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CHAPTER 11 Matrix Algebra. Networks
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conjugate of the product of complex numbers is the product of the conjugates, and that the conjugate of a is a2 and the conjugate of a2 is a, the above equations become " " " " Ia I1 I2 Io " " " " Ib a2 I1 aI2 Io " " " " I c aI 1 a2 I 2 I o Thus eq. (550) becomes PT srp: Va Vb 2 3 " I1 76 " 7 1 5 4 I2 5 32 1 " Io
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551
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Now, in the above equation, since the phase voltage matrix is a row matrix it can also be expressed as the TRANSPOSE of the corresponding column matrix ; doing this, the above row matrix can be put into the following form 3 32 31 2 3 2 02 * V1 Va V1 V2 Vo 1 1 1 B6 7 76 7C 6 7 6 2 2 Va Vb Vc 4 Vb 5 4 aV1 a V2 Vo 5 @4 a a 1 54 V2 5A Vc
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a2 V1 aV2 Vo
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and thus, upon making use of the reversal rule (eq. (492)), we have that 3 2 1 a a2 7 6 Va Vb Vc V1 V2 Vo 4 1 a2 a 5 1 Thus eq. (551) becomes PT srp: V1 V2 2 32 1 a2 76 2 a 54 a a 1 1 1 3 " I1 76 " 7 1 54 I2 5 1 " Io 32
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which, since a3 1, a4 a, and 1 a a2 0, becomes 32 3 2 " 3 0 0 I1 76 " 7 6 PT srp: V1 V2 Vo 4 0 3 0 54 I2 5 3 V1 " 0 0 3 Io thus, nally, " " " PT srp: 3V1 I1 3V2 I2 3Vo Io
3 " I1 6"7 V o 4 I2 5 " Io 552
The meaning of eq. (552) is as follows. In section 10.9 we found that the TOTAL POWER produced in a balanced three-phase system is three times the power per phase (eq. (438)). Equation (552) shows that the total power produced in an unbalanced three-phase system is equal to the simple sum of the powers separately produced by the positive-sequence, negative-sequence, and zero-sequence systems; that is, as far as power is concerned, each system acts independently of the other two. Note that this is an unexpected result, because the principle of superposition does not generally apply to power calculations. (See note of caution following problem 73 in section 5.7.)
* See Fig. 309.
Binary Arithmetic. Switching Algebra
12.1 Analog and Digital Signals. Binary Arithmetic
An ANALOG type of signal has in general a continuous range of amplitude values, such as is illustrated in Fig. 310. A DIGITAL signal, on the other hand, is an ordered sequence of discontinuous pulsetype signals that can have only a limited number of di erent levels of amplitude. If only two di erent levels are allowed, or can be detected, the digital signal is said to be a BINARY ( BY nary ) type signal, the word binary meaning two-valued. The two di erent levels of a binary signal can be said to represent the on and o conditions of the signal, or the presence or absence of a pulse, and can be denoted by 1 and 0, as in Fig. 311.
Fig. 310. Analog signal.
Fig. 311.
Binary digital signal.
In Fig. 311 T is the measured time allotted to one unit of information, which is called a bit ; thus, 9 bits of information are represented in Fig. 311. The signal is said to be binary digital because its two di erent states can be represented by the digits 1 and 0, as shown.
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CHAPTER 12 Binary Arithmetic
It is true that, in the real world, most signals originate in analog form; for instance, the outputs of microphones, TV cameras, and most other sensing devices, are in analog form. You might therefore very well ask Of what use are digital-type signals; why use a digital system at all . In answer to the question, one very important reason is that the internal operations of DIGITAL COMPUTERS are handled in the form of binary digital signals. This is because a digital computer uses integrated circuits containing thousands of transistors operating in the binary on or o mode. Another reason is that it is sometimes bene cial to rst transform an analog signal into a coded binary-type signal before it is fed into a transmission system or channel. This can have a very good e ect if the channel is noisy, because at the receiver it is then only necessary to detect whether a pulse is PRESENT or NOT PRESENT. If this can be done, the original analog signal can then be completely recovered from the binary coded signal, even if the binary signal is mixed with so much noise that it would not be possible to recover the signal if it were in analog form.* In binary work, especially in regard to digital computers, it is necessary to be uent in binary arithmetic, which let us introduce as follows. A digit is a single symbol representing a whole, or integral, quantity. A number is a quantity represented by a group of digits. The number of di erent digits a number system uses is called the base or the radix, R, of the system. Thus the familiar decimal system has the radix ten, using the ten digits 0 through 9. In all practical number systems the value of a digit in a number depends not only on the digit itself but also upon the position of the digit in the number. Consider, as an example, the quantity represented by the decimal system number 2684:735 As you know, the number to the left of the decimal point is the whole or integral part of the quantity, while the number to the right of the decimal point is the fractional part of one unit. Note that the digits in the above decimal number have the following values: the digit 2 has the value 2 103 2000: the digit 6 has the value 6 102 600: the digit 8 has the value 8 101 the digit 4 has the value 4 10
0 1 2
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