# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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noise signal is of no interest as the signals are only observed in some finite time

interval.

The average power in a random noise signal A(/) observed in the time interval

— t < t < т is given by

P = lira fT {\\{t)\2) dt (A.34)

Г—‘СС 9 T 1

= iim ^- / (\A(f)\2)df (A.35)

Г—TO i.T J-nс

If A( i) is represented as a Fourier series in the time interval -r < t < r then the average power is given by

1 f" / ~

P = lira — I l У] Л(exp!

1 ^ ~r ^P\ = — r-0

V V* ovnf_ , fh

, -1*- S ■' Jj

P2-— :o

,s >

г—

P\--'X> p2 = -oo

x lim / exp[-j2~(p2 - Pi)&] dt (A.37)

r—2T J^r

264

A. Mathematical concepts

1 for p2 = pi 0 otherwise

(Л.38)

Insertion of Equation (A.3a) into (A.3gives

P

(A.39)

Thus the total average power is determined by summing the ensemble average of the magnitude squared Fourier series coefficients at all the relevant frequencies. From Equation (A.39) an average power density is defined as

where it is assumed that r is chosen so large that there exists an integer p such that pf = /. It should be noted that / in Equation (A.40) runs over both positive and negative frequencies.

A.4 References

[1] Haus, H. A. (Chairman of IR.E Subcommittee 7.9 on Noise) et al.: “Representation of noise in linear twoports”, Proc. IRE, vol. 48, pp. 69-74, 1960.

[2] Lighthill, M. J.: “Introduction to Fourier analysis and generalized functions'’, Cambridge University Press, London, 1958.

[3] Papoulis, A.: “Probability, random variables, and stochastic processes”, McGraw-Hill,

p'(f) = <|л(/)|2;

(A.40)

USA, 1965.

в

Expressions for reflection coefficients and exchangeable powers

In the first section of this appendix the reflection coefficients looking into an n-port and also looking into the terminating immittances are derived. In the second section the exchangeable power gain from an arbitrary port is derived.

B.l Derivation of reflection coefficients

Figure B.l: Reflection roeffkients at port

The reflection coefficients connected to port, i are shown in Figure B.l. The reflection coefficient S'^ is given by

B-

S'a = -f- fli.ii

The power waves .4, and Ht are expressed by the port voltage and currcnt which leads to

265

266

В. Expressions for reflection coefficients and exchangeable powers

= V- - Z--1' ,B_2)

V: + Z,I; 4 '

where Zt is the reference impedance at port i. The input impedance at port i is Z,r and then

Zln,; f, + Z; I; Zin, l ~

Zin,i "f" Zi

The reflection coefficient Гг.; is derived from the equation

ZT, - ^ Zt.i + z;

(B.3)

/1, = Гг,; В; + Bts (B.4)

This means that

Гг, = fj (В - 5)

'IbT[=o

is correct, as Br,i - 0 when Vjtl — 0.

Vi+Zilil Zr.. / •• /, /

r'‘ - Vi - z-i,\Vti=0 - -zT,u-z;h

(B.6)

Comparing Equations (B.3) and (B.6) it is seen that S';i has the reference impedance conjugated in the numerator and Гт; in the denominator. Thus care should be shown when using complex reference impedance.

Conjugate match is obtained when

Zt, 1 = Z{T. ^ or Гг, — 61:

BO T»i r*i rA q »-»+ r> t* т o v« iTr^irn nvnvnccnrl nvpVi *3 rin’Q'lK | n

* md iAlV-AUClit yv d v с ca^hjooCu U j

power

As the incident power wave Bj i in Figure B.i is an independent node, can be derived from

A{ — Г Г 1 B; -f- B-J•

В.2. Incident power wave expressed by exchangeable oower

267

and from Equation (B.7)

Вт,, = Ai\g_0 (B.S)

The condition B, = 0 is fulfilled when port i does not reflect power. This means that S't = 0 or that Zin,i — Z' as shown in Figure B.2.

Figure B.2: Reflection free match at port i. The incident power wave at port i, .4;. can now be written [1]

A, = Вт,,

V + г, I,

(B.'J)

2 v/jReii/ijj

From Figure B.2 Vj and I, are expressed by Vj, and then inserted in Equation (B.9):

Вт, =

г: т, ,

—7^ yT,, + 7Т-Г

2 л/jRe[Z,j

Re[Z;] ^T.i

v'|Re[2,]i i'r,i ( Zt,i t Z")

Now (ji?T,:|2/ is derived:

< !Яг.;|‘/ = {Br.iBj:)

Re^ZiKIVV,,!2)

1ВД1\zTA + z'\-

Re[Z,j < jVV.ii2) l ReiZr.;: Rel Z,,

symbol o, is introduced as

Re[Z,] ( 1 for Re[Z,] > 0

|Re[Z,]j “ I -I for RefZ,] < 0

(B.11)

268

В. Expressions for reflection coefficients and exchangeable powers

The second term is the termination's exchangeable power:

P = ( i^.,12} 19s

гй‘ 4Re[ZT,1]

The third term is related to thp rpfiprtion ropffirient of t.hp termination by

4 Rc[ZTti] Re[Zi] (ZTs + Zf ,■) + Z?)

\ZT, + ZfP ~ | ZT,i + Ztf

Zx:,Zi — Zj-^Z* — ZjxZ{ — Z-f, Z"

|Zr,i +

~ Zt- )[Zj’j т Z, i — (Zj\j — Z,)(Z^ — Z*)

!Zr,. + z*j2

1^,- - z,!‘

= i

,Zr, + Z-P

1 - |rT,j- (13.13)

Tims

( t-Br,ij2) = P, • PeS, ■ (1 - ITr.d2) (B.14)

As the left side of Equation (B.14) is always positive the signs of the right side add up to be positive as well. This is shown in Table B.l.

Re[Zr,{] Re[Z,j 1-|Гг,'|2 P'Si Pt <!Bt,|2}

> 0 > 0 > 0 > 0 > 0 > 0

> 0 < 0 < 0 > 0 < 0 > 0

< 0 > 0 < 0 < 0 > 0 > 0

< 0 < 0 > 0 < 0 < 0 > 0

Table B.l: Signs of (j-Вт,;!2) and of its three terms as a function of the signs of ReSZr.i] and Re[Z,-].

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