# Practical RF System design - Egan W.F

ISBN 0-471-20023-9

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f \Znk + Z22k-\2

and the corresponding power is

Pf =

\Znk + Ro |2 Rllkâ€” \Znk + Z22k-\2 Ro

watts.

The ratio of powers and, therefore of power gains, are therefore

ga

pa 1 Z11k + Z22kâ€” |2/R22kâ€”

gf Pf \Zuk + Ro\2/Ro This equals Eq. (11), as claimed in Section N.4.

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ENDNOTE

'Equation (57) indicates that vâ€ž and iâ€ž in Fig. 3.6 are completely correlated when they represent the isolated noise source in Fig. 3.2. This is verified by circuit analysis, which shows that ek will be produced across Riik, independently of the driving source impedance, if in = ek/Riik and vn = ekZiik/R'ik. But then vn = inZ\\k = inZY, so vn and in are completely correlated.

APPENDIX P

Practical RF System Design. William F. Egan

Copyright Â© 2003 John Wiley & Sons, Inc.

ISBN: 0-471-20023-9

IM PRODUCTS IN MIXERS

This development supports Section 7.3 and is also pertinent to Chapter 4.

In mixing, a relatively weak RF signal,

Va = A Cos Pa(t), (1)

where

p(t) = Mt + d, (2)

is basically raised to the nth power and shifted in phase by mpb(t), resulting in a term given by Eq. (7.8). It follows that, if va should consist of two sinusoids, the results in the mixer output would also appear in this simple form, so that an input signal

V = V1 + V2 = kl cos pi(t) + k2 cos P2(t) (3)

would produce an output proportional to

vn = (Vi + V2)n (4)

with phases additionally shifted by mpb(t). (This is, in general, a fixed phase

shift plus a frequency shift.) Our objective is to determine the amplitudes of

the IMs, resulting from this process, relative to the linear terms. These can be determined by expanding Eq. (4). This expansion is well known to be

n

Vn = (Vi + V2)n =J2 c(n, iWV-1, (5)

i=0

where c(n, i) is the binomial coefficient, the values of which are given in Table P.l.

345

346 APPENDIX P IM PRODUCTS IN MIXERS TABLE P.1 Binomial Coefficients

i

Expanding, we obtain

v"(p(t)) = Y"' c(n,i)kl^k2L 1 cos1 p1(t) cos" 1 p2(t). (6)

i=0

When the signals both have amplitude A, this becomes

n

vn(p(t)) c(n,i)An cosi p1(t) cosn-i p2(t). (7)

i=0

This process applies directly to the IMs of Chapter 4, which are not frequency converted. But it also applies to IMs in mixers if we account for the addition of mpb(t) to the phase, in which case we need vn[p(t) + mpb(t)]. However, since we are interested in the relative amplitude of the terms, we will not bother to show mpb(t) explicitly.

The nth-order spurs are produced by the terms with i = 0 or i = n,

vn = An cosnpj(t), (8)

where j is i or 2. This term can be expanded to give

3 cos pj (t) + cos[3pj (t)]\ n_3

= Aâ€ž|------- J 22 [cosn~3pj(t). (10)

n

APPENDIX P IM PRODUCTS IN MIXERS

347

Repeating this process n â€” 3 more times gives

+ (11)

This last term is the nth-order spur with amplitude given in the spur-level table. The terms not shown are at lower frequencies. This frequency Mj also occurs in other terms in Eq. (4.1) but with amplitudes raised to higher powers. As a result, at sufficiently low signal levels the term shown in Eq. (11) dominates.

For n odd, we can see from Table P.1 that the strongest IMs have i = (n Â± 1)/2. These are also the most troublesome producing frequencies near M1(t) and M2(t). We can expand the corresponding term in Eq. (7), by applying the process that we used to obtain Eq. (11) to each of its two cosines, to give

Vn,(nÂ±1)/2(p(t)) = Anc[n, (n â€” 1)/2] cos(n â€”1)/2 P1(t) cos(n+1)/2 P2(t) (12)

cos \(n â€” 1)P1 (t)/2lcos \(n + 1)p2(t)/2l = â€¢â€¢â€¢ + Anc[n, (n - l)/2]---------^--------------- 2â€ž_2 J (13)

cos([(n â€” 1)P1(t) + (n + 1)p2(t)]/2)

, + cos([(n â€” 1)p1(t) â€” (n + 1)p2(t)]/2)

A c\n, (n â€” 1)/2] â€”--------------------------------------------------------------â€”------------------------------------------------------- (14)

= A/[Â», (n ~ l)/2] | cos({<jo2(l) - + n[<p2(t) + ^i(f)]}/2) 1

2"_1 j + cos({<j0i(O + Viit) + n[<P2(t) â€” ^i(f)]}/2) J

The amplitude of these IMs, relative to the nth-order spur from Eq. (11) is the binomial coefficient

r = c(n, â– (16)

For n = 3, Eq. (15) is

,33 L__/0.5[p2(t) â€” p1(t)] V /0.5[p1(t) + p2(t)]

U3,(1Â°r2) A 4 |C0S V+IA^W +<Pi(t)y +CÂ°S V+lAfeW-<Pi(0]

(17)

3

= A3-{cos[^i(f) + 2^2(0] + cos{^2(0 + \<P2 (t) â€” Wi (?)])}, (18)

representing the IMs f and c or IMs d and g in Fig. 4.6, depending on whether p2 is a parameter of a or b.

With n even, the largest IM will be for i = n/2 and the equivalent of Eq. (15) will be

c (nâ€™ o) f /Â« \ (n \l

n,n/2(<p(t)) ~ A 2n_f |cos^-[^2(0 +<Pl(t)]j +COS y-[_(p2(t) â€” | â€¢

(19)

v

348

APPENDIX P IM PRODUCTS IN MIXERS

The amplitude, relative to the spur is

r = c (n> i) â– ^

We can represent both Eq. (16) and (20) by saying that the ratio of the largest nth-order IMs to the nth order spur is

r = c[n, int(n/2)]. (21)

For n = 2, Eq. (19) is represented by c and e in Fig. 4.2.

The RF is shifted by the LO phase and frequency pb(t) to produce the IF. After the IMs are created, they are shifted by the same amount to accompany

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