = ^ .^, 90° behind the cur-
355:
356:
357: 36 ALTERNATING-CURRENT PHENOMENA
358:
359: ji
360:
361: rent; and may be represented by — o'— 779 or — jxj, where
362:
363: Ztt/u
364:
365: ^1 ~ o — Tr* i^ ^^6 condensive reactance or condensance of the
366: Z irjL
367:
368: condenser.
369:
370: Condensive reactance is of opposite sign to inductive reactance;
371: both may be combined in the name reactance.
372:
373: We therefore have the conclusion that
374: If r = resistance and L — inductance,
375:
376: thus X = 2 TcJL = inductive reactance.
377:
378: If C = capacity, Xi = ^ — 77-, = condensive reactance,
379:
380: Z — r -{- j(x — Xi) is the impedance of the circuit.
381: Ohm's law is then re-established as follows:
382:
383: E = ZI, I = y, Z = -J-
384:
385: The more general form gives not only the intensity of the wave

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Admittance As Reciprocal Impedance

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Field	Value
Equation ID	`ac-admittance-reciprocal`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Admittance, Conductance, Susceptance
Source location	Chapter VIII, Admittance, Conductance, Susceptance, cleaned OCR lines 48-222
Original form	`Y = 1 / Z = g - jb`
Modern form	`Y = 1 / Z; sign convention depends on reactance convention`
Candidate status	source-located candidate
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Admittance, Conductance, Susceptance, lines 48-222

46: ADMITTANCE, CONDUCTANCE, SUSCEPTANCE 55
47:
48: 49. In alternating-current circuits, instead of the term resist-
49: ance we have the term impedance, Z = r -\- jx, with its two
50: components, the resistance, r, and the reactance, x, in the formula
51: of Ohm's law, E = IZ. The resistance, r, gives the component
52: of e.m.f. in phase with the current, or the power component
53: of the e.m.f., Ir; the reactance, x, gives the component of the
54: e.m.f. in quadrature with the current, or the wattless component
55: of e.m.f., Ix; both combined give the total e.m.f.,
56:
57: Iz = iVr^ + x^.
58: Since e.m.fs. are combined by adding their complex expressions,
59: we have:
60:
61: The joint impedance of a number of series-connected impedances
62: is the sum of the individual impedances, when expressed in com-
63: plex quantities.
64:
65: In graphical representation impedances have not to be added,
66: but are combined in their proper phase by the law of parallelo-
67: gram in the same manner as the e.m.fs. corresponding to them.
68:
69: The term impedance becomes inconvenient, however, when
70: dealing with parallel-connected circuits; or, in other words, when
71: several currents are produced by the same e.m.f., such as in
72: cases where Ohm's law is expressed in the form,
73:
74: / = I
75: . Z
76:
77: It is preferable, then, to introduce the reciprocal of impe-
78: dance, which may be called the admittance of the circuit, or
79:
80: Z
81:
82: As the reciprocal of the complex quantity, Z = r -{- jx, the
83: admittance is a complex quantity also, or Y = g — jh; it con-
84: sists of the component, g, which respresents the coefficient of
85: current in phase with the e.m.f., or the power or active com-
86: ponent, gE, of the current, in the equation of Ohm's law,
87:
88: I =YE ={g- jh)E,
89:
90: and the component, h, which represents the coefficient of current
91: in quadrature with the e.m.f., or wattless or reactive component,
92: hE, of the current.
93:
94: g is called the conductance, and h the susceptance, of the cir-
95: cuit. Hence the conductance, g, is the power component, and
96:
97:
98: 56 ALTERNATING-CURRENT PHENOMENA
99:
100: the susceptance, h, the wattless component, of the admittance,
101: Y = g ~ jb, while the numerical value of admittance is
102:
103: y = Vg' + h^;
104:
105: the resistance, r, is the power component, and the reactance,
106: X, the wattless component, of the impedance, Z = r -^ jx, the
107: numerical value of impedance being
108:
109: z = Vr^ + x^.
110:
111: 50. As shown, the term admittance implies resolving the cur-
112: rent into two components, in phase and in quadrature with the
113: e.m.f., or the power or active component and the wattless or
114: reactive component; while the term impedance implies resolving
115: the e.m.f. into two components, in phase and in quadrature
116: with the current, or the power component and the wattless or
117: reactive component.
118:
119: It must be understood, however, that the conductance is not
120: the reciprocal of the resistance, but depends upon the reactance
121: as well as upon the resistance. Only when the reactance x = 0,
122: or in continuous-current circuits, is the conductance the recip-
123: rocal of resistance.
124:
125: Again, only in circuits with zero resistance (r = 0) is the
126: susceptance the reciprocal of reactance; otherwise, the suscep-
127: tance depends upon reactance and upon resistance.
128:
129: The conductance is zero for two values of the resistance:
130:
131: 1. If r = oo^ or a: = co ^ since in this case there is no current,
132: and either component of the current = 0.
133:
134: 2. If r = 0, since in this case the current in the circuit is in
135: quadrature with the e.m.f., and thus has no power component.
136:
137: Similarly, the susceptance, b, is zero for two values of the
138: reactance:
139:
140: 1. If a; = 00, or r = oo .
141:
142: 2. Ux = 0.
143:
144: From the definition of admittance, Y = g — jb, as the recip-
145: rocal of the impedance, Z = r -\- jx,
146: we have
147:
148: \^
149: f
150:
151: or, multiplying numerator and denominator on the right side by
152: (r - jx),
153:
154: h — r — jx
155:
156: ■ ^ ~ 3 - (r+jx) (r - jx)'
157:
158:
159: ADMITTANCE, CONDUCTANCE, SUSCEPTANCE 57
160: hence, since
161:
162:
163: (r +
164:
165: jx) (r -
166:
167: jx) =
168:
169: _ J.2 _j_ 2^2
170:
171: =
172:
173: ^^
174:
175: jb =
176:
177: r
178:
179: -^r
180:
181: X
182:
183: r
184:
185: 22
186:
187: . aj
188:
189: J.2 _|_ -J.2
190:
191: ■2 + a;2
192:
193: •^2^
194:
195: or
196:
197:
198: 9 ^2 _^ 2-2
199:
200: 0 = o 1 o 2'
201:
202:
203: and conversely
204:
205:
206: r =
207:
208:
209: J.2
210:
211: g'
212:
213: + 62
214: 6
215:
216: 2/'
217: ^^
218:
219: •" ~ ^2 _|_ 52 - ^2
220:
221: By these equations, the conductance and susceptance can be
222: calculated from resistance and reactance, and conversely.
223: Multiplying the equations for g and r, we get
224:

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Conductance and Susceptance From Impedance

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Field	Value
Equation ID	`ac-admittance-components`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	needs chapter routing
Source location	Chapter VIII relation plus algebraic reciprocal; existing public derivation
Original form	`g = r / (r^2 + x^2), b = x / (r^2 + x^2)`
Modern form	`For Z = R + jX and Y = G - jB: G = R/(R^2+X^2), B = X/(R^2+X^2)`
Candidate status	mathematical reconstruction from source relation
Links	Equation page - Source overview - External source

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AC Power Equation

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Field	Value
Equation ID	`ac-power-factor-equation`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Introduction
Source location	Chapter I, Introduction, cleaned OCR lines 248-254
Original form	`P_o = ei cos theta`
Modern form	`P = VI cos phi`
Candidate status	source-located candidate
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Introduction, lines 248-254

246: more fully discussed in Chapter VIII.
247:
248: In alternating-current circuits the power equation contains
249: a third term, which, in sine waves, is the cosine of the angle of
250: the difference of phase between e.m.f. and current:
251:
252: Po = ei cos d.
253: Consequently, even if e and i are both large, Po may be very
254: small, if cos d is small, that is, 6 near 90°.
255:
256: Kirchhoff's laws become meaningless in their original form,

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Effective Resistance From Real Power

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Field	Value
Equation ID	`ac-effective-resistance-power`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Introduction, Effective Resistance And Reactance
Source location	Chapter I, Introduction, cleaned OCR lines 236-246 and Chapter XII lines 64-85
Original form	`P = i^2 r`
Modern form	`R_eff = P / I^2`
Candidate status	source-located candidate
Links	Equation page - Source overview - Source text 1 - Source text 2 - Workbench 1 - Workbench 2 - External source

OCR context for Introduction, lines 236-246

234: later chapters.
235:
236: 5. In Joule's law, P = i^r, r is not the true ohmic resistance,
237: but the "effective resistance;" that is, the ratio of the power
238: component of e.m.f. to the current. Since in alternating-cur-
239: rent circuits, in addition to the energy expended iii the ohmic re-
240: sistance of the conductor, energy is expended, partly outside,
241: partly inside of the conductor, by magnetic hysteresis, mutual
242: induction, dielectric hysteresis, etc., the effective resistance,
243: r, is in general larger than the true resistance of the conductor,
244: sometimes many time larger, as in transformers at open sec-
245: ondary circuit, and is no longer a constant of the circuit. It is
246: more fully discussed in Chapter VIII.
247:
248: In alternating-current circuits the power equation contains

OCR context for Effective Resistance And Reactance, lines 64-85

62: is the effective suscepta7ice of the circuit.
63:
64: While the true ohmic resistance represents the expenditure
65: of power as heat inside of the electric conductor b}^ a current
66: of uniform density, the effective resistance represents the total
67: expenditure of power.
68:
69: Since in an alternating-current circuit, in general power is
70: expended not only in the conductor, but also outside of it,
71: through hysteresis, secondary currents, etc., the effective resist-
72: ance frequently differs from the true ohmic resistance in such
73: way as to represent a larger expenditure of power.
74:
75: In dealing with alternating-current circuits, it is necessarj-,
76: therefore, to substitute everywhere the values "effective re-
77: sistance," "effective reactance," "effective conductance," and
78: "effective susceptance," to make the calculation applicable to
79: general alternating-current circuits, such as inductive reactances
80: containing iron, etc.
81:
82: While the true ohmic resistance is a constant of the circuit,
83: depending only upon the temperature, but not upon the e.m.f.,
84: etc., the effective resistance and effective reactance are, in gen-
85: eral, not constants, but depend upon the e.m.f., current, etc.
86: This dependence is the cause of most of the difficulties met in
87: dealing analytically with alternating-current circuits containing

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Capacity Susceptance

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Field	Value
Equation ID	`ac-dielectric-capacity-susceptance`
Source	Theory and Calculation of Alternating Current Phenomena
Chapter	Dielectric Losses
Source location	Chapter XIV, Dielectric Losses, cleaned OCR lines 209-221 and 595-599
Original form	`b = 2 pi f C`
Modern form	`B_C = omega C = 2 pi f C`
Candidate status	source-located candidate
Links	Equation page - Source overview - Source text 1 - Workbench 1 - External source

OCR context for Dielectric Losses, lines 209-221

207: the corresponding values of the second layer.
208:
209: It is then :
210:
211: yA
212: g = -y- = electric conductance
213:
214: kA
215: C = -J- = electrostatic capacity of the layer
216:
217:
218: of dielectric, hence:
219:
220: 2 irfk A
221: b = 2irfC = — J — = capacity susceptance, and
222:
223:

OCR context for Dielectric Losses, lines 595-599

593: part of circuit:
594:
595: where g is the effective conductance of the dielectric circuit, or
596: the energy component of the admittance, representing the energy
597: consumption by leakage, dielectric hysteresis, corona, etc., and h
598: = 2 tt/C is the capacity susceptance. Instead of the admittance
599: Y, its reciprocal, the impedance Z = r — jx, may be used.
600:
601: The main differences between the dielectric and the electro-

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