A Method for Predicting Intermodulation Product Levels

Abstract

Recent designs of H F radio transmission systems have employed high power multichannel RF power amplifiers that feed a single broadband antenna system. These amplifier/antenna combinations operate with bandwidths that encompass the entire H F band and simultaneously amplify as many as 20 to 25 independent signals of arbitrary carrier frequency. The advantages of such an H F radio system include more effective utilization of assets, frequency agility, and greater efficiency. Along with the advantages, however, comes an increase in the number of intermodulation (IM) products that are generated when the multiple signals are amplified within the high power multichannel RF amplifier. The system designer must be knowledgeable of the impact of these intermodulation products. He must be able to determine what interference will be caused (if any) and to do this, he must know both the frequency and the level of the intermodulation products. The method for calculating the frequency of intermodulation products is well known. The means for predicting the level of the intermodulation products is less familiar to the system designer, especially when multiple signals of various power levels are involved and when intermodulation products of several orders and types are of concern. This paper introduces a means of providing a cursory prediction of the level of the intermodulation products generated by a high power multi­ channel RF amplifier and makes an effort to contribute to more effective frequency management techniques. D ESC R IPTIO N O F TH E IM PR OD UCT The transfer characteristic of the power amplifier can be expressed in the familiar form of an exponential series: N v0 = 2 an Vj eq. (1 ) n =1 Where v0 is the amplifier output signal, vj is the input signal and an are the non-linearity coefficients. When the input signal, vj, is the sum of several sustained carrier signals, the input voltage will be of the form: J Vj = 2 Ej cos W;t eq. (2 ) i=i When vi is raised to the various exponents as indicated by eq. (1), intermodulation products (IPr) are generated and take the form: IPr A r Kr EA a E g ^ E q 7 cos 27t (oA±|3B±7 C± . . .) t eq. (3 ) where r is the order of the intermodulation product, i.e., r= a + /? + 7 + ... A r is the transfer function coefficient Ea , E b , E c , etc. are the amplitudes of signals A, B, C, etc. Kr is a trigonometric expansion coefficient determined by the type of intermodulation product, (see Table I). If Ar is known, then the level of IP r may be calculated. However, seldom is this the case, and attempts at measuring Ar have been unsuccessful in the past. TH E PR ED IC TIO N FORM U LA We wish to have a means to calculate the amplitude coefficient of IPr without a numerical value for A r. This suggests that a ratio be used. A r appears in every intermodulation product of order r regardless of type, i.e., intermodulation products of type A + B + C , 2A4-B, 3A, etc. all contain the transfer function coefficient, A 3. Therefore, if we measured a level of one of the 3rd order intermodulation products, we will be able to write a ratio to predict any of the other 3rd order products. This is shown as follows. Let capital letters represent the conditions of equal amplitude signals fully loading the power amplifier under a reference condition and let lower case letters represent the conditions of unequal amplitude signals partially loading the power amplifier and representing the case to be predicted. More specifically, IPr = ArKrEA" Eg*5' Ec 7’ . . . cos 2tr (ctA+/3B+7C+ . . ,)t eq. (4) represents the equal amplitude, fully loaded case and ipr = Ar k r ea a eg^ ec7 , . . Cos 2n (aA+/3B+YC± . . .)t eq. (5) Where Ej is the amplitude and Wj is the frequency of the j**1 signal. 408 CH2116-2/85/0000-408 $1.00 © 1985 IEEE is the unequal amplitude, partially loaded case. « , /?, y, etc. can be different from cl, /S', y ,etc., however a + /?+ 7 + ... = r = a'+ @'+y'+ ... IPr will be used to represent the measured reference from which other data will be extrapolated. One measurement of IPr must be made for each intermodulation order of interest because other orders have different transfer characteristic coefficients. A further condition is that the fully loaded amplifier be operated in a region where eq. ( 1 ) and eq. (3) hold true for all input amplitudes. Under these conditions, then Amplitude of ipr ipr A r kr eaa ec7. .. Amplitude of I Pr IPr A r K r E^a Eg Eq7 ... / k r \ / ea “ e b , 3ec 7 ■ ■ \ iPr = l r ( k J v e a ^' e b ^ Ec 7'.../ eq. (6) expressed in decibels, then kr imr= IMr+ 20 log — + 20 log K r ea eb ec • E a “ E q 7 eq. (7)