US3868587A

US3868587A - Constant phase distributed impedance

Info

Publication number: US3868587A
Application number: US227715A
Authority: US
Inventors: Amos Nathan; Reed K Even
Original assignee: Individual
Current assignee: Individual
Priority date: 1971-10-19
Filing date: 1972-02-22
Publication date: 1975-02-25
Anticipated expiration: 1992-02-25

Abstract

An electrical device is described herein which has an input impedance of substantially constant phase angle over a limited frequency range. It consists of a distributed circuit having a capacitive and resistive medium between two electrodes, of appropriate properties. Implementations can use thin film techniques or monolythic techniques.

Description

United States Patent [1 1 Nathan et al.

[4 1 Feb. 25, 1975 CONSTANT PHASE DISTRIBUTED IMPEDANCE [76] Inventors: Amos Nathan, Haifa, Israel; Reed K.

Even, Middletown, NJ.

[22] Filed: Feb. 22, 1972 [21] Appl. No.: 227,715

[30] Foreign Application Priority Data Oct. 19, 1971 Great Britain 48502/71 [52] U.S. Cl 330/38 M, 330/185, 333/70 CR [51] Int. Cl. 1103f 3/14 [58] Field of Search 333/70 CR, 70 S, 21 R,

[56] References Cited UNITED STATES PATENTS 1/1949 Boghosian et al. 333/70 CR X 7/1965 Barditch et al 333/70 CR 3,233,196 2/1966 Osafune et al. 333/70 CR 3,371,295 2/1968 Bourgault et al 333/70 CR 3,432,778 3/1969 Ertel 333/70 CR FOREIGN PATENTS OR APPLICATIONS 1,349,378 12/1963 France 330/70 S OTHER PUBLICATIONS Electronics, Sept. 4, 1959, pp. 44-49, Vol. 32 No. 36, Network Design of Microcircuits, by Hagar.

Primary Examiner-Nathan Kaufman [57] ABSTRACT An electrical device is described herein which has an input impedance of substantially constant phase angle over a limited frequency range. It consists of a distributed circuit having a capacitive and resistive medium between two electrodes, of appropriate properties. 1mplementations can use thin film techniques or monolythic techniques.

7 Claims, 11 Drawing Figures PPJENTEU FEB 2 51975 sum 1 UF 2 CONSTANT PHASE DISTRIBUTED IMPEDANCE This invention relates to distributed RC structures whose input impedance has a substantially frequency independent, i.e. constant, phase angle over a range of frequencies.

The phase angle is defined as the angle between electrical current and voltage in the steady state sinusoidal mode of operation, or, equivalently, as the angle between them if they are expressed as complex quantities, as well known in the description of alternating currents. Impedance is defined as the ratio of complex voltage and complex current. R stands for resistance and C denotes capacitance.

It is well known in the art how to implement impedances of substantially constant phase angle over some frequency range in lumped electrical circuits. Such impedances will, for short, be called in the sequel constant phase impedance." For example, in the following paper: R. Morrison, RC constant argument driving point admittances," IRE Transactions of Circuit Theory, volume CT-6, pages 310-317, September, 1959, RC ladder circuits are described for this purpose.

This invention provides constant phase impedances in devices using distributed structures.

It is thus an object of the invention to provide constant phase distributed impedance.

It is a further object of the invention to provide constant phase impedance operating over a wider frequency range or with a better phase constancy than prior art lumped circuits.

Other objects of the invention are implementations of constant phase impedance in thin film and in monolythic devices.

The invention possesses other objects and features of advantage, some of which of the foregoing will be set forth in the following description of the preferred form of the invention which is illustrated in the drawings accompanying and forming part of this specification. It is to be understood, however, that variations in the showing made by the said drawings and description may be adopted within the scope of the invention as set forth in the claims.

FIG. 1 is a perspective view of an embodiment of a device according to the invention;

FIG. 2 is a sectional cut through such a device and also defines a conceptual system of coordinates;

FIG. 3 is a schematic diagram of an equivalent circuit for a small section of the device;

FIG. 4 is a section through the device of FIG. 1 and also includes a schematic diagram of a compensating lumped RC network;

FIG. 5 is a section through another embodiment of the invention according to FIG. 4;

FIGS. 6 and 7 are sectional cuts through yet further embodiments of the invention;

FIGS. 8, 9, and 10 are schematic diagrams showing the use of the invention for the provision of constant phase transfer function devices;

FIG. 11 is a curve typical of phase deviations from the desired constant value, in devices according to the invention.

The theory and practice of devices according to the invention will now be described. v

In FIGS. 1 and 2, l and 2 are conductive electrodes, Q is a dielectric layer, and Q a resistive layer. The device is accessible through terminals shown in FIG. 2, which are conductively connected to electrodes 1 and g. The constant phase impedance is produced across terminals 5 xyz is a Cartesian system of coordinates. The section of the device between coordinates x and .r dx is equivalent to the series combination of capacitance c(x)d.r and conductance g(x)dx, as shown diagramatically in FIG. 3. Conductance g(.r) is the reciprocal of resistance r(x), i.e., g(x) l/r(.r). In this instance, conductance, capacitance, and resistance are to be understood as specific values, i.e. per unit length." The admittance of such a section is then given by H y(j g( /I g( )/(j where cu is the angular frequency, defined as 2 11- times frequency.

If the structure extends from x x, to x x (in the x direction) then the admittance between terminals is givenby n 22 YUw)=f y(jw,x)dx

Admittance is defined as the reciprocal of impedance. Defining where 1 (x) is an arbitrary differentiable function and (x) is the inverse function, defined through then the admittance function m, U 1( )l n( corresponds to a structure extending in the x direction from x (x to x (x that likewise realizes the admittance Y (j w). The transition from y to y'through use of 17(x) will be called a scale transformation. Thus, any realization c rresponding to some function y (j w, x) is equivalent/to a family of realizations generated from it by all possible scale transformationsj here denotes the square root of l.

If the structure is infinite (in the x direction) then a constant phase distributed structure results from the laws:

where R and C are constant resistance and capacitance, respectively, and a and b are constants. Defining k a/b the constant phase angle is e= l/l+k'1r/2=1r/2 b/a-i-b andthe inputadrnittance can be shown to be given by As remarked above, it is still possible to perform scale transformations and thereby obtain equivalent realizations.

The relations are transformed through a scale transformation into 3 If the device were infinite, then, with the relations c(x) C, if; 'y(x) R, e

- and the definitions there follows, as can be shown,

where L L are finite, and the corresponding structure extends from x L, to x L in the x-direction. Two specific examples will now be given, each of length 2L, with the structure extending from L to +L. (a) With 17(x) =L sgn X In (I IxI/L; (x)= L sgnx there follows (b) With I n(x) 2/90 L tan (/n'x/L); (x) 2/90 L tan /mx/L) I there follows 7 FIGTGTot to scalei 1s a gerieial view of a device embodying one implementation of the invention and FIG. 4 is the corresponding sectional view. FIG. 2 is a section through a more general embodiment of the invention and also defines an associated conceptual coordinate system. In the device according to FIG. 4, the required laws for C(x) and r(x) are produced by having variable width of structure, w(x), in the y direction and variable height, h(x), of the resistive layer, in the z direction. A dielectric layer, 2, of dielectric constant e and constant thickness d has variable width w(x) and is adjacent to electrode 1. Resistive layer f thickness h(x) fill the space between the said dielectric layer and a second electrode 2. The part of the structure between x x, and x x satisfies the laws found above for r(x) and C(x), provided that, in this interval,

In the embodiment of FIG. 4, in parallel with this structure, there are connected between electrodes 1 and 2 capacitor 9 and resistor 10 for the improvement of the constancy of phase angle. Alternatively, resistor and capacitor can be replaced by distributed structures. For example, as shown in FIG. 5, the required capacitance extends from x to X1 and is shown at 7, and the required resistance extends from x to x, and is shown at 8. In this example, the latter is in series with a large capacitance formed through dielectric medium 6, in order to provide infinite impedance at zero frequency, which is at times of advantage. The width and height of the structure in the regions x to x, and x to x, are constant, in this example. The effect on the input impedance of the parts of structure from x to x, and from x to x, is to compensate for the cut off infinite lengths at both ends of the variable resistance variable capacitance part of the structure extending from x, and x rather than from minus to puls infinity. If properly chosen, these additions improve performance.

Denoting the dielectric constant of layer 3 in FIGS. 2, 3, and 4 by e and the resistivity of layer 4 by p there holds By use of a scale transformation, this structure can be transformed into another one of constant width. For this purpose we take the function 7( In m l- (4 2; where ,u is a constant, and obtain for c(x) and r(x) the relations 0 7( 0 w and for the end points of the shaped structure (formerly x to x respectively.

Similarly, a scale transformation with (v 0; p. a) /2vx 1) x ,u. e'" /2 where u, v are constants, transforms C(X) C eax into 0 (ILX which is a linear function of x.

Note also that 11(x) x corresponds to r(x) r(x), y( y( For a device according to the invention and extending from x x, to x x and such that let the frequency range of operation be defined by L A a These relations permit the easy approximate calculation of the required values of the compensating lumped capacitance and resistance. y 'y must be varied around unity in order to obtain the best results for given specifications.

A specific example of the invention embodying the principles and descriptions as hereinbefore set forth is as follows: For a phase angle of (b =45 and a length of structure corresponding to -ax ax 3, with 'y, 1.9 and 7 1.7; and operating range for w of between 0.07 (n and 10.0 there resulted a phase angle error of the order of not more than 1.

In one embodiment of the invention according to FIG. 4 the resistive layer is composed of cermet with a resistivity of the order of p 1 ohm meter and the capacitance is composed of tantalum oxide with a capacitance per unit area of the order of c 10 ,uF/m Taking 4) 45 which corresponds to 1r/4 radians, it follows that a b and p 0. We take x, l0 meters and x meters, so that the length of the structure is x -x 2 centimeters and a=b= l l5/meter. In this example, the maximum value of h is 10 meters and its minimum value is 10 meters. Electrodes l and 2 are themselves made of tantalum and the capacitive layer is produced by oxidation. W, 6.3 l0" meters, so that w(x) increases from a minimum value of 2X10 meters to a maximum value of 2X10 meters. h,, 10' meters. The device performs properly around an angular frequency of the order of or 1.60 Megahertz, approximately. The compensating capacitor and resistor have the approximate values c. E 1.73 p. F; R E 0.058 ohms,

respectively. In another embodiment the compensating capacitor and resistor are as above, but the compensating resistor is placed series with a capacitor having a capacitance that presents a virtual short circuit at the operating frequencies; i.e., it is sufficiently great.

A further embodiment of the invention corresponding to FIG. 4 uses a silicon wafer doped on one side so as to form electrode 2, the doping on this side being such as to provide good conductivity. The rest of the wafer, with the exception of the opposite face, has constant doping and variable height and implements the required law of resistance, and the other face is uniformly oxidized, forming silicon oxide. 1 is an adjacent metal electrode, and the capacitance is formed between it and the bulk of the wafer through a layer of silicon oxide. Typical values are: Resistivity of the silicon resistor: p 10 to 10 am, MOS capacitor: 0 3 l0 ai /m The capacitor formed according to the above description is called an M08 capacitor. Maximum height of the resistive layer: 2.5 10" meters. The middle angular frequency w is approximately 4 l0 secor about 60 kHz. lf gallium arsenite is used instead of Silicon as basic material, the middle frequency is about 1 kHz, for a ratio of maximum to minimum height of the variable height part of the device of 100 to I.

It is also possible to use a layer of constant height and variable doping in order to implement the required law of resistance. Variable height and doping can also be used for the formation of the required resistance function.

FIGS. 6 and 7 relate to a yet further implementation of the invention. 12 is an insulating substrate, such as glass, for example. The electrodes are 1 and 2. Electrode 1 is oxidized to form a dielectric layer 3 of constant thickness. Electrode 2 is oxidized in one example. In another example it is not oxidized. The electrodes are made, in one example of the invention, of tantalum and the dielectric consists of tantalum oxide. The structure is covered by a resistive layer 4. The resistance law between the electrodes is controlled by variations in the distance between them. These variations are not shown in the FIGURES. The comb like arrangement in FIG. 7 merely provides a convenient method to accommodate a great length of structure in a limited space, or with a limited length. If the structure of FIG. 6 is used, then FIG. 7 corresponds to a view from above of the section shown in FIG. 6. In the example of FIG. 7 the length of structure is only somewhat greater than one sixth its effective length, because there are exactly six slots between the teeth of the comb. Layer 4 is not shown in FIG. 7, nor are the variations in slot width in dicated.

In FIGS. 8, 9, and 10 it is shown how the invention can be used for the provision of constant phase transfer functions. In FIG. 8, l5 and 16 are input and output terminals, respectively. 19 is a high gain operational amplifier, l3 is the input impedance and 17 is the feedback impedance. In this example, 17 is aresistor and 13 is a constant phase impedance according to the invention. The device provides the substantially constant phase transfer function between

terminals

15 and 16. In the device corresponding to FIG. 9 the input impedance is provided by resistor 20 and the feedback im pedance is provided by a constant phase impedance 14 according to the invention. In FIG. 10 both input impedance l3 and feedback impedance 18 are constant phase impedances according to the invention, corresponding, however, to different phase angles. In this latter example the operating ranges of impedances l3 and 18 must be substantially the same and the phase angle between input terminal 15 and output terminal 16 is given by the difference between the phase angles of 13 and 18.

It is also possible to replace

resistors

17 and 20 by different impedances. For example, replacing resistor 17 by a capacitor, the transfer phase angle is shifted by ninety degrees.

FIG. 11 is a typical performance curve of an imped ance according to the invention. The curve corresponds to a device for a phase shift of 9. and also for a phase shift of 81, the two curves being co-incident. The curves are obtained for such a device if compensating resistors and capacitors are used as hereinbefore specified with values of coefficients y, Y2 l. The scale of abscissae is logarithmic. The ordinates are the deviation of the actual phase from the desired phase dz.

What we claim is:

1. An electrical device having an input impedance of substantially constant phase over a finite range of frequencies comprising:

impedance means having a first and a second surface,

the maximal distance therebetween being small compared with the later dimensions thereof;

first and second electrically conductive means substantially covering and contiguous to said first and second surfaces, respectively;

one of said lateral dimensions defining length;

where (the law of) said impedance means is constituted so that its impedance per unit length (of said impedance means) is substantially given by e is the basis of natural logarithms, x is a coordinate measuring said length, and R are predetermined non-vanishing constant capacitance and resistance, respectively, a and b are predetermined constants such that their product does not vanish, 17(x) is a predetermined non-constant function of x, d/dx 1 (x) is the derivative of 1;(x) with respect to x,j is the square root of l, and w is'the angular frequency;

said input impedance being the impedance between said plural conductive means.

2. The device as recited in claim 1 wherein said impedance means is comprised of dielectric means having said first surface and resistive means (adjacent to) contiguous with said dielectric means and having said second surface and so constitrited that the laws of capacitance C(x) and resistance r(x) per unit length of said dielectric means and said resistive means are substantially determined by respectively.

3. The device as recited in claim 2 wherein "1 (x) x and c(x) and r(x) are substantially given by exponential relations as functions of said coordinate x.

4. The device as recited in claim (5) 2 wherein said dielectric (layer) means has substantially constant height d and the required law of capacitance C(X) is implemented through variations with x of width w(x) (thereof) of said impedance means (and) substantially according to the relation wherein e is a constant and wherein the width of said layer is likewise w(x)) and the required law of resistance r(x) is implemented through variations with x of (its) height h'(x) of said resistive means substantially according to the relation wherein p is a constant; height being measured perpendicularly to said lateral dimensions and width being defined as the lateral extension perpendicular to said length dimension.

5. The device as recited in claim 2 wherein said plural conductive means are comprised of metal electrodes, and said dielectric means is comprised of a layer of metal oxide oxidized onto one face of one of said electrodes.

6. The device as claimed in claim 2 (and including) wherein said device is composed of a Silicon wafer having a first and a second face and doped on one of said faces to provide a low-resistance electrode, said electrode forming said first conductive means; and second face of said wafer oxidized to provide said dielectric (layer) means; a metal electrode (placed adjacent) contiguous (thereto) therewith to provide said second conductive means; (a second doped layer) the material of said wafer between said first electrode and said dielectric (layer) means doped to provide said resistive means.

7. The device as recited in claim 1 and including a lumped electrical network having first and second connections, connected (at one end) thereat to said first (conductive means and connected at a second end to said) and second conductive means, respectively, for

extending said frequency range.

Claims

1. An electrical device having an input impedance of substantially constant phase over a finite range of frequencies comprising: impedance means having a first and a second surface, the maximal distance therebetween being small compared with the later dimensions thereof; first and second electrically conductive means substantially covering and contiguous to said first and second surfaces, respectively; one of said lateral dimensions defining length; where (the law of) said impedance means is constituted so that its impedance per unit length (of said impedance means) is substantially given by z(x) (Roe b (x) + 1/j omega Co ea (x))/(d/dx eta (x) ), e is the basis of natural logarithms, x is a coordinate measuring said length, Co and Ro are predetermined non-vanishing constant capacitance and resistance, respectively, a and b are predetermined constants such that their product does not vanish, eta (x) is a predetermined non-constant function of x, d/dx eta (x) is the derivative of eta (x) with respect to x, j is the square root of -1, and omega is the angular frequency; said input impedance being the impedance between said plural conductive means.

2. The device as recited in claim 1 wherein said impedance means is comprised of dielectric means having said first surface and resistive means (adjacent to) contiguous with said dielectric means and having said second surface and so constitrited that the laws of capacitance c(x) and resistance r(x) per unit length of said dielectric means and said resistive means are substantially determined by c(x) Co e a (x) d kappa (x)/dx gamma (x) Ro e b (x) / (d eta (x)/dx ) respectively.

3. The device as recited in claim 2 wherein eta (x) x and (c(x) Co e ax ) gamma (x) Ro e bx ) c(x) and r(x) are substantially given by exponential relations as functions of said coordinate x.

4. The device as recited in claim (5) 2 wherein said dielectric (layer) means has substantially constant height d and the required law of capacitance c(x) is implemented through variations with x of width w(x) (thereof) of said impedance means (and) substantially according to the relation w(x) 1/ epsilon d. c(x) wherein epsilon is a constant ( and wherein the width of said layer is likewise w(x)) and the required law of resistance r(x) is implemented through variations with x of (its) height h(x) of said resistive means substantially according to the relation h(x) 1/ Rho r(x). w(x) wherein Rho is a constant; height being measured perpendicularly to said lateral dimensions and width being defined as the lateral extension perpendicular to said length dimension.

7. The device as recited in claim 1 and including a lumped electrical network having first and second connections, connected (at one end) thereat to said first (conductive means and connected at a second end to said) and second conductive means, respectively, for extending said frequency range.