Complex Quantities Concordance
Complex Quantities
Section titled “Complex Quantities”Concordance status: generated from processed OCR/PDF text. Treat these as source-location aids until each passage is checked against the scan.
156 hits
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10 sources
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48 sections
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Matched Aliases
Section titled “Matched Aliases”complex quantities, complex quantity, imaginary quantities, imaginary quantity
Source Distribution
Section titled “Source Distribution”| Source | Hits | Sections |
|---|---|---|
| Theory and Calculation of Alternating Current Phenomena | 41 | 9 |
| Theory and Calculation of Alternating Current Phenomena | 41 | 10 |
| Theory and Calculation of Alternating Current Phenomena | 31 | 10 |
| Theory and Calculation of Transient Electric Phenomena and Oscillations | 16 | 7 |
| Engineering Mathematics: A Series of Lectures Delivered at Union College | 12 | 2 |
| Theory and Calculation of Electric Circuits | 5 | 2 |
| Theory and Calculation of Electric Apparatus | 4 | 3 |
| Elementary Lectures on Electric Discharges, Waves and Impulses, and Other Transients | 2 | 2 |
| Theoretical Elements of Electrical Engineering | 2 | 1 |
| Elementary Lectures on Electric Discharges, Waves and Impulses, and Other Transients | 2 | 2 |
Section Hits
Section titled “Section Hits”Representative Source Snippets
Section titled “Representative Source Snippets”Chapter 5: Symbolic Method - 13 hit(s)
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... eriod; that is, leading the wave by one-quarter period. Similarly — Multiplying by — j jneans lagging the wave by one-quarter period. Since j^ = - 1, it is j = v^^=n:; and j is the imaginary unit, and the sine wave is represented by a complex imaginary quantity or general number, a ^- jb. As the imaginary unit, j, has no numerical meaning in the system of ordinary numbers, this definition of j = V — 1 does not contradict its original introduction as a distinguishing index. For the Algebra of Complex Quantities ...Chapter 1: The General Number - 11 hit(s)
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... rature with each other can be expressed by the plus si^n, and the result of combination thereby expressed by OB^-BP = 3+2j. THE GENERAL NUMBER. 17 Such a combination of an ordinary number and a quadra- ture number is called a general number or a complex quantity. The quadrature number jh thus enormously extends the field of usefulness of algebra, by affording a numerical repre- sentation of two-dimensional systems, as the plane, by the general number a-\-jh. They are especially useful and impor- tant in electri ...Chapter 5: Symbouc Mbthod - 10 hit(s)
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... ing the wave through one-quarter period. Fig. 24, Similarly, — Multiplying by — / means advancing the wave through -one-quarter period. since y^ = ~ 1, y = V— 1 ; that is, — j is the imaginary unity and the sine wave is represented by a complex imaginary quantity ^ a -\- jb. As the imaginary unit j has no numerical meaning in the system of ordinary numbers, this definition ofy = V— 1 does not contradict its original introduction as a distinguish- ing index. For a more exact definition of this complex imaginary q ...Chapter 30: Quartbr-Fhase System - 9 hit(s)
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... ual distribution of load, but are liable to become un- balanced at unequal distribution of load ; the three-wire quarter-phase system is unbalanced in voltage and phase, even at equal distribution of load. APPENDICES APPENDIX I. ALGEBRA OF COMPLEX IMAGINARY QUANTITIES. INTRODUCTION. 267. The system of numbers, of which the science of algebra treats, finds its ultimate origin in experience. Directly derived from experience, however, are only the absolute integral numbers ; fractions, for instance, are not directly d ...Chapter 5: Symbolic Method - 9 hit(s)
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... ing the wave through one-quarter period. Fig. 24. Similarly, — Multiplying by — j means advancing the wave through one-quarter period. since y'2 = — 1, j = V— 1 ; that is, — j is the imaginary unit, and the sine wave is represented by a complex imaginary quantity, a -+- jb. As the imaginary unit j has no numerical meaning in the system of ordinary numbers, this definition of/ = V— 1 does not contradict its original introduction as a distinguish- ing index. For a more exact definition of this complex imaginary qu ...Chapter 32: Quarter-Phase System - 9 hit(s)
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... al distribution of load, but are liable to become un- balanced at unequal distribution of load ; the three-wire quarter-phase system is unbalanced in voltage and phase, even at equal distribution of load. APPENDICES. APPENDIX I. ALGEBRA OF COMPLEX IMAGINARY QUANTITIES. INTRODUCTION. 296. The system of numbers, of which the science of algebra treats, finds its ultimate origin in experience. Directly derived from experience, however, are only the absolute integral numbers ; fractions, for instance, are not directly d ...Chapter 7: Admittance, Conductance, Susceftance - 5 hit(s)
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... combined give the total E.M.F., — Iz = lWr' + x\ Since E.M.Fs. are combined by adding their complex ex- pressions, we have : The joint impedance of a number of series-connected impe- dances is the sum of the individual impedances , when expressed in complex quantities. In graphical representation impedances have not to be added, but are combined in their proper phase by the law of parallelogram in the same manner as the E.M.Fs. corre- sponding to them. The term impedance becomes incon- venient, however, when dealing ...Chapter 7: Admittance, Conductance, Susceptance - 5 hit(s)
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... .M.F., Ix ; both combined give the total E.M.F., — Since E.M.Fs. are combined by adding their complex ex- pressions, we have : The joint impedance of a number of series-connected impe- dances is the sum of the individual impedances, when expressed in complex quantities. In graphical representation impedances have not to be added, but are combined in their proper phase by the law of parallelogram in the same manner as the E.M.Fs. corre- sponding to them. The term impedance becomes inconvenient, however, when dealing ...Chapter 2: Long-Distance Transmission Line - 5 hit(s)
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... (8) By equation (1), E ld' Y~dl' and substituting herein equation (8) gives E -A,™ + A*-" , (9) 286 TRANSIENT PHENOMENA or, substituting (7), E =\/A1e+vl+A,e-vi . (10) The integration constants A1 and A2 in (8), (9), (10), in general, are complex quantities. The coefficient of the exponent, F, as square root of the product of two complex quantities, also is a complex quantity, therefore may be written V = a - jp, (11) and substituting for F, Z and Y gives (a - j/?)2 = (r - jx) (g - jb), or (a2 - /?2) ...Chapter 8: Admittance, Conductance, Susceptance - 4 hit(s)
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... e produced by the same e.m.f., such as in cases where Ohm's law is expressed in the form, / = I . Z It is preferable, then, to introduce the reciprocal of impe- dance, which may be called the admittance of the circuit, or Z As the reciprocal of the complex quantity, Z = r -{- jx, the admittance is a complex quantity also, or Y = g — jh; it con- sists of the component, g, which respresents the coefficient of current in phase with the e.m.f., or the power or active com- ponent, gE, of the current, in the equation of O ...Chapter 18: Polyphase Induction Motors - 4 hit(s)
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... ective e.m.f. generated by the magnetic field per primary circuit. Counting the time from the moment where the rising mag- netic flux of mutual induction, <J> (flux interlinked with both electric circuits, primary and secondary), passes through zero, in complex quantities, the magnetic flux is denoted by $ = - i$, and the primary generated e.m.f., E = - e; where e = \/2 xn/$ 10~* may be considered as the "active e.m.f. of the motor," or "counter e.m.f." Since the secondary frequency is sf, the secondary induced e.m ...Chapter 13: Ths Alternating^Cnrrent Traxsfobmer - 4 hit(s)
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... 73 ohms ; that is, about one-tenth as large as assumed. Thus the changes of the values of E^y E^, etc., under the different conditions will be very much smaller. A/. TERAA TINC-CVRRENT PHENOMKAA Symbolic Method. 124- In symbolic representation by complex quantities the transformer problem appears as follows : The exciting current, /„, of the transformer depends upon the primary K.M.K., which dcpendance can be rc|> resented by an admittance, the " primary admittance," Y^=^ g^ ■\- j b^, of the transformer. rig. 9 ...Chapter 15: Induction Motob - 4 hit(s)
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... = effective E.M.F. induced by the mag- netic field per primary circuit. Counting the time from the moment where the rising magnetic flux of mutual induction * (flux interlinked with both electric circuits, primary and secondary) passes through zero, in complex quantities, the magnetic flux is denoted by and the primary induced E.M.F., j5 = — ^; where e = V2 TTfiN^ 10~* may be considered as the " Active E.M.F. of the motor." Since the secondary frequency is s Ny the secondary induced E.M.F. (reduced to primary syst ...Chapter 12: Power, And Double Frequency Quantities In General - 4 hit(s)
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... ints, these points representing the abso- lute values of potential (with regard to any reference point chosen as co-ordinate center) and their connection the dif- ference of potential in phase and intensity. Algebraically these vectors are represented by complex quantities. The impedance, admittance, etc., of the circuit is a complex quantity also, in symbolic denotation. Thus current, E.M.F., impedance, and admittance are related by multiplication and division of complex quantities similar as current, E.M.F., resistance, ...Chapter 14: The Alternating-Current Transformer - 4 hit(s)
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... ms ; that is, about one-tenth as large as assumed. Thus the changes of the values of E0, Elt etc., under the different conditions will be very much smaller. 204 ALTERNATING-CURRENT PHENOMENA. Symbolic Method. 134. In symbolic representation by complex quantities the transformer problem appears as follows : The exciting current, 700, of the transformer depends upon the primary E.M.F., which dependance can be rep- resented by an admittance, the " primary admittance," °f tne transformer. Fig. 105. The resista ...Chapter 16: Induction Motor - 4 hit(s)
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... = effective E.M.F. induced by the mag- netic field per primary circuit. Counting the time from the moment where the rising magnetic flux of mutual induction & (flux interlinked with both electric circuits, primary and secondary) passes through zero, in complex quantities, the magnetic flux is denoted by and the primary induced E.M.F., 240 ALTERNATING-CURRENT PHENOMENA. where e= V2irrt7V<I>10-8 maybe considered as the "Active E.M.F. of the motor," or " Counter E.M.F." Since the secondary frequency is s N, the seco ...Chapter 18: Oscillating Currents - 4 hit(s)
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... lar decrement of the oscillating wave. The oscillating wave can be represented by the equation, E = e€"***'^«cos(« - 6). In the example represented by Figs. 130 and 131, we have A = 0.4, a = 0.1435, a = 8.2°. Impedance and Admittance 184. In complex imaginary quantities, the alternating wave, z = e cos (0 — 6)^ is represented by the symbol, fl = e(cos d — j sin ^) = ei — je2» By an extension of the meaning of this symbolic expression, the oscillating wave, JS? = tt"*** cos {<t> — 6), can be expressed by the symbol, ...Chapter 14: The Osni!Raij Aiitebnatina-Cubbent Tbakbfobmsb - 3 hit(s)
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... ^^^ + ^"^"^ ("■' "•'•''"^ I • z, = <7 1+ — ^ ^0 —7-^0 tfTi — y^^i + (^0 -JXo)(go +J^o) /Returning now to the general alternating-current trans- former, we have, by substituting (n + r)« -t- ^ (x, + xy = V, and separating the real and imaginary quantities, £,= -noe\ h + -^^(ro(r, + r)-^sxo(x, + x)) 1136] ALTERNATING'CURRENT TRANSFORMER. 201 + (^0^0 +^o^o) + y -/-iC-f '•o(-^i +^)-^o(n + r)) V\77. + (^0^0 - ^o--^o^o)1 I • Neglecting the exciting current, or rather considering it as a separate ...