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Design of HF wideband power transformers APPLICATION NOTE ECO6907 1998 Mar 23 2 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 CONTENTS 1 INTRODUCTION 2 TRANSFORMER SPECIFICATION 3 INFLUENCE OF THE CORE ON PERFORMANCE 3.1 Primary Inductance 3.2 Core Losses 4 INFLUENCE OF THE TRANSMISSION LINE ON PERFORMANCE 4.1 Resistive Loss and Power Handling 4.2 Mismatch loss 5 COMPENSATION TECHNIQUES 5.1 Compensation at Low Frequencies 5.2 Compensation at High Frequencies 6 TRANSFORMER CONFIGURATION 6.1 Phase Reversing Transformer 6.2 Balanced to Unbalanced Transformer 6.3 Symmetrical 1 : 4 Impedance Transformer 6.4 Asymmetrical 1 : 4 Impedance Transformer 6.5 Symmetrical 9 : 1 Impedance Transformer 6.6 Asymmetrical 1 : 9 Impedance Transformer 6.7 Single-ended Hybrid 6.8 Push-pull Hybrid 6.8.1 Impedance Step-up Type 6.8.2 Impedance Step-Down Type 7 PRACTICAL EXAMPLES 7.1 12.5 Ω Balanced to 50 Ω Unbalanced Transformer 7.2 50 Ω Unbalanced to 5.55 Ω Balanced Transformer 8 REFERENCES 1998 Mar 23 3 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 1 INTRODUCTION Transmission line power transformers can be used to perform a variety of functions, among which are phase reversal, balanced to unbalanced coupling, impedance transformation and hybrid functions. Such transformers find many applications in wide-band power amplifiers for both s.s.b. transmitters in the h.f. region and f.m. transmitters in the lower v.h.f. region. The properties of a practical h.f. power transformer are discussed here and their effect on transformer performance is analysed. Since losses must be kept low, in practice the transformer will use a ferrite core. Further, we have limited the discussion to cores without an air-gap since these have a low stray magnetic field, a high permeability, and can cover the power range (up to 80 W) dealt with here. Data (dimensions, permeability values etc.) on all core types can be found in our Data Handbook “Soft Ferrites”, MA01. A glance through the Handbook will show the wide range of materials, dimensions and types from which the designer may choose. It must be remembered, of course, that when cores constructed in two parts (pot-cores and cross-cores, for example) are used, the type without an air-gap must be selected. Throughout we have aimed at giving practical solutions to the problems posed by material and design limitations. In particular, compensating techniques for extending the frequency range of a number of transformer configurations are discussed. To give an idea of some application possibilities, practical examples in several transformer configurations have been worked, using transformer cores from our range of ferrites. 2 TRANSFORMER SPECIFICATION The transformer design considerations dealt with in this publication are: • Maximum power level to be handled • Frequency range • Input and output impedance • Allowable reflection and resistive losses. How a transformer can meet the above considerations for a particular application is analysed in the following three sections. The first two sections deal with the influence of the core and transmission line respectively on transformer performance, and the third with mismatch compensation techniques. 3 INFLUENCE OF THE CORE ON PERFORMANCE 3.1 Primary Inductance This inductance determines the amount of reflection at the low frequency end of the band. It can be calculated using the formula: L = µoµrn2 A/l in which: L = inductance in H µo = 4 pi 10-7 (rationalised M.K.S. units) µr = relative permeability A = average ferrite cross section in m2 l = average length of the lines of force in m n = number of turns between the input connections. 1998 Mar 23 4 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 In a simple example, like the phase reversing transformer, this relation holds. Other cases may require a transformation (see Section 7.1). If degrading of performance at the high end of the band is to be avoided, the value of L must not be higher than really necessary. A good practical value is: L = 4R/ωmin in which: R = midband input resistance in Ω ωmin = 2pi times the minimum frequency in Hz. Where requirements are severe the compensation technique described in Section 5.1 may be used. 3.2 Core Losses The losses caused by the core material will be represented here as a resistance (Rp) in parallel with the input. This resistance depends on: • The sort of ferrite material • The frequency • The quantity L/µr • The maximum flux density Bmax. In the small signal case (Bmax → 0), Rp can be calculated with the aid of curves of the type shown in Fig.1(1). In these curves a comparison is made between different core materials based on equal core dimensions and equal number of turns. It can be seen that 4C4 and 4C6 are the best materials for frequencies above approximately 2.5 MHz. In the high v.h.f. region IZ2 ferroxplana shows interesting properties as can be seen from the same figure. The power handling capability of a transformer is closely dependent on the behaviour of Rp as a function of Bmax. For the section of the B-H curve with which we are dealing, Bmax can be calculated using the formula: Bmax = Vmax/ω × A × n in which: Bmax = maximum flux density in T(2) ω = 2pi times frequency in Hz A = ferrite cross section in m2 n = number of turns Vmax = maximum value of voltage across n turns in V. (1) The curves of Figs 1 to 7 have been drawn from measurements on single samples of the ferrite materials. Thus the average curves may differ somewhat from those shown. (2) The letter T stands for Tesla, the unit of magnetic flux density in the SI unit system. The following relationship holds: 1T = 1 Wb/m2 = 1 Vsec/m2 = 10000 gauss. 1998 Mar 23 5 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 In Figs 2 to 7 the quantity µrRp/L is given for different ferrite materials as a function of the product Bmax × ƒ with the frequency as a parameter. The product Bmax × ƒ has been chosen because, for most transformers, its value remains constant for changing frequency. From Figs 2 to 7 it can be seen that Rp decreases as Bmax increases, especially at lower frequencies. This forms the primary limit on the power handling capability of these transformers. If 4C4 material (Fig.5) is used in the h.f. region, the Bmax × ƒ product must not be higher than approx. 2 × 104 T.Hz. Combining this with the choice of L according to the second equation in Section 3.1, we find that the power loss caused by the core material will be no more than 1%. At frequencies of 30 MHz and higher it seems that higher Bmax × ƒ products, perhaps up to 105 T.Hz, can be used. For IZ2 ferroxplana this has already been confirmed by measurements at 165 MHz. A very conservative choice of the Bmax value must also be avoided because this leads to a greater length of the transmission line and consequently more loss at the high end of the band. 4 INFLUENCE OF THE TRANSMISSION LINE ON PERFORMANCE 4.1 Resistive Loss and Power Handling The power loss in the transmission line depends on: • The type of line • The frequency • The length. Data on power loss in some 50 Ω coaxial cables is given in Fig.8. This power loss and the allowable maximum cable temperature restrict the power handling of the cable. The maximum power which can be transmitted depends on the type of cable and the frequency; data is given in Fig.9. Fig.1 Curves of µrRp/L plotted against frequency for our various ferrites. The curves have been plotted for small signal conditions (Bmax → 0). handbook, full pagewidth MGL235 1011 1012 µrRp L (sec−1) f (MHz)102 1031 10 1010 3H1 4A4 4B1 4C6 4C4 1Z2 1998 Mar 23 6 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 4.2 Mismatch loss Another kind of loss caused by the transmission line can occur if the characteristic impedance of this line is not the required value. This results in a mismatch being maximum at the high frequency end of the band. The amount of mismatch depends on: • The ratio between the length of the line and the wavelength on the line • The ratio between the required and the actual value of the characteristic resistance. Fig.2 3H1 material. handbook, full pagewidth MGL236 1011 1012 µrRp L (sec−1) Bmax ·f (T·Hz)10 4 105102 103 1010 1.6 MHz 5 MHz 15/28 MHz 1998 Mar 23 7 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 Fig.3 4A4 material. handbook, full pagewidth MGL237 1011 1012 µrRp L (sec−1) Bmax ·f (T·Hz)10 4 105102 103 1010 1.6 MHz 5 MHz 15 MHz 28 MHz Fig.4 4B1 material. handbook, full pagewidth MGL238 1011 1012 µrRp L (sec−1) Bmax ·f (T·Hz)10 4 105102 103 1010 10 MHz 15 MHz 28 MHz 4 MHz 1.6 MHz 1998 Mar 23 8 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 Fig.5 4C4 material. handbook, full pagewidth MGL239 1011 1012 µrRp L (sec−1) Bmax ·f (T·Hz)10 4 105102 103 1010 15 MHz 5 MHz 28 MHz 1.6 MHz Fig.6 4C6 material. handbook, full pagewidth MGL240 1011 1012 µrRp L (sec−1) Bmax ·f (T·Hz)10 4 105102 103 1010 1.6 MHz 1998 Mar 23 9 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 Fig.7 1Z2 material. handbook, full pagewidth MGL241 1011 1012 µrRp L (sec−1) Bmax ·f (T·Hz)10 5 106103 104 1010 165 MHz Fig.8 Curves of power loss plotted against frequency for three 50 Ω coaxial cables. A: diameter = 1.7 mm. B: diameter = 2.8 mm. C: diameter = 5 mm. handbook, full pagewidth MGL242 1 10−1 1 10 A loss (dB/m) f (MHz)102 103 10−2 B C 1998 Mar 23 10 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 From transmission line theory the input impedance is given by: in which: Rin = midband input resistance r = ratio between the actual and the required characteristic resistance β = 2pi/λ λ = wavelength on the line (for 50 Ω coaxial cables approx. 67-70% of wavelength in free space) l = length of the line. If the deviation of Zin from the required value is unacceptably large it is in many cases possible to make use of the compensation technique described in Section 5.2. Fig.9 Curves of power handling capability plotted against frequency for two 50 Ω coaxial cables. A: diameter = 2.8 mm. B: diameter = 5 mm. handbook, full pagewidth MGL243 102 103 1 10 power handling capability (W) f (MHz)102 103 10 B A Zin Rin 1 jrtanβ l+ 1 j1 r --- tanβ l+ -----------------------------×= 1998 Mar 23 11 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 5 COMPENSATION TECHNIQUES 5.1 Compensation at Low Frequencies Fig.10 Schematic diagram. handbook, halfpage R R E MGL218 Fig.11 Low frequency equivalent diagram. handbook, halfpage R RL E MGL219 Fig.12 Equivalent diagram with compensation. handbook, halfpage R RL E MGL220 CL CL Fig.13 Schematic diagram with compensation. handbook, halfpage R R E MGL221 CL CL 1998 Mar 23 12 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 Compensation will be illustrated by means of the phase reversing transformer. The schematic diagram is given in Fig.10 and the equivalent diagram for the low frequency end of the band is shown in Fig.11. For compensation we add two equal capacitors CL such that a high-pass T-filter section is formed (Fig.12). According to filter theory: The original diagram of Fig.10 is now transformed to the new one of Fig.13. If L is dimensioned according to eq.(2) the input impedance without compensation at the lowest frequency is R//(+ j4R). With compensation the input impedance is (.999R//+j264R), illustrating that the mismatch has been reduced to a negligible level. For some types of transformer, the capacitor at the output must have a different value from that at the input. With a l : n2 impedance transformer, for instance, it must be n2 times smaller than the capacitor at the input. Sometimes two or more transformers must be connected in cascade. In such a case low frequency compensation is possible if a high-pass pi-filter section is used (Fig.14). When the parallel inductance of the transformers at the interconnecting point are both approximately equal to L, the capacitance of CL must be L/2R2. 5.2 Compensation at High Frequencies This is only necessary when the characteristic resistance of the transmission line differs from the required value. A situation often met with in practice is that in which the required characteristic resistance is lower than that of the available line. Taking the simple case of the phase reversing transformer (Fig.15) with a required characteristic resistance equal to R, we find that compensation for an actual value equal to r × R can be made as follows. In parallel with the load resistance, we connect a capacitor of such a value that at the highest frequency the real part of the input admittance becomes 1/R. The resulting imaginary part of the input admittance is tuned out by means of a capacitor in parallel with the input. Both capacitors turn out to have the same value; given by in which ωmax = 2pi times the maximum frequency. The schematic diagram is shown in Fig.15. CL 2L R 2⁄= Fig.14 Equivalent diagram of cascaded transformers with compensation. handbook, halfpage R RL E MGL222 L CL CH 1 ωmaxrRtanβ l ---------------------------------- 1 1 r2 1–   tan 2β l––{ }= 1998 Mar 23 13 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 The result of this compensation is an exact match at the maximum frequency. There will be however, a slight mismatch at lower frequencies which is many times smaller than that at the maximum frequency without compensation. A combination of high and low frequency compensation is of course possible. 6 TRANSFORMER CONFIGURATION Because of the variety of the existing configurations it is hardly possible to give a complete survey. Therefore a restriction will be made to some principally different types. 6.1 Phase Reversing Transformer This type has already been discussed in the previous section. 6.2 Balanced to Unbalanced Transformer The schematic diagram is shown in Fig.16. This type can be considered as a modification of the phase reversing transformer. The primary inductance in this case is 4 times the inductance of the winding between the points A and B because of the voltage division. If low frequency compensation is used, a capacitor equal to 2CL must be placed in series with each input connection (see Section 5.1) to preserve symmetry, and one capacitor equal to CL in series with the output (point B). Fig.15 Phase reversing transformer with high frequency compensation. handbook, halfpage R R E MGL223 CHCH Fig.16 Balanced to unbalanced transformer. handbook, halfpage BA R MGL224 +1/2 V − 1/2 V +V 1998 Mar 23 14 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 6.3 Symmetrical(1) 1 : 4 Impedance Transformer The schematic diagram is given in Fig.17. Two cables each having a characteristic resistance of 2R, can be wound on a common core. The direction of the windings follows from the voltage division. Low frequency compensation can be made with a capacitor equal to 2CL (see Section 5.1) in each of the input leads and a capacitor equal to 1/2CL in each of the output leads. 6.4 Asymmetrical 1 : 4 Impedance Transformer If, in the 1 : 4 impedance transformer of Section 6.3, the points A and B are connected to earth it is no longer possible to wind the two lines on a common core. In fact the lower line may be wound on an ‘air core’ because there is no voltage difference between the points A and B. The logical next step is to omit the lower line completely (with a consequent slight phase difference between the lines). Then we get the transformer shown in Fig.18. The characteristic resistance of the transmission line has again an optimum value of 2R. But even if this value is chosen the input impedance is not constant as a function of frequency. From theory (see Ref.5) it is: in which r = ratio between the actual characteristic resistance and 2R. (1) Terminology in normal use is employed to describe the various transformer configurations. However the terms ‘symmetrical’, ‘balanced’ and ‘push-pull’ are used here synonymously to mean ‘antiphase port signals having equal amplitudes with respect to ground’, while likewise ‘asymmetrical’, ‘unbalanced’ and single-ended’ are synonymous with ‘one port terminal grounded’. Strictly defined, the terms ‘asymmetrical, and ‘unbalanced’ also apply to unequal port terminal values with respect to ground. Fig.17 Symmetrical 1 : 4 impedance transformer. handbook, halfpage BA 4R0 MGL225 +1/2 V − 1/2 V − V + V R Zin R 2cosβ l jrsinβ l+ 1 cosβ l j1 r ---sinβ l+ + --------------------------------------------------×= 1998 Mar 23 15 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 If r > 1 high frequency compensation is sometimes possible. A capacitor C1 given by: must then be connected across the input and a capacitor C2 given by: must be connected across the output. 6.5 Symmetrical 9 : 1 Impedance Transformer The schematic diagram is given in Fig.19. The two transmission lines have an optimum characteristic resistance of 3R. They can again be wound on a common core, and the third line omitted as in the previous case. The input impedance is given by: in which r is the ratio between the actual characteristic resistance and 3R. If r > 1 high frequency compensation is sometimes possible. A capacitor C1 given by: must than be connected across the low impedance side, and a capacitor C2, given by: must be connected across the high impedance side. Fig.18 Asymmetrical 1 : 4 impedance transformer. handbook, halfpage 4RV MGL226 V − 2V R C1 1 cosβ l 1 cosβ l+( ) 2 r2sin2β l––+ ωmaxrRsinβ l ------------------------------------------------------------------------------------------------= C2 2 cosβ l 1 cosβ l+( ) 2 r2sin2β l––+ 4ωmaxrRsinβ l ------------------------------------------------------------------------------------------------= Zin 9R 4 5cosβ l j6rsinβ l+ + 9cosβ l j6 r ---sinβ l+ ------------------------------------------------------×= C1 4 5cosβ l 9 3cos2β l 4cosβ l 2+ +   36r2sin2β l––+ 6ωmaxrRsinβ l ------------------------------------------------------------------------------------------------------------------------------------------= C2 9cosβ l 9 3cos2β l 4cosβ l 2+ +   36r2sin2β l–– 54ωmaxrRsinβ l --------------------------------------------------------------------------------------------------------------------------------= 1998 Mar 23 16 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 6.6 Asymmetrical 1 : 9 Impedance Transformer In this transformer (see Fig.20), two transmission lines are required, each having an optimum characteristic resistance of 3R. Although these lines can be wound on a common core, the upper line must have twice the number of turns of the lower line, because of the voltage division. The third line has again been omitted. High frequency compensation follows the same principle as that used in previous sections. Fig.19 Symmetrical 9:1 impedance transformer. handbook, halfpage R MGL227 +1/2 V +1/6 V +1/6 V − 1/2 V − 1/6 V − 1/6 V 9R Fig.20 Asymmetrical 1 : 9 impedance transformer. handbook, halfpage 9R 2V 3V V MGL228 V R 1998 Mar 23 17 Philips Semiconductors Design of HF wideband power transformers Application Note ECO6907 6.7 Single-ended Hybrid This circuit (see Fig.21) permits the combination of two signals in a common load (R/2) in such a wa