DE20207425U1 - Diplexer with unbalanced Butterworth filters - Google Patents

Description

200205628

Description

Diplexer with asymmetrical Butterworth filters

The invention relates to a diplexer which comprises at least one input connected to a source impedance, a first output connected to a first load impedance and a second output connected to a second load impedance, a first filter arranged between the input and the first output with a first pass frequency range and a second filter arranged between the input and the second output with a second pass frequency range different from the first. Such a diplexer is known from US 5,180,999 .

A diplexer is used in particular as a crossover in high frequency technology. As described in US 5,180,999, for example, an input signal of the diplexer is split into two or more partial signals, each with a different passband frequency range, and each is fed to different outputs of the diplexer. Accordingly, a diplexer has two or more branches, each of which contains a frequency-selective filter. In particular, these frequency filters are purely passive, for example as ladder networks made up of an alternating interconnection of passive longitudinal and transverse reactances.

Preferably, a diplexer should provide a sharp separation between the different pass-through frequency ranges. In addition, it is desirable to match the source impedance of the circuit arrangement upstream of the diplexer as closely as possible in order to exclude reflection of the input signal as far as possible.

In order to meet these requirements, among others, the frequency-selective and bi-directional transmission system known from US 3,806,813 has

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The amplifier device for the functional amplification has two special diplexers. The frequency-selective circuits used in the two diplexer branches are each designed as pseudo-complementary elliptical filters. With this type of diplexer, an input impedance can be achieved that has very good, but still not exact, homogeneity over the frequency range of interest and thus also not perfect impedance matching.

The object of the invention is therefore to provide a diplexer of the type described at the outset, which ensures an exact source impedance adaptation for all frequencies and also a sharp separation between the pass-through frequency ranges.

To solve the problem, a diplexer according to the features of patent claim 1 is specified.

The diplexer according to the invention is characterized in that the source impedance and the two load impedances have the same impedance value and that the first filter is designed as an asymmetrical Butterworth filter with an order of at least 3 and the second filter is designed as an asymmetrical Butterworth filter with an order of at least 3 that is complementary to the first filter.

The invention is based on the finding that a diplexer which is loaded with an electrical impedance of the same value both at the input and at its two outputs, i.e. is connected symmetrically, has a uniform input impedance for all frequencies, especially if the two filters of the diplexer are each designed as asymmetrical Butterworth filters.

A Butterworth filter is basically known from filter theory. It is described, for example, in "Electronic Filter Design Handbook"*, by A. B. Williams and F. J. Taylor,

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McGraw-Hill Inc., 3 ^rd Edition, pp. 2.34-2.35, table 11-2. The use of such a Butterworth filter in a diplexer is also not unusual in principle. The essential fact of the invention, however, is that with the symmetrical circuit described above, Butterworth filters designed for asymmetrical circuitry lead to a uniform input impedance of the diplexer for all frequencies.

In this context, an asymmetrical Butterworth filter is understood to mean that the filter coefficients of the Butterworth filter tabulated in the usual standard works on filter theory are selected for an asymmetrical circuit, i.e. a short-circuit circuit at the input and a different circuit with an impedance greater than 0 Ω, for example the line characteristic impedance or an infinite resistance (= open circuit), at the two outputs.

The choice of filter coefficients is therefore in direct contradiction to the actual physical circuitry of the diplexer. Nevertheless, it is precisely this choice of coefficients that leads to the aforementioned favorable behavior in relation to the input impedance of the diplexer. Due to the uniform input impedance of the diplexer, this favorable configuration also allows the diplexer input impedance to be adapted for all frequencies to the source impedance of the upstream circuit arrangement. In addition to Butterworth filters, so-called Bessel filters or Chebyshev filters are also commonly used in filter theory. However, none of these other filter types can achieve a comparably favorable behavior in terms of impedance adaptation.

The respective passband frequency ranges of the two diplexer filters are chosen to be complementary to each other. This means that they essentially cover non-overlapping parts of the total frequency range. In a transition region

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There is a slight overlap between the two filter frequency responses between the two pass-through frequency ranges. The higher the filter order of the Butterworth filter is selected, the sharper the separation between the two pass-through frequency ranges is. From a filter order of 3, a very good separation between the two pass-through frequency ranges is achieved. Together with the favorable choice of filter coefficients described above in accordance with the asymmetrical load condition, this achieves both impedance matching for all frequencies and a sharp separation between the pass-through frequency ranges of the two filters used in the diplexer.

Advantageous embodiments of the diplexer according to the invention emerge from the dependent claims.

Preferably, the passband frequency range of the first filter is designed for low frequencies and that of the second filter for high frequencies. The first filter is therefore a low-pass filter and the second filter is a high-pass filter.

A variant is also possible in which the first filter has a bandpass characteristic and the second filter has a complementary bandstop characteristic.

A diplexer with the properties described above can also be used in magnetic resonance (MR) technology. It is used there, for example, as a mixer termination. A mixer is required in MR technology, for example, in the receiving branch in order to convert a received MR signal (e.g. of frequency 64MHz) to a lower and therefore easier to process intermediate frequency position, e.g. at 1MHz. In order for a ring mixer that is usually used to work very well, it should be terminated with a uniform impedance for all frequencies, for example a line impedance of 50Ω. The diplexer precisely fulfills this requirement.

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Preferred, but in no way restrictive, embodiments of the diplexer will now be explained in more detail with reference to the drawing. For clarity, the drawing is not to scale and certain features are shown schematically. In detail, the following show:

Figure 1 shows a circuit arrangement with a diplexer with two asymmetrical, complementary 3rd order Butterworth filters, Figure 2 shows the frequency response of the two output voltages

and the input reflection factor of the circuit arrangement of Figure 1,

Figure 3 shows a circuit arrangement with a diplexer with two asymmetrical, complementary

4th order Butterworth filters and

Figure 4 the frequency response of the two output voltages

and the input reflection factor of the circuit arrangement of Figure 3.
20

Corresponding parts in Figures 1 to 4 are provided with the same reference numerals.

Figure 1 shows a circuit arrangement 100 with a diplexer 110. The diplexer 110 comprises a first branch with a first filter 20 and a second branch with a second filter 30. The diplexer 110 also has an input E and a first output A1 at the end of the first branch and a second output A2 at the end of the second branch.

At input E, the diplexer 110 is connected to a signal source 10, which can be interpreted as an equivalent circuit for a largely arbitrary upstream network. The signal source 10 consists of an ideal voltage source 11 with a vanishing internal resistance and a source impedance ZQ. The signal source 10 calls at

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Input E of the diplexer 110 produces an input voltage UE. In the example, this is a unit voltage.

The two outputs A1 and A2 of the diplexer 110 are connected to a first and a second load impedance ZL1 and ZL2, respectively. A first and a second output voltage ULI and UL2 are dropped across them.

The source impedance ZQ and the two load impedances ZL1 and ZL2 each have the same impedance value. In particular, this uniform impedance value is equal to the characteristic impedance of the circuit arrangement 100. Typical values of such a characteristic impedance that is usually used are 50 Ω or 75 Ω. In the example in Figure 1, a value of 1 Ω is chosen in analogy to the notation used in the tables and textbooks. However, conversion to another impedance value, e.g. 50 Ω, is easily possible.

The first and second filters 20 and 30 are each designed as third-order Butterworth filters. The first filter 20 is a low-pass filter in the form of a purely passive ladder network with longitudinal inductances L21 and L23 and the transverse capacitance C22 arranged between them. The second filter 30, on the other hand, is a high-pass filter, also in the form of a purely passive ladder network, but with longitudinal capacitances C31 and C33 and with a transverse inductance L32 arranged between them. The two filters 20 and 30 are complementary to one another. This means that their respective pass frequency ranges do not overlap, apart from a transition region, and both together cover the entire frequency range of interest.

The dimensioning of the reactive components (= longitudinal/transverse inductances/capacitances) used to construct the two filters 20 and 30 is carried out according to the principles given in every standard work on filter theory, such as in "Electronic Filter Design Handbook", by A. B. Williams and

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FJ Taylor, McGraw-Hill Inc., 3 ^rd Edition, pp. pp. 2.34-2.35, table 11-2, listed filter coefficients. As is common practice in filter theory, both a standardized frequency f and standardized impedance values are used. In the example shown in Figure 1, the frequency standardization is carried out with reference to lrad=l/(27i:) Hz. The impedance standardization is carried out with reference to the uniform impedance value of the source impedance ZQ and the two load resistances ZLl and ZL2 (= 1Ω).

Although the circuit arrangement 100 provides for a symmetrical connection of the diplexer 110, the filter coefficients taken from the standard work on filter theory belong to a Butterworth filter that is designed for an asymmetrical connection, i.e. a short-circuit connection at the input E and a connection with the standard impedance at the two outputs Al and A2. The coefficients for the standardized transverse/longitudinal inductances/capacitances are provided in accordance with the following Table 1:

Coefficients
of the filter 20 Coefficient-
Values Coefficients
of the filter 30 Coefficient-
Values L21 3/2 C31 2/3 C22 4/3 L32 3/4 L23 H C33 2

The following coefficient relationship exists between the first filter 20, which is designed as a low-pass filter, and the second filter 30, which is designed as a high-pass filter:

C3v = L3v =

LIV with &ngr; - 1, 3, V = 1, 3, . 1 with 2, 4, Clv 2, 4, V =

... (n-1) for even &eegr; and

. &eegr; for odd &eegr; .. &eegr; for even &eegr; and

... (n-1) for odd n.

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This relationship results from the fact that the two filters 20 and 30 are Butterworth filters that complement each other. The special choice of asymmetrical Butterworth filters with a simultaneous, actually symmetrical wiring of the diplexer 110 leads to an unexpectedly particularly advantageous behavior of the input impedance ZE of the diplexer 110. In this case, there is an impedance matching of the input impedance ZE to the source impedance ZQ that applies to all frequencies f.

This particularly favorable behavior is clear from the two diagrams shown in Figure 2. In the upper of the two diagrams, the magnitude curves of the two output voltages ULI and UL2 are shown semi-logarithmically in relation to the input voltage UE present at the input E of the diplexer 110, which in this case has been assumed to be a unit voltage. The characteristic low-pass and high-pass behavior of the first filter 20 (solid line) and the second filter 30 (dashed line) is clear. With the low-pass and high-pass filters used, the two frequency responses cross at 3 dB and a frequency f = 1 rad = 1/(2π) Hz «0.159 Hz.

The behavior shown in the lower diagram of Figure 2 is decisive for the constancy of the input impedance ZE of the diplexer 110 and thus for the impedance matching. The input reflection factor 20 Ig (ZE - ZQ/ZE + ZQ) is plotted. The value of the input reflection factor varies between -300 and -500 dB. This is an extremely low value, which is also only due to the resolution of the computer used for the simulation. In fact, there is a vanishing input reflection factor for all frequencies. The impedance matching for all frequencies f mentioned above is therefore actually given.

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·

If a sharper separation between the passband frequency range of the first filter 20 and that of the second filter 30 is desired while still maintaining impedance matching for all frequencies, corresponding higher-order Butterworth filters can be selected. For fourth-order Butterworth filters, the first filter 20 would be supplemented by a transverse capacitance C24 (not shown in Figure 1) and the second filter 30 by a transverse inductance L34 (also not shown in Figure 1). The coefficients of the Butterworth low-pass and high-pass filters formed in this way are summarized in the following Table 2:

Coefficients
of the filter 20 Coeffi cient-
Values Coefficients
of the filter 30 Coefficient-
Values L21 2V2 C31 1 C22 /2+ V2 L32 L21 L23 (1 + V2V2 + V2 C33 1 C24 2V2 L34 C22 2 1 V2 + V2 123 V2+V2 1 2·(δ+λ/2) C24

There are also other applications than the circuit arrangement 100 shown in Figure 1, which performs a low-pass/high-pass division of the frequency range. In an alternative diplexer application, a band-pass/band-stop division is provided instead.

A circuit arrangement 200 with such a frequency division is shown in Figure 3. Within the framework of filter theory, it is generally known to convert a low-pass/high-pass configuration into a band-pass/band-stop configuration. The longitudinal/transverse inductances/capacitances are then converted into longitudinal/transverse series/parallel resonant circuits. The low-pass to band-pass and high-pass to band-pass

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Lock transformations are well known in filter theory (see “Electronic Filter Design Handbook*, by AB Williams and FJ Taylor, McGraw-Hill Inc., 3 ^rd Edition, pp. 4.1-4.3, 5.5.4-5.5.7, 6.1-6.3).
5

After these transformations, the circuit arrangement 200 shown in Figure 3 results with the diplexer 210, which has a first filter 40 designed as a bandpass filter and a second filter 50 designed as a bandstop filter. The first filter 40 contains two longitudinal series resonant circuits, which are formed by the longitudinal inductance L41 and the longitudinal capacitance C42 or by the longitudinal inductance L45 and the longitudinal capacitance C46, as well as two transverse parallel resonant circuits, which are formed from the parallel inductance L43 and the parallel capacitance C44 or from the parallel inductance L47 and the parallel capacitance C48. The complementary second filter 50, on the other hand, contains two longitudinal parallel resonance circuits, which are formed from the longitudinal capacitance C51 and the longitudinal inductance L52 or from the longitudinal capacitance C55 and the longitudinal inductance L56, as well as two transverse series resonance circuits, which are formed from the transverse capacitance C53 and the transverse inductance L54 or from the transverse capacitance C57 and the transverse inductance L58.

From the coefficients in Table 2 and taking into account a scaling factor of 0.5 for the relative bandwidth, the coefficients listed in Table 3 below result for the components of the two filters 40 and 50 shown in Figure 3:

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Coefficients
of the filter 40 Coefficient-
Values Coefficients
of the filter 50 Coefficient-
Values L41 471 C51 &lgr;/2772 C42 72+71 L52 TI L43 /2+~72 C53 TI C44 4&lgr;/2 L54 λ/2 +TI L45 TI C55 (1+Ti)· T2+T2 C46 (idiagr; + TI)·72 + 72 L56 4T2 L47 (&igr; + 7&Igr;)·λ/2&Tgr;72 C57 4&lgr;/2 C48 TI L58 (l + λ/2") · λ/2 + λ/2 4 &Lgr;^&Tgr;&Idigr; T2+T2 1 V2+72 72+71 4 72+71 1+71 4·(&igr;+7&Igr;) /2 + 7λ 4·(&igr;+7&Igr;) 72+71 72+71 1+71

The filters 40 and 50 used in the diplexer 210 of Figure 3 are also asymmetrical Butterworth filters, so that the favorable impedance matching of the input impedance ZE of the diplexer 210 to the source impedance ZQ is restored. This is evident from the two diagrams in Figure 4. In the upper diagram in Figure 4, the magnitude frequency response of the two output voltages ULI and UL2 is again shown semi-logarithmically. The bandpass behavior of the first filter 40 (solid line) and the bandstop behavior of the second filter 50 (dashed line) are clearly expressed. The -3 dB bandwidth of the first filter, which is designed as a bandpass, is 1/(4π) Hz, i.e. 50% of the center frequency of the first filter 40, which is at f = 1/(2π) Hz. The crossing points of the frequency responses of the two filters 40 and 50 are at f = 3/{87c) Hz «0.119 Hz and at f = 5/(8π) Hz «0.199 Hz.

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The second diagram shows, just as in the circuit arrangement 100 of Figures 1 and 2, an input reflection factor whose frequency response fluctuates between -300 dB and -500 dB depending on the resolution. The diplexer 210 of the circuit arrangement 200 also has a vanishing input reflection factor and thus an impedance matching to the source impedance ZQ that applies to all frequencies f.

In addition to the diplexers 110 and 210 shown in Figures 1 and 3, other diplexers with advantageous impedance matching applicable to all frequencies are possible. For this purpose, asymmetrical and mutually complementary Butterworth filters are to be provided for the filters in the two diplexer branches. The Butterworth filters can also have any higher order than the filters shown in Figures 1 and 3. The coefficients of these filters can be taken from the known filter tables. The higher the order of the filters used, the sharper the separation between the pass-through frequency ranges, while at the same time providing favorable impedance matching for all frequencies f.

Claims (3)

1. Diplexer comprising at least a) an input (E) connected to a source impedance (ZQ), a first output (A1) connected to a first load impedance (ZL1) and a second output (A1) connected to a second load impedance (ZL2), b) a first filter ( 20 , 40 ) arranged between the input (E) and the first output (A1) with a first pass frequency range and c) a second filter ( 30 , 50 ) arranged between the input (E) and the second output (A2) with a second pass frequency range different from the first characterized in that a) the source impedance (ZQ) and the two load impedances (ZL1, ZL2) have the same impedance value and b) the first filter ( 20 , 40 ) is designed as an asymmetrical Butterworth filter with an order of at least 3 and the second filter ( 30 , 50 ) is designed as an asymmetrical Butterworth filter with an order of at least 3 that is complementary to the first filter ( 20 , 40 ). 2. Diplexer according to claim 1, wherein the first filter ( 20 ) is a low-pass filter and the second filter ( 30 ) is a high-pass filter. 3. Diplexer according to claim 1, wherein the first filter ( 40 ) is a bandpass filter and the second filter ( 50 ) is a bandstop filter.