Freescale Semiconductor AN721 Application Note Rev. 1.1, 10/2005
NOTE: The theory in this application note is still applicable, but some of the products referenced may be discontinued. Impedance Matching Networks Applied to RF Power Transistors
By: B. Becciolini
1. INTRODUCTION Rp XP Some graphic and numerical methods of impedance matching will be reviewed here. The examples given will refer to high frequency power amplifiers. IND. Although matching networks normally take the form of f filters and therefore are also useful to provide frequency f f f fS fP discrimination, this aspect will only be considered as a S P CAP. corollary of the matching circuit. Matching is necessary for the best possible energy transfer from stage to stage. In RF-power transistors the Figure 1. Input Impedance of RF-Power Transistors as input impedance is of low value, decreasing as the power a Function of Frequency increases, or as the chip size becomes larger. This impedance must be matched either to a generator — of LS RBB generally 50 ohms internal impedance — or to a preceding stage. Impedance transformation ratios of 10 or even 20 are not rare. Interstage matching has to be made between two ZIN CC CDE CTE RE complex impedances, which makes the design still more difficult, especially if matching must be accomplished over a wide frequency band. Where: RE = emitter diffusion resistance 2. DEVICE PARAMETERS CDE,CTE = diffusion and transition capacitances of the emitter junction 2.1 INPUT IMPEDANCE RBB = base spreading resistance C = package capacitance The general shape of the input impedance of RF-power C L = base lead inductance transistors is as shown in Figure 1. It is a large signal S parameter, expressed here by the parallel combination of Figure 2. Equivalent Circuit for the Input Impedance of a resistance Rp and a reactance Xp (Ref. (1)). RF-Power Transistors The equivalent circuit shown in Figure 2 accounts for the behavior illustrated in Figure 1. QIN is an important parameter to consider for broadband With the presently used stripline or flange packaging, matching. Matching networks normally are low-pass or most of the power devices for VHF low band will have their pseudo low-pass filters. If QlN is high, it can be necessary to use band-pass filter type matching networks and to allow Rp and Xp values below the series resonant point fs. The input impedance will be essentially capacitive. insertion losses. But broadband matching is still possible. Most of the VHF high band transistors will have the series This will be discussed later. resonant frequency within their operating range, i.e. be 2.2 OUTPUT IMPEDANCE purely resistive at one single frequency fs, while the parallel resonant frequency fp will be outside. The output impedance of the RF-power transistors, as Parameters for one or two gigahertz transistors will be given by all manufacturers’ data sheets, generally consists beyond fs and approach fp. They show a high value of Rp of only a capacitance COUT. The internal resistance of the and Xp with inductive character. transistor is supposed to be much higher than the load and A parameter that is very often used to judge on the is normally neglected. In the case of a relatively low internal broadband capabilities of a device is the input Q or QlN resistance, the efficiency of the device would decrease by defined simply as the ratio Rp/Xp. Practically QlN ranges the factor: around 1 or less for VHF devices and around 5 or more for 1+RL/RT microwave transistors.
Freescale Semiconductor, Inc., 1993, 2005, 2009. All rights reserved. AN721 RF Application Information Freescale Semiconductor 1 where RL is the load resistance, seen at the collector-emitter 4. MATCHING NETWORKS terminals, and RT the internal transistor resistance equal to: In the following matching networks will be described by 1 , order of complexity. These are ladder type reactance T,(CTC +CDC) networks. defined as a small signal parameter, where: The different reactance values will be calculated and determined graphically. Increasing the number of reactances = transit angular frequency T broadens the bandwidth. However, networks consisting of C +C = transition and diffusion capacitances TC DC more than four reactances are rare. Above four reactances, at the collector junction the improvement is small. The output capacitance COUT, which is a large signal 4.1 NUMERICAL DESIGN parameter, is related to the small signal parameter CCB,the collector-base transition capacitance. 4.1.1 Two-Resistance Networks Since a junction capacitance varies with the applied voltage, COUT differs from CCB in that it has to be averaged Resistance terminations will first be considered. Figure 4 over the total voltage swing. For an abrupt junction and shows the reactive L-section and the terminations to be assuming certain simplifications, COUT =2CCB. matched. Figure 3 shows the variation of COUT with frequency. X2 COUT decreases partly due to the presence of the collector lead inductance, but mainly because of the fact that the R R X R = 1 base-emitter diode does not shut off anymore when the 1 1 2 n operating frequency approaches the transit frequency fT. n>1
Figure 4. Two-Reactance Matching Network COUT Matching or exact transformation from R2 into R1 occurs at a single frequency fo. At fo,X1 and X2 are equal to:
R2 1 X1 = R1 =R1 R1 -- R 2 n--1 f n--1 Figure 3. Output Capacitance C as a Function X2 = R2 (R1 -- R 2)=R1 OUT 1 of Frequency At fo:X1 X2 =R1 R2
X1 and X2 must be of opposite sign. The shunt reactance is 3. OUTPUT LOAD in parallel with the larger resistance. In the absence of a more precise indication, the output The frequency response of the L-section is shown in Figure 5, where the normalized current is plotted as a load RL is taken equal to: function of the normalized frequency. 2 [VCC -- V CE(sat)] RL = 2P OUT I2 1.0 2 with VCE(sat) equal to 2 or 3 volts, increasing with frequency. Io 4 2 5 Qo =1 The above equation just expresses a well-known relation, 0.8 6 but also shows that the load, in first approximation, is not 8 10 4 Qo =2 related to the device, except for VCE(sat). The load value 0.6 5 is primarily dictated by the required output power and the 6 8 peak voltage; it is not matched to the output impedance of 0.4 10 Q =3 the device. o At higher frequencies this approximation becomes less 0.5 0.7 0.9 1.1 1.3 1.5 exact and for microwave devices the load that must be f/fo LOW-PASS — fo/f HIGH-PASS presented to the device is indicated on the data sheet. This R1 X2 parameter will be measured on all Freescale RF-power devices in the future. Strictly speaking, impedance matching is accomplished E X1 R2 only at the input. Interstage and load matching are more I2 impedance transformations of the device input impedance and of the load into a value RL (sometimes with additional n=R1/R2 reactive component) that depends essentially on the power Q = n--1 demanded and the supply voltage. o Figure 5. Normalized Frequency Response for the L-Section in Low-Pass or High-Pass Form AN721 RF Application Information 2 Freescale Semiconductor If X1 is capacitive and consequently X2 inductive, then: f R f 1 o 2 o X X X =X -- X X1 =-- R1 =-- R1 R2 int ext ext 2 int f R1 -- R 2 f n--1
f f n--1 and X2 = R2 (R1 -- R 2)= R1 fo fo 1 X X The normalized current absolute value is equal to: X = 1 int R1 X X ext int ext Xin -- X1 I2 2n = Io f4 f2 (n -- 1)2 -- 2 +(n+1)2 fo fo Figure 6. Termination Reactance Compensation
nE 4.1.1.1 Use of transmission lines and inductors where Io = , and is plotted in Figure 5 (Ref. (2)). 2 R 1 In the preceding section, the inductance was expected If X1 is inductive and consequently X2 capacitive, the only to be realized by a lumped element. A transmission line can change required is a replacement of f by fo and vice-versa. be used instead (Fig. 7). The L-section has low pass form in the first case and high-pass form in the second case. R1 L R1 Zo(,L)
TheQofthecircuitatfo is equal to:
X2 R1 C R2 C R2 Qo = = =n--1 R2 X1 For a given transformation ratio n, there is only one (a) (b) possible value of Q. On the other hand, there are two symmetrical solutions for the network, that can be either a Figure 7. Use of a Transmission Line in the L-Section low-pass filter or a high-pass filter. The frequency fo does not need to be the center As can be seen from the computed selectivity curves frequency, (f1 +f2)/2, of the desired band limited by f1 (Fig. 8) for the two configurations, transmission lines result and f2. in a larger bandwidth. The gain is important for a In fact, as can be seen from the low-pass configuration transmission line having a length L = /4 ( =90) and a of Figure 5, it may be interesting to shift f toward the high o characteristic impedance Zo= R1 R2. It is not significant band edge frequency f2 to obtain a larger bandwidth w, where for lines short with respect to /4. One will notice that there 2(f1 +f2) is an infinity of solutions, one for each value of C, when using w= transmission lines. f2 -- f 1 This will, however, be at the expense of poorer harmonic 4.1.2 Three-Reactance Matching Networks rejection. The networks which will be investigated are shown in Example: Figure 9. They are made of three reactances alternatively For a transformation ratio n = 4, it can be determined from connected in series and shunt. the above relations: A three-reactances configuration allows to make the quality factor Q of the circuit and the transformation ratio n Bandwidth w 0.1 0.3 =R/R independent of each other and consequently to Max insertion losses 0.025 0.2 2 1 choose the selectivity between certain limits. X /R 1.730 1.712 1 1 For narrow band designs, one can use the following If the terminations R1 and R2 have a reactive component formulas (Ref. (5) AN267, where tables are given): X, the latter may be taken as part of the external reactance Network (a): as shown in Figure 6. This compensation is applicable as long as XC1 =R1/Q Q must be first selected
XINT R1 R1/R2 QINT = or AN721 RF Application Information Freescale Semiconductor 3 POUT rel. 1.0 .9 .8 (b) .7 .6 .5 .4 (a) .3 .2 .1 f/fo 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 RG L RG Z() C RL RL (a) (b) TRANSFORMATION RATIO n = 10 Figure 8. Bandwidth of the L-Section for n = 10 (a) with Lumped Constants (b) with a Transmission Line (/4) (a) (b) Network (c): XL XL1 XL2 XL1 =Q R1 Q must be first selected R 2 R1 XC1 XC2 R2 R1 XC1 2 R1 (1 + Q ) XC2 =A R2 A= -- 1 R2 B 2 XC1 = B=R1 (1 + Q ) (c) Q--A X X L C2 The network which yields the most practical component values, should be selected for a given application. R2 >R1 The three-reactance networks can be thought of as being R1 XC1 R2 formed of a L-section (two reactances) and of a compensation reactance. The L-section essentially performs the impedance transformation, while the additional reactance Figure 9. Three-Reactance Matching Networks compensates for the reactive part of the transformed impedance over a certain frequency band. Network (b): Figure 10 shows a representation in the Z-plane of the circuit of Figure 9 (a) split into two parts R1 -- C 1 -- L 1 and XL1 =R1Q Q must be first selected C2 -- R 2. 2 XL2 =R2 BA=R1 (1 + Q ) A A XC1 = B= -- 1 Q+B R2 AN721 RF Application Information 4 Freescale Semiconductor L1 Again, if one of the terminations has a reactive component, the latter can be taken as a part of the matching network, provided it is not too large (see Fig. 6). R1 C1 C2 R2 L2 C1 Z Z 2 R1 R1 C1 R2(G2) C2 R1(G1) Z = +j L1 -- =R +jX 2 2 2 2 2 2 1+ R1 C1 1+ R1 C1 2 2 R2 j C2 R2 Y Y Z = -- =R +jX 2 2 2 2 2 2 1+ C R 1+ C2 R2 2 2 2 2 C1 R1 jC1 Y =G +jB = -- 2 2 2 2 2 2 1+ C1 R1 1+ C1 R1 L1 R R1 x = -- -- 1 R C R 2 R R 2 jL2 1 1 1 Y =G +jB = +jC2 -- 2 2 2 2 2 2 R2 + L2 R2 + L2 x,--x 2 x = R R2 -- R B =-- G/R -- G 2 B =(1/kR -- G )1/GR -- 1 x 1 2 2 2 k=L2/C2R2 -- x M M B (f1) N B R M R2 R1 G or G Figure 10. Z-Plane Representation of the Circuit of G1 G2 Figure 9(a) Exact transformation from R1 into R2 occurs at the points B .-- B of intersection M and N. Impedances are then conjugate or Z =R +jX and Z =R +jX with R =R and X =--X. Figure 11. Y-Plane Representation of the Circuit The only possible solution is obtained when X and -- X of Figure 9(c) are tangential to each other. For the dashed curve, 4.1.3 Four-Reactance Networks representing another value of L1 or C1, a wider frequency band could be expected at the expense of some ripple inside Four-reactance networks are used essentially for the band. However, this can only be reached with four broadband matching. The networks which will be considered reactances as will be shown in section 4.1.3. in the following consist of two two-reactance sections in With a three-reactance configuration, there are not cascade. Some networks have pseudo low-pass filter enough degrees of freedom to permit X =--X and character, others band-pass filter character. In principle, the simultaneously obtain the same variation of frequency on former show narrower bandwidth since they extend the both curves from M to point N. impedance transformation to very low frequencies Exact transformation can, therefore, only be obtained at unnecessarily, while the latter insure good matching over a one frequency. wide frequency band around the center frequency only (see The values of the three reactances can be calculated by Fig. 14). making dX dX a)X2 M X3 b) X2 M X4 X =--X,R =R and = -- . dR dR The general solution of these equations leads to R1 X1 X4 R2 R1 X1 X3 R2 complicated calculations. Therefore, computed tables should be used. Z1 Z2 One will note on Figure 10 that the compensation c) d) reactance contributes somewhat to impedance X2 M X1 X2 M X4 transformation, i.e. R varies when going from M to R2. The circuit of Figure 9 (b) is dual with respect to the first X X one and gives exactly the same results in a Y-plane R1 1 R2 R1 3 R2 X X representation. 3 4 Circuit of Figure 9 (c) is somewhat different since only one intersection M exists as shown in Figure 11. Narrower R1 =n.R2 frequency bands must be expected from this configuration. The widest band is obtained for C1 = . Figure 12. Four-Reactance Networks AN721 RF Application Information Freescale Semiconductor 5 The two-reactance sections used in above networks have the selectivity curve will be different from the curves given either transformation properties or compensation properties. in Figure 14. Figure 15 shows the selectivity for networks Impedance transformation is obtained with one series (a) and (b) when the source resistance R1 is infinite. reactance and one shunt reactance. Compensation is made From Figure 15 it can be seen that network (a) is more with both reactances in series or in shunt. sensitive to R1 changes than network (b). If two cascaded transformation networks are used, transformation is accomplished partly by each one. 0+j1 With four-reactance networks there are two frequencies, f1 and f2, at which the transformation from R1 into R2 is exact. These frequencies may also coincide. 0.28 + j0.45 NORMALIZED IMP. Q For network (b) for instance, at point M, R1 or R2 is 0.1 + j0.16 2 (1 -- j1.6) NORMALIZED ADM. Q transformed into R1R2 when both frequencies fall together. 1 (2.8 -- j4.5) (a) At all points (M), Z1 and Z2 are conjugate if the transformation b is exact. 3 b1 0.1 + j0.147 b In the case of Figure 12 (b) the reactances are easily, Q1 (0.316 -- j4.65) 3 0.316 -- j0 4 4 calculated for equal frequencies: x x’ 1+j0 00.1 2 R1 n--1 x x2 b1 X1 = ,X2 =R1 n n--1 0.28 -- j0.14 (2.8 + j1.4) (b) X X =R R =X X 1 4 1 2 2 3 0.316 -- j0.465 Q 2 (1 + j1.47) R1 R1 X3 = ,X4 = n--1 n R =1 n( n--1) 1 R2 =0.1 For network (a) normally, at point (M), Z1 and Z2 are complex. This pseudo low-pass filter has been computed (a) X1 = 0.624 X3 = 0.169 X1 =0.68 X3 = 0.215 elsewhere (Ref. (3)). Many tables can be found in the ( ( ( ( literature for networks of four and more reactances having B1 =1.6 B3 =5.9 B1 =1.47 B3 =4.65 Tchebyscheff character or maximally-flat response (Ref. (3), X2 =0.59 X4 = 0.160 X2 = 0.465 X4 = 0.147 (4) and (6)). Q1 =1.6=Q2 Q1 =Q2 =1.47 Figure 13 shows the transformation path from R1 to R2 for networks (a) and (b) on a Smith-Chart (refer also to (a)X2 X4 (b) X2 X4 section 4.2, Graphic Design). Case (a) has been calculated using tables mentioned in Ref. (4). R1 X1 X3 R2 R1 X1 X3 R2 Case (b) has been obtained from the relationship given above for X1 ...X4. Both apply to a transformation ratio equal to 10 and for R1 =1. Figure 13. Transformation Paths for Networks There is no simple relationship for X1 ...X4 of network (a) and (b) (b) if f1 is made different from f2 for larger bandwidth. Figure 14 shows the respective bandwidths of network (a) and (b) for the circuits shown in Figure 13. ATTEN. If the terminations contain a reactive component, the -- 0 -- 1 computed values for X1 or X4 may be adjusted to (a) compensate for this. -- 2 -- 3 (b) For configuration (a), it can be seen from Figure 13, that -- 4 in the considered case the Q’s are equal to 1.6. -- 5 (b) For configuration (b) Q , which is equal to Q ,isfixed 1 2 -- 6 (a) for each transformation ratio. -- 7 -- 8 n 2 4 8 10 16 Q = n--1 -- 9 Q1 =Q2 0.65 1 1.35 1.46 1.73 -- 1 0 f/fo .3 .4 .5 .6 .7 .8 .9 1 1.1 1.3 1.5 1.7 1.9 The maximum value of reactance that the terminations may have for use in this configuration can be determined Figure 14. Selectivity Curves for Networks (a) and (b) from the above values of Q. of Figure 13 If R1 is the load resistance of a transistor, the internal transistor resistance may not be equal to R1. In this case AN721 RF Application Information 6 Freescale Semiconductor REL. ATTEN. L2 L1 C1 2 1 R R 2 C2 2 0 (G2) (G1) -- 1 -- 2 -- 3 Y Y -- 4 R1 j(L1 -- 1 / c1) -- 5 Y =G +jB = -- (a) (b) (a) (b) 2 2 2 2 -- 6 R1 +(L1 -- 1 / c1) R1 +(L1 -- 1 / c1) -- 7 -- 8 R2 L2 Y =G +jB = +j C2 -- -- 9 2 2 2 2 2 2 R2 + L2 R2 + L2 -- 1 0 f/fo .4 .5 .6 .7 .8 .9 1 1.1 1.3 1.5 1.7 G B = -- G 2 R1 Figure 15. Selectivity Curves for Networks (a) and (b) 1 L of Figure 13 with Infinite R k= 2 1 B =(1/kR2 -- G ) -- 1 2 GR2 C2 R2 As mentioned earlier, the four-reactance network can also B be thought of as two cascaded two-reactance sections; one G G2 used for transformation, the other for compensation. 1 G Figure 16 shows commonly used compensation networks, M B together with the associated L-section. N The circuit of Figure 16 (a) can be compared to the M three-reactance network shown in Figure 9(c). The difference is that capacitor C2 of that circuit has been replaced by a L-C circuit. The resulting improvement may be seen by comparing Figure 17 with Figure 11. B COMPENSATION NETWORK (a) Figure 17. Y-Plane Representation of the Circuit of Figure 16(a) When M and N coincide in, M, the new dX/df = dX/df R2 R 1 condition can be added to the condition X =--X (for three-networks) and similarly R =R and dR/df = dR/df. If f1 is made different from f2, a larger bandwidth can be (b) C.N. achieved at the expense of some ripple inside the band. Again, a general solution of the above equations leads to still more complicated calculations than in the case of R2 R1 three-reactance networks. Therefore, tables are preferable (Ref. (3), (4) and (6)). The circuit of Figure 16(b) is dual of the circuit of R1 =n.R2 Figure 14(a) and does not need to be treated separately. It (c) C.N. C.N. gives exactly the same results in the Z-plane. Figure 16(c) shows a higher order compensation requiring six reactive elements. R 2 R1 The above discussed matching networks employing compensation circuits result in narrower bandwidths than the former solutions (see paragraph 4.1.3) using two transformation sections. A matching with higher order Figure 16. Compensation Networks Used with compensation such as in Figure 16(c) is not recommended. a L-Section Better use can be made of the large number of reactive elements using them all for transformation. When the above configurations are realized using short By adding one reactance, exact impedance portions of transmission lines, the equations or the usual transformation is achieved at two frequencies. It is now tables no longer apply. The calculations must be carried out possible to choose component values such that the point on a computer, due to the complexity. However, a graphic of intersection M occurs at the same frequency f1 on both method can be used (see next section) which will consist curves and simultaneously that N occurs at the same essentially in tracing a transformation path on the Z-Y-chart frequency f2 on both curves. Among the infinite number of using the computed lumped element values and replacing possible intersections, only one allows to achieve this. it by the closest path obtained with distributed constants. The AN721 RF Application Information Freescale Semiconductor 7 bandwidth change is not significant as long as short portions 1.0 of lines are used (Ref. (13)). n=4 0.9 4.1.4 Matching Networks Using Quarter-Wave Transformers 0.8 At sufficiently high frequencies, where /4-long lines of practical size can be realized, broadband transformation can 0.7 1 2 3 easily be accomplished by the use of one or more /4-sections. o/ Figure 18 summarizes the main relations for 0.6 0.8 1.0 1.2 1.4 (a) one-section and (b) two-section transformation. Figure 20. Selectivity Curves for One, Two and Three A compensation network can be realized using a /2-long /4-Sections transmission line. 4.1.5 Broadband Matching Using Band-Pass Filter Type Networks. High Q Case. (a) The above circuits are applicable to devices having low R nR 1 2 input or output Q, if broadband matching is required. R2 Z R1 R1 Generally, if the impedances to be matched can be Z= R1 R2 =R2 n = n represented for instance by a resistor R in series with an inductor L (sometimes a capacitor C) within the band of interest and if L is sufficiently low, the latter can be (b) 4 incorporated into the first inductor of the matching network. 3 Z1 =R1 R2 This is also valid if the representation consists of a shunt 4 combination of a resistor and a reactance. R2 Z2 Z1 R1 3 Z2 =R1 R2 Practically this is feasible for Q’s around one or two. For higher Q’s or for input impedances consisting of a series or at A, R = R1 R2 parallel resonant circuit (see Fig. 2), as it appears to be for A large bandwidths, a different treatment must be followed. Let us first recall that, as shown by Bode and Fano (Ref. Figure 18. Transformation Networks Using /4-Long (7) and (8)), limitations exist on the impedance matching of Transmission Lines a complex load. In the example of Figure 21, the load to be matched consists of a capacitor C and a resistor R in shunt. Figures 19 and 20 show the selectivity curves for different transformation ratios and section numbers. RG Exponential lines MATCHING Exponential lines have largely frequency independent V ZIN NETWORK C R transformation properties. (LOSSLESS) The characteristic impedance of such lines varies exponentially with their length I: kl Z=Zo e Figure 21. General Matching Conditions where k is a constant, but these properties are preserved only if k is small. The reflection coefficient between transformed load and generator is equal to: ZIN -- R g = ZIN +Rg 1.0 = 0, perfect matching, 0.9 = 1, total reflection. The ratio of reflected to incident power is: 0.8 Pr = 2 Pi 0.7 n=6 4 2 The fundamental limitation on the matching takes the form: o/ 0.6 0.8 1.0 1.2 1.4 1 ln d ≤ Bode equation RC Figure 19. Selectivity Curves for Two /4-Section =o Networks at Different Transformation Ratios and is represented in Figure 22. AN721 RF Application Information 8 Freescale Semiconductor 1 Thus, devices having relatively high input Q’s are usable ln for broadband operation, provided the consequent higher attenuation or reflection introduced is acceptable. The general shape of the average insertion losses or attenuation a (neglecting the ripple) of a low-pass impedance S= RC matching network is represented in Figure 24 as a function of 1/Q for different numbers of network elements n (ref. (3)). c For a given Q and given ripple, the attenuation decreases if the number n of the network element increases. But above Figure 22. Representation of Bode Equation n = 4, the improvement is small. For a given attenuation a and bandwidth, the larger n the The meaning of Bode equation is that the area S under the smaller the ripple. curve cannot be greater than /RC and therefore, if matching For a given attenuation and ripple, the larger n the larger is required over a certain bandwidth, this can only be done at the bandwidth. the expense of less power transfer within the band. Thus, Computations show that for Q < 1 and n 3the power transfer and bandwidth appear as interchangeable attenuation is below 0.1 db approximately. The impedance quantities. transformation ratio is not free here. The network is a true It is evident that the best utilization of the area S is low-pass filter. For a given load, the optimum generator obtained when is kept constant over the desired band impedance will result from the computation. Before impedance transformation is introduced, a c and made equal to 1 over the rest of the spectrum. Then conversion of the low-pass prototype into a band-pass filter -- type network must be made. Figure 25 summarizes the main RC =e c relations for this conversion. within the band and no power transfer happens outside. r is the conversion factor. A network fulfilling this requirement cannot be obtained For the band-pass filter, QlN max or the maximum possible in practice as an infinite number of reactive elements would input Q of a device to be matched, has been increased by be necessary. the factor r (from Figure 25, QIN max =r QIN max). If the attenuation a is plotted versus the frequency for Impedance inverters will be used for impedance practical cases, one may expect to have curves like the ones transformation. These networks are suitable for insertion into shown in Figure 23 for a low-pass filter having Tchebyscheff a band-pass filter without affecting the transmission character. characteristics. a a 2 a 1 LOW-PASS BAND-PASS amax2 amin2 amax1 a o b amin1 wc = 2 1 2 r= o Figure 23. Attenuation versus Angular Frequency for C1 Different Bandwidths with Same Load L1 L3 L1 R5 R5 For a given complex load, an extension of the bandwidth Ro C C R 2 4 o C from 1 to 2 is possible only with a simultaneous increase 2 L2 of the attenuation a. This is especially noticeable for Q’s exceeding one or two (see Figure 24). cL1 1 QIN max = L =r.L C1 = 1 1 2 Ro o L1 dB oL1 1 4 QIN max = C =r.C L2 = 2 2 2 Ro o C2 3 etc. n=2 a 2 3 Figure 25. Conversion from Low-Pass into 1 Band-Pass Filter 0 Figure 26 shows four impedance inverters. It will be 0.1 11/Q noticed that one of the reactances is negative and must be Figure 24. Insertion Losses as a Function of 1/Q combined in the band-pass network with a reactance of at AN721 RF Application Information Freescale Semiconductor 9 least equal positive value. Insertion of the inverter can be (4) Convert the element values found by step (3) into made at any convenient place (Ref. (3) and (9)). series or parallel resonant circuit parameters, (5) Insert the impedance inverter in any convenient INVERTER EQUIVALENT TO: place. In the above discussions, the gain roll-off has not been C C 1--1/n 1/n2 -- 1 / n taken into account. This is of normal use for moderate bandwidths (30% for ex.). However, several methods can be employed to obtain a constant gain within the band a) nC C despite the intrinsic gain decrease of a transistor with frequency. Tables have been computed elsewhere (Ref. (10)) for n:1 matching networks approximating 6 db/octave attenuation nC C versus frequency. Another method consists in using the above mentioned network and then to add a compensation circuit as shown b) (1 -- n)C (n2 -- n ) C for example in Figure 27. L2 C2 n:1 L(1 -- 1/n) L(1/n2 -- 1/n) R=ZIN = CONSTANT vs. f. R1 L1C1 =L2C2 c) R R =R L/n L 1 2 =1/L C =1/L C L1 C1 b 1 1 2 2 (HIGH BAND EDGE) n:1 L/n Q2 = bL2/R = Q1 = bC1R L d) L L 1--n n2 -- n Figure 27. Roll-Off Compensation Network n:1 Resonance b is placed at the high edge of the frequency band. Choosing Q correctly, roll-off can be made 6 db/octave. Figure 26. Impedance Inverters The response of the circuit shown in Figure 27 is expressed by: 1 , where < b When using the band-pass filter for matching the input b 2 1+Q2 -- impedance of a transistor, reactances L1 C1 should be b made to resonate at o by addition of a convenient series reactance. This must be equal to /b for 6 db/octave compensation. As stated above, the series combination of Ro, L1 and At the other band edge a, exact compensation can be C1 normally constitutes the equivalent input network of a obtained if: transistor when considered over a large bandwidth. This is b 2 -- 1 a good approximation up to about 500 MHz. a In practice the normal procedure for using a bandpass Q= a b 2 filter type matching network will be the following: -- b a (1) For a given bandwidth center frequency and input impedance of a device to be matched e.g. to 50 ohms, 4.1.6 Line Transformers first determine QlN from the data sheet as oL1 The broadband properties of line transformers make them Ro very useful in the design of broadband impedance matching networks (Ref. (11) and (12)). after having eventually added a series reactor for A very common form is shown by Figure 28. This is a centering, 4:1 impedance transformer. Other transformation ratios like (2) Convert the equivalent circuit RoL1 C1 into a 9:1 or 16:1 are also often used but will not be considered low-pass prototype Ro L1 and calculate QlN using the here. formulas of Figure 25, The high frequency cut-off is determined by the length (3) Determine the other reactance values from tables of line which is usually chosen smaller than min/8. Short (Ref. (3)) for the desired bandwidth, lines extend the high frequency performance. AN721 RF Application Information 10 Freescale Semiconductor Z0 =Rg/2 Rg Rg R = L 4 Figure 28. 4:1 Line Transformer Line Transformer on P.C. Board (Figure 29) The low frequency cut-off is determined first by the length 35 of line, long lines extending the low frequency performance Cu 66 Ribbon thickness of the transformer. Low frequency cut-off is also improved Electrical tape by a high even mode impedance, which can be achieved EE3990 by the use of ferrite material. With matched ends, no power Permacel New is coupled through the ferrite which cannot saturate. ISOLATION Brunswick For matched impedances, the high frequency attenuation ISOLATION Cu a of the 4:1 transformer is given by: (1 + 3 cos 2I/)2 +4sin2 2I/ a= FERRITE 4(1+cos2I/)2 For I = /4, a = 1.25 or 1 db; for I = /2, a = . The characteristic impedance of the line transformer must be equal to: Zo = Rg R . Figures 29 and 30 show two different realizations of 4:1 25 mm transformers for a 50 to 12.5 ohm-transformation designed for the band 118 -- 136 MHz. The transformers are made of two printed circuit boards or two ribbons stuck together and connected as shown in -- Stick one ribbon (1.5 mm wide) against the other (2.5) Figures 29 and 30. Total length per ribbon -- 9 cm. Turns 3.5 EPOXY 0.2 mm THICK Figure 30. 4:1 Copper Ribbon Line Transformer 35 with Ferrite in mm 15 22 Cu 10 61.5 68.5 Line Transformer on Ferrite (Figure 30) STICK ONE AGAINST THE OTHER 4.2 GRAPHIC DESIGN The common method of graphic design makes use of the Impedance-Admittance Chart (Smith Chart). It is applicable to all ladder-type networks as encountered in matching circuits. Matching is supposed to be realized by the successive algebraic addition of reactances (or susceptances) to a given start impedance (or admittance) until another end impedance (or admittance) is reached. Impedance chart and admittance chart can be superimposed and used alternatively due to the fact that an immittance point, defined by its reflection coefficient with respect to a reference, is common to the Z-chart and the Y-chart, both being representations in the -plane. Figure 29. 4:1 Line Transformer on P.C. Board AN721 RF Application Information Freescale Semiconductor 11 Z--Rs i = r +j i +1 Z+Rs b=--0.5 g=0.5 Gs -- Y INDUCTIVE = g=1 b<0 Gs +Y b=--1 r 1 b=0 Rs = = Characteristic impedance of the line -- 1 +1 G SHORT b=1 s OPEN More precisely, the Z-chart is a plot in the -plane, while CAPACITIVE the Y-chart is a plot in the -- -plane. The change from the g=0 b>0 to -- -plane is accounted for in the construction rules given -- 1 b= 0.5 below. g>0 Figure 31 and 32 show the representation of normalized -- -PLANE Z and Y respectively, in the -plane. The Z-chart is used for the algebraic addition of series Y =y= g+jb reactances. The Y-chart is used for the algebraic addition Gs 2 2 of shunt reactances. 1--r -- i For the practical use of the charts, it is convenient to make g= (1 + )2 + 2 the design on transparent paper and then place it on a usual r i INDUCTIVE -- 2 i Smith-chart of impedance type (for example). For the b= 2 2 g addition of a series reactor, the chart will be placed with (1 + r) + i “short” to the left. For the addition of a shunt reactor, it will 1--g2 -- b 2 CAPACITIVE be rotated by 180 with “short” (always in terms of r = impedance) to the right. (1 + g)2 +b2 i -- 2 b b i = +1 (1 + g)2 +b2 RECTANGULAR COORDINATES x=0.5 r=1 INDUCTIVE Radius: 1/(1 + g) gcircles r=0 x>0 x=1 Center at: r =--g/(1+g); i =0 r=0.5 r 0 -- 1 x=0 +1 SHORT x=--1 Radius: 1/b OPEN bcircles Center at: =--1, =--1/b CAPACITIVE r i x=--0.5 x<0 -- 1 Figure 32. Representation of the Normalized Y-Values in the -Plane r>0 The following design rules apply. They can very easily -PLANE be found by thinking of the more familiar Z and Y Z representation in rectangular coordinates. =z=r+jx x For joining two impedance points, there are an infinity of Rs 2 2 solutions. Therefore, one must first decide on the number 1--r -- i r= of reactances that will constitute the matching network. This 2 2 (1 -- r) + i INDUCTIVE number is related essentially to the desired bandwidth and 2 i the transformation ratio. x= r 2 2 (1 -- r) + i Chart Using CAPACITIVE r2 +x2 -- 1 to be Curve of r = (r + 1)2 +x2 Addition of Used Direction Constant 2x RECTANGULAR COORDINATES Series R Z Open (in terms of x i = admittance) (r + 1)2 +x2 Series G Y Short (in terms of b Radius: 1/(1 + r) admittance) rcircles r Series C (+ 1/j C) Z ccw r Center at: r = , i =0 1+r Shunt C (+ jC) Y cw g Radius: 1/x Series L (+ jL) Z cw r xcircles Center at: r =1 , i =1/x Shunt L (+ 1/jL) Y ccw g Figure 31. Representation of the Normalized Z Values in the -Plane AN721 RF Application Information 12 Freescale Semiconductor Secondly, one must choose the operating Q of the circuit, The collector load must be equal to: which is also related to the bandwidth. Q can be defined at [Vcc -- Vce (sat)]2 (28 -- 3)2 each circuit node as the ratio of the reactive part to the real or = 15.6 ohms . part of the impedance at that node. The Q of the circuit, which 2xPout 40 is normally referred to, is the highest value found along the path. The collector capacitance given by the data sheet is Constant Q curves can be superimposed to the charts 40 pF, corresponding to a capacitive reactance of 22.7 ohms. and used in conjunction with them. In the -plane Q-curves The output impedance seen by the collector to insure the are circles with a radius equal to required output power and cancel out the collector capacitance must be equal to a resistance of 15.6 ohms in 1 parallel with an inductance of 22.7 ohms. This is equivalent 1+ Q2 to a resistance of 10.6 ohms in series with an inductance of 7.3 ohms. and a center at the point 1/Q on the imaginary axis, which The input Q is equal to, 1.1/1.94 or 0.57 while the output is expressed by: Q is 7.3/10.6 or 0.69. x 2 i 1 2 1 It is seen that around this frequency, the device has good Q== 2 + + =1+ . 2 2 r i 2 broadband capabilities. Nevertheless, the matching circuit r 1-- r -- i Q Q will be designed here for a narrow band application and the The use of the charts will be illustrated with the help of effective Q will be determined by the circuit itself not by the an example. device. The following series shunt conversion rules also apply: Figure 34 shows the normalized impedances (to 50 ohms). X 0.022 0.146 R R X 0.039 0.212 R 1 R R= G = = R2 R R2 +X2 NORMALIZED INPUT IMPEDANCE LOAD IMPEDANCE 1+ X2 Figure 34. Normalized Input and Output Impedances X 1 X for the 2N5642 X= -- B = = X2 X R2 +X2 1+ Figure 35 shows the diagram used for the graphic design R2 of the input matching circuit. The circuit Q must be larger Figure 33 shows the schematic of an amplifier using the than about 5 in this case and has been chosen equal to 10. 2N5642 RF power transistor. Matching has to be achieved AtQ=5,C1 would be infinite. The addition of a finite value at 175 MHz, on a narrow band basis. of C1 increases the circuit Q and therefore the selectivity. The normalized values between brackets in the Figure are VCC =28V admittances (g + jb). 0+j1 RFC L4 PARALLEL CAPACITANCE C2 (b2) C1 L3 Q-- 0.039 + j0.39 CIRCLE (0.25 -- j2.5) 1+j1.75 (10) (0.25 -- j0.42) SERIES 50 RFC 50 INDUCTANCE L3 SERIES C2 C0 C5 (x3) CAPACITANCE C1 (x1) 01+j00.039 + j0.022 2N5642 Figure 35. Input Circuit Design Figure 33. Narrow-Band VHF Power Amplifier At f = 175 MHz, the following results are obtained: L3 = 50x3 = 50 (0.39 -- 0.022) = 18.5 ohms The rated output power for the device in question is 20 W ∴ L3 = 16.8 nH at 175 MHz and 28 V collector supply. The input impedance 1 1 C2 = b2 = (2.5 -- 0.42) = 0.0416 mhos at these conditions is equal to 2.6 ohms in parallel with -- 50 50 200 pF (see data sheet). This converts to a resistance of 1.94 ohms in series with a reactance of 1.1 ohm. ∴ C2 = 37.8 pF AN721 RF Application Information Freescale Semiconductor 13 1 Rated output power: = 50x1 =50 1.75 = 87.5 ohms C1 30 W for 8 W input at 175 MHz. From the data sheet it appears that at 125 MHz, 30 W output will be achieved with ∴ C1 = 10.4 pF about 4 W input. Figure 36 shows the diagram for the output circuit, Output impedance: designed in a similar way. [V -- V )]2 100 0+j1 cc ce (sat = = 1.67 ohms 2xPout 60 Cout = 180 pF at 125 MHz 0.212 + j0.146 5.2 CIRCUIT SCHEMATIC 12.5 V 1+j0 0 C7 SERIES INDUCTANCE L4 (x4) PARALLEL CAPACITANCE C5 RFC2 0.212 -- j0.4 (b5) (1 + j1.9) C6 Figure 36. Output Circuit Design Here, the results are (f = 175 MHz): RFC1 L3 C5 L4 =50 x4 =50 (0.4 + 0.146) = 27.3 ohms L = 24.8 nH ∴ 4 L2 1 1 C5 = b5 = 1.9 = 0.038 mhos 50 50 C4 T2 T1 C1 ∴ C5 = 34.5 pF 50 50 The circuit Q at the output is equal to 1.9. Theselectivityofamatchingcircuitcanalsobe C1 = 300 pF (chip) T1,T2 seeFigure30 determined graphically by changing the x or b values L2 = 4.2 nH (adjust) RFC1 =7turns according to a chosen frequency change. The diagram will L3 = 8 nH (adjust) 6.3 mm coil diameter give the VSWR and the attenuation can be computed. C4 = 130 pF (chip) 0.8 mm wire diameter The graphic method is also useful for conversion from C5 = 750 pF (chip) RFC2 = 3 turns on ferrite bead a lumped circuit design into a stripline design. The C6 =2.2FC7 =0.68F immittance circles will now have their centers on the 1 + jo Figure 37. Circuit Schematic point. At low impedance levels (large circles), the difference 5.3 TEST RESULTS between lumped and distributed elements is small. W 5. PRACTICAL EXAMPLE 30 PIN =2W The example shown refers to a broadband amplifier stage using a 2N6083 for operation in the VHF band 118 -- 136 MHz. The 2N6083 is a 12.5 V-device and, since 20 amplitude modulation is used at these transmission 1W frequencies, that choice supposes low level modulation POUT associated with a feedback system for distortion 10 compensation. 0.5 W Line transformers will be used at the input and output. Therefore the matching circuits will reduce to two-reactance MHz networks, due to the relatively low impedance transformation 0 ratio required. 118 124 128 132 136 f 5.1 DEVICE CHARACTERISTICS Input impedance of the 2N6083 at 125 MHz: Figure 38. POUT vs Frequency Rp = 0.9 ohms Cp = -- 390 pF AN721 RF Application Information 14 Freescale Semiconductor % Acknowledgments PIN =2W The author is indebted to Mr. T. O’Neal for the fruitful 70 discussions held with him. Mr. O’Neal designed the circuit 60 shown in Figure 37; Mr. J. Hennet constructed and tested 1W the lab model. 50 40 0.5 W 6. LITERATURE 30 1. Freescale Application Note AN282A “Systemizing RF 20 Power Amplifier Design” 2. W. E. Everitt and G. E. Anner “Communication 10 Engineering” McGraw-Hill Book Company, Inc. MHz 3. G. L. Matthaei, L. Young, E. M. T. Jones “Microwave 118 124 128 132 136 Filters, Impedance-Matching Networks and Coupling f Structures” McGraw-Hill Book Company, Inc. 4. G. L. Matthaei “Tables of Chebyshew Figure 39. vs Frequency Impedance-Transforming Networks of Low-Pass Filter db. Form” Proc. IEEE, August 1964 5. Freescale Application Note AN267 “Matching Network 0 Designs with Computer Solutions” P refl. 6. E. G. Cristal “Tables of Maximally Flat P -- 5 in PIN =2W Impedance-Transforming Networks of Low-Pass-Filter -- 1 0 Form” IEEE Transactions on Microwave Theory and Techniques Vol. MTT 13, No 5, September 1965 1W -- 1 5 Correspondence 0.5 W 7. H. W. Bode “Network Analysis and Feedback Amplifier -- 2 0 Design” D. Van Nostrand Co., N.Y. 8. R. M. Fano “Theoretical Limitations on the Broadband MHz Matching of Arbitrary Impedances” Journal of Franklin 118 124 128 132 136 Institute, January -- February 1950 f 9. J. H. Horwitz “Design Wideband UHF-Power Amplifiers” Electronic Design 11, May 24, 1969 Figure 40. P /P vs Frequency REFL. IN 10. O. Pitzalis, R. A. Gilson “Tables of Impedance Matching Networks which Approximate Prescribed Attenuation Versus Frequency Slopes” IEEE Transactions on Microwave Theory and Techniques Vol. MTT-19, No 4, April 1971 11. C. L. Ruthroff “Some Broadband Transformers” Proc. IRE, August 1959 12. H. H. Meinke “Theorie der H. F. Schaltungen” München, Oldenburg 1951 118 - 136 MHz Amplifier (see Figure 37) Before Coil Adjustment AN721 RF Application Information Freescale Semiconductor 15 How to Reach Us: Information in this document is provided solely to enable system and software implementers to use Freescale products. There are no express or implied copyright Home Page: licenses granted hereunder to design or fabricate any integrated circuits based on the freescale.com information in this document. Web Support: Freescale reserves the right to make changes without further notice to any products freescale.com/support herein. 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