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Electronic Applications of the Smith Chart

Introduction

1.1 Graphical vs. Mathematical Representations

The physical laws governing natural phenomena can generally be represented either mathematically or graphically. Usually the more complex the law the more useful is its graphical representation. For example, a simple physical relationship such as that expressed by Ohm's law does not require a graphical representation for its comprehension or use, whereas laws of spherical geometry which must be applied in solving navigational problems may be sufficiently complicated to justify the use of charts for their more rapid evaluation. The ancient astrolabe, a Renaissance version of which is shown in Fig. 1.1, provides an interesting example of a chart which was used by mariners and astronomers for over 20 centuries, even though the mathematics was well understood.

The laws governing the propagation of electromagnetic waves along transmission lines are basically simple; however, their mathematical representation and application involves hyperbolic and exponential functions (see Appendix A) which are not readily evaluated without the aid of charts or tables. Hence these physical phenomena lend themselves quite naturally to graphical representation.

Tables of hyperbolic functions published by A. E. Kennelly  in 1914 simplified the mathematical evaluation of problems relating to guided wave propagation in that period, but did not carry the solutions completely into the graphical realm.

Fig. I.1. A Renaissance version of the oldest scientific instrument in the world. (Danti des Renaldi, 1940.)

I.2 The Rectangular Transmission Line Chart

The progenitor of the circular transmission line chart was rectangular in shape. The original rectangular chart devised by the writer in 1931 is shown in Fig. 1.2. This particular chart was intended only to assist in the solution of the mathematics which applied to transmission line problems inherent in the design of directional shortwave antennas for Bell System applications of that period; its broader application was hardly envisioned at that time.

The chart in Fig. I.2 is a graphical plot of a modified form of J. A. Fleming's 1911 "telephone" equation , as given in Chap. 2 and in Appendix A, which expresses the impedance characteristics of high-frequency transmission lines in terms of measurable effects of electromagnetic waves propagating thereon, namely, the standing-wave amplitude ratio and wave position. Since this chart displays impedances whose complex components are "normalized," i.e., expressed as a fraction of the characteristic impedance of the transmission line under consideration, it is applicable to all types of waveguides, including open-wire and coaxial transmission lines, independent of their characteristic impedances. In fact, it is this impedance normalizing concept which makes such a general plot possible.

Although larger and more accurate rectangular charts have subsequently been drawn, their uses have been relatively limited because of the limited range of normalized impedance values and standing-wave amplitude ratios which can be represented thereon. This stimulated several attempts by the writer to transform the curves into a more useful arrangement, among them the chart shown in Fig. 7.7 which was constructed in 1936.

I.3 The Circular Transmission Line Chart

The initial clue to the fact that a conformal transformation of the circular orthogonal curves of Fig. I.2 might be possible was provided by the realization that these two families of circles correspond exactly to the lines of force and the equipotentials surrounding a pair of equal and opposite parallel line charges, as seen in Fig. I.1. It was then a simple matter to show that a bilinear conformal transformation [55,109] would, in fact, produce the desired results (see Appendix B), and the circular form of chart shown in Fig. I.3, which retained the normalizing feature of the rectangular chart of Fig. I.2, was subsequently devised and constructed. All possible impedance values are representable within the periphery of this later chart. An article describing the impedance chart of Fig. I.3 was published in January, 1939 .

During World War II at the Radiation Laboratory of the Massachusetts Institute of Technology, in the environment of a flourishing microwave development program, the chart first gained widespread acceptance and publicity, and first became generally referred to as the Smith Chart.

Descriptive names have in a few instances been applied to the Smith Chart (see glossary) by other writers; these include "Reflection Chart," "Circle Diagram (of Impedance)," "Immittance Chart," and "Z-plane Chart." However, none of these are in themselves sufficiently definitive to be used unambiguously when comparing the Smith Chart with similar charts or with its overlay charts as discussed in this text. For these reasons, without wishing to appear immodest, the writer has decided to use the more generally accepted name in the interest of both clarity and brevity.

Imped. Along Trans. Line vs. Standing Wave Ratio (r) and Distance (D), in Wavelengths, to Adjacent Current (or voltage) Min. or Max. Point

R

Dist. to Following Imin or Emax    R/Z0            Dist. to Following Imax or Emin

- (jX/Z0)               Normalized Reactance         + (jX/Z0)

Fig. I.2. The original rectangular transmission line chart.

Fig. I.3. Transmission line calculator. (Electronics, January, 1939.)

Drafting refinements in the layout of the impedance coordinates were subsequently made and additional scales were added showing the relation of the reflection coefficient to the impedance coordinates, which increased the utility of the chart. These changes are shown in Fig. I.4. A second article published in 1944 incorporated these improvements . This later article also described the dual use of the chart coordinates for impedances and/or admittances, and for converting series components of impedance to their equivalent parallel component values.

In 1949 the labeling of the chart impedance coordinates was changed so that the chart would display directly either normalized impedance or normalized admittance. This change is shown in the chart of Fig. 2.3. On this later chart the specific values assigned to each of the coordinate curves apply, optionally, to either the impedance or to the admittance notations.

In 1966 additional radial and peripheral scales were added to portray the fixed relationship of the complex transmission coefficients to the chart coordinates, as shown in Fig. 8.6.

I.4 Orientation of Impedance Coordinates

The charts in Figs. I.2 and I.3 as originally plotted have their resistance axes vertical. It became apparent shortly after publication of Fig. I.3, as thus oriented, that a horizontal representation of the resistance axis was preferable since this conformed to the accepted convention represented by the Argand diagram in which complex numbers (x ± iy) are graphically represented with the real (x) component horizontal and the imaginary (y) component vertical.

Therefore, subsequently published Smith Charts have generally been shown, and are shown throughout the remainder of this book, with the resistance (R) axis horizontal, and the reactance (± jX) axis vertical; inductive reactance (+ jX) is plotted above, and capacitive reactance (- jX) below the resistance axis.

I.5 Overlays for the Smith Chart

Axially symmetric overlays for the Smith Chart were inherent in the first chart, as represented by the peripheral and radial scales for the chart coordinates. These overlays include position and amplitude ratio of the standing waves, and magnitude and phase angle of the reflection coefficients. Additionally, overlays showing attenuation and reflection functions were represented by radial scales alone (see Fig. I.4).

Fig. I.4. Improved transmission line calculator (Electronics, January, 1944.)

In the present text 26 additional general-purpose overlays (both symmetrical and asymmetrical) for which useful applications exist and which have been devised for the Smith Chart are presented.

Fig. 3.1. Construction of normalized resistance circles for Smith Chart of unit radius.

Fig. 3.2. Construction of normalized reactance circles for Smith Chart of unit radius.

Vector Convention

Legend

i - Incident

r - Reflected

R - Resultant

Fig. 5.1. Vector representation of phase relations for voltages on impedance coordinates, or currents on admittance coordinates,

Smith Chart when SWR = 3.0 (refer to Table 5.1).

A relative phase lag of one vector over another is indicated by a negative sign (-) on the lagging vector, whereas a relative phase lead of one vector over another is indicated by a positive sign (+).

In accordance with the above convention, Fig. 5.1 shows a Smith Chart upon which eight specific vector representations of the voltage or current on the impedance or admittance coordinates, respectively, are plotted.

Network Impedance Transformations

Fig. 10.2. Impedances in unshaded areas of Smith Charts, represented by eight circular boundaries, are transformable to a pure resistance Z0 with specific L·type circuits indicated (transforming effect of each circuit element is indicated by a heavy line with arrows).

...for erasable chalk marking are also commercially available . (See Fig. 14.12.) These are printed with white characters and can be rolled up and down on a curtain roller. The blackboard chart is intended for basic instruction in a large classroom, and consequently designations are in a bold type and a coarse coordinate grid is employed. This chart is not suitable for accurate solution of specific problems.

Fig. 14.12. Blackboard Smith Chart for classroom use .

14.11 Mega-Charts

14.11.1 Paper Smith Charts

Regular size (8 1/2 x 11 in.) Smith Charts in the following several forms, each of which is described herein, are commercially available . These "Mega-Chart" forms are printed in red ink on 151b. (approximately 7lbs/ 1,000 sheets) translucent master paper, and packaged in clear plastic envelopes of 100 sheets each, either padded or loose:

1. Standard Smith Chart - Form 82-BSPR (9-66) (see Fig. 8.6).

2. Expanded Center Smith Charts ­ Form 82-SPR (2-49) (see Fig. 7.2).

3.Highly Expanded Center Smith Charts - Form 82-ASPR (see Fig. 7.3).

Also, Smith Charts with coordinates having negative real parts (negative Smith Charts) are available in the same paper packaging, printed in green ink, in the following form:

1. Negative Smith Chart - Form 82-CSPR (see Fig. 12.3).

14.11.2 Plastic Laminated Smith Charts

All of the above chart forms (except -the negative Smith Chart) are available  laminated to a thickness of 0.025 in., with a matte finish on the front for erasable pen marking. Abbreviated instructions for use of Smith Charts are printed on the back.

14.11.3 Instructions for Smith Charts

Abbreviated sets of instructions for use the Smith Chart, containing an explanation of the chart coordinates and radial scales and printed on single sheets, are available commercially  printed on 50 lb offset paper. These may be used for classroom instruction.

Glossary - Smith Chart Terms

The terms which appear on Smith Charts as coordinate designations, radially scaled parameters, peripheral scale captions, etc., are individually defined and reviewed in this glossary. A more complete discussion of these terms is found in applicable sections of the text.

Although relevant to all Smith Charts, these terms are specifically associated with the basic chart forms printed in Chaps. 6, 8 and 12, and enlarged chart forms described in Chap. 7. Other Smith Charts with which these terms are specifically associated include the normalized current and voltage overlay in Chap. 4, and the charts with dual (polar and rectangular) coordinate transmission and reflection coefficients in Chap. 8.

In the definitions which follow, certain qualifying words and phrases are omitted when, in the context in which the terms are used, these words and phrases will be understood to apply. For example, the phrase "at a specified frequency" will apply to many of the definitions, and the phrases "normalized input impedance of a uniform waveguide" and "normalized input admittance of a uniform waveguide" will generally be understood to be meant by the shorter terms, "waveguide impedance" and "waveguide admittance," respectively.

This glossary supplements definitions which have been formulated and published by the Institute of Radio Engineers (IRE)  (presently the Institute of Electrical and Electronics Engineers, IEEE), and by the American Standards Association (ASA-C42.65-1957) (presently the United States of America Standards Institute, USASI), and upon which usage of such terms in this text is based.

Angle of Reflection Coefficient, Degrees

At a specified point in a waveguide, the phase angle of the reflected voltage or current wave relative to that of the corresponding incident wave. The relative phase angle of the reflection coefficient; i.e., the total angle reduced to a value less than ± 180°, is generally indicated by this term. This relative phase angle has a fixed relationship to a specific combination of waveguide impedances or admittances and, accordingly, to a specific locus on the impedance or admittance coordinates of a Smith Chart, this locus being a radial line.

Note: The angles of both the voltage and the current reflection coefficients are represented on Smith Charts by a single linear peripheral scale, with designated values ranging between 0 and ± 180°. The angle of the voltage reflection coefficient is directly obtainable for any point on the impedance coordinates by projecting the point radially outward to this peripheral scale, labeled "Angle of Reflection Coefficient, Degrees." Similarly, the angle of the current reflection coefficient is directly obtainable for any point on the admittance coordinates. At any specified point along a waveguide the angle of the current reflection coefficient always lags that of the voltage reflection coefficient by 180°.

Angle of Transmission Coefficient, Degrees

At a specified point along a waveguide, the phase angle of the transmitted wave relative to that of the corresponding incident wave. The. "transmitted" wave is the complex ratio of the resultant of the incident and reflected wave to the incident wave. The angle of the transmission coefficient has a fixed relationship to a specific combination of waveguide impedances or admittances and, accordingly, to a specific locus on the impedance or admittance coordinates of a Smith Chart, this locus being a straight line stemming from the origin of the coordinates.

Note: The angles of both the voltage and the current transmission coefficients are represented on Smith Charts by a single linear angle scale at the periphery, referenced to the origin of the impedance or admittance coordinates, and ranging between 0 and ±90°. The angle of the voltage transmission coefficient is directly obtainable for any point on the impedance coordinates by projecting the point along a straight line stemming from the origin of the impedance coordinates to the intersection of the peripheral scale labeled "Angle of Transmission Coefficient, Degrees." Similarly, the angle of the current transmission coefficient is directly obtainable for any point on the admittance coordinates.

Attenuation (1 dB Maj. Div.)

The losses due to dissipation of power within a waveguide and/or the radiation of power therefrom when the waveguide is match-terminated. On Smith Charts attenuation is expressed as a ratio, in dB, of the relative powers in the forward-traveling waves at two separated reference points along the waveguide.

Note: On a Smith Chart, "attenuation' is a radially scaled parameter. The attenuation scale is divided into dB (or fraction of dB) divisions which are not designated with specific values, with an arbitrarily assignable (floating) zero point. The number of attenuation scale units (dB) radially separating any two impedance or admittance points on the impedance or admittance coordinates of a Smith Chart is a measure of the attenuation in the length of waveguide which separates the two reference points.

Coordinate Components

The normalized rectangular components of the equivalent series or parallel input impedance or admittance of a waveguide or circuit. which are represented on Smith Charts by two captioned families of mutually orthogonal circular curves comprising the coordinates of the chart.

Note 1: Coordinate components on the three Smith Charts printed in red on translucent sheets in the back cover envelope are:

Chart A

1. the equivalent series circuit impedance coordinates: resistance component R/Z0 and inductive (or capacitive) reactance component ± j X/Z0.

2. the equivalent parallel circuit admittance coordinates: conductance component G/Y0 and inductive (or capacitive) susceptance component -/+ jB/Y0.

Chart B

1. the equivalent parallel circuit impedance coordinates: parallel resistance component R/Z0 and parallel inductive (or capacitive) reactance component ± jX/Z0.

2. the equivalent series circuit admittance coordinates: series conductance components G/Y0 and series inductive (or capacitive) component -/+ jB/Y0.

Chart C

1. the equivalent series circuit impedance or shunt circuit admittance coordinates with negative real parts: negative resistance component -R/Z0 and negative conductance component -G/Y0.

Note 2: The inductive reactance and inductive susceptance coordinate components represent equivalent primary circuit elements which are capable of storing magnetic field energy only. The resistance component and the conductance component of the coordinates represent equivalent primary circuit elements which are capable of dissipating electromagnetic field energy. The negative resistance component and the negative conductance component of the coordinates represent equivalent circuit elements which are capable of releasing electromagnetic field energy, as would be represented by the equivalent circuit of a generator.

The families of orthogonal circular curves representing the real and imaginary compo­nents of the waveguide or circuit impedance and/or admittance, and comprising the main body of a Smith Chart. The designated values of the curves are normalized with respect to the characteristic impedance and/or the characteristic admittance of the waveguide, and the entire range of possible values lies within a circle. Enlarged portions of Smith Chart coordinates are sometimes used to represent or display a portion of the total area of the coordinate system, thereby providing improved accuracy or readability.

Note: Most commonly, Smith Chart impedance or admittance coordinates express components of the equivalent series circuit impedance or parallel circuit admittance. However, a modified form of Smith Chart expresses components of the equivalent parallel circuit impedance or series circuit admittance. A coordinate characteristic which is common to all Smith Charts is that a complex impedance point on the impedance coordinates and a complex admittance point on the admittance coordinates which is diametrically opposite, and at equal chart radius, always represent equivalent circuits.

Negative Real Parts

On the Smith Chart form in Fig. 12.5, a designation of the sign of the normalized resistance component of the impedance, or the normalized conductance component of the admittance coordinates.

Note 1: See "Coordinate Components (Chart C)."

Note 2: A Smith Chart whose impedance or admittance coordinates are designated with negative real parts is useful in portraying conditions along a waveguide only when the returned power is greater than the incident power.

Normalized Current EQUATIONS HERE

The rms current which would exist at a specified point along a hypothetical waveguide having a characteristic impedance of one ohm (or a characteristic admittance of one mho) and transmitting one watt of power to a load. This current is the vector sum of the incident and reflected currents at the point.

Note 1: The actual current at any specified power level in a waveguide is obtainable from the normalized current by multiplying it by the square root of the ratio of the power and characteristic impedance (or by the square root of the product of the power and characteristic admittance).

Note 2: A plot of normalized current and/or normalized voltage is provided as an overlay for Smith Chart impedance or admittance coordinates in Fig. 4.2.

Normalized Voltage EQUATONS HERE

The rms voltage which would exist at a specified point along a hypothetical waveguide having a characteristic impedance of one ohm (or a characteristic admittance of one mho) and transmitting one watt of power to a load. This voltage is the vector sum of the incident and reflected voltages at the point.

Note 1: The actual voltage at any specified power level in a waveguide is obtainable from the normalized voltage by multiplying it by the square root of the product of the power and characteristic impedance (or by the square root of the ratio of the power and characteristic admittance).

Note 2: See Note 2 in definition for normalized current.

Percent Off Midband Frequency n  ·Δƒ

Captions for peripheral scales near the pole regions on expanded Smith Charts, which relate specific values of the frequency deviation, from the resonant or antiresonant frequency, to the impedance or admittance characteristics of open- and short-circuited stub transmission lines n quarter wavelengths long. Δƒ is the deviation from the midband frequency in percent.

Peripheral Scales

The four scales encircling the impedance or admittance coordinates of the Smith Chart, individual graduations on each of which are applicable to a straight line locus of points on the impedance or admittance coordinates.

Note: Each graduation on each of the three outermost of these scales is applicable to all points on the impedance or admittance coordinates which are radially aligned therewith; each graduation on the innermost of these is applicable to all points on the coordinates which are in line with the graduations and the point of origin of the impedance or admittance coordinates.

A set of guided wave parameters represented by a corresponding number of scales whose overall lengths equal the radius of a Smith Chart, and which are used to measure the radial distance between the center and the perimeter of the impedance or admittance coordinates, at which point a specific value of the parameter exists.

Note 1: Radially scaled parameter values are mutually related to each other as well as to a circular locus of normalized impedances or admittances centered on these coordinates (see Chap. 14, Par. 14.8).

Note 2: The use of radial scales to represent radially scaled parameter values avoids the need to superimpose families of concentric circles on the impedance or admittance coordinates which (if all parameters were thus represented) would completely obscure the coordinates.

Reflection Coefficients E or I

At a specified point in a waveguide, the ratio of the amplitudes of the reflected and incident voltage or current waves. If the waveguide is lossless the magnitude of the "Reflection Coefficients E or I" is independent of the reference position. If it is lossy the magnitude will diminish as the reference position is moved toward the generator.

Note 1: At any specified reference position along any uniform waveguide the magnitude of the voltage reflection coefficient is equal to that of the current reflection coefficient.

Note 2: On a Smith Chart the "Reflection Coefficients E or I" is a radially scaled parameter.

Reflection Coefficient P

At a specified point in a waveguide the ratio of reflected to incident power.

Note 1: In a uniform lossless waveguide the "Reflection Coefficient P" is independent of the reference position.

Note 2: When expressed in dB the "Power Reflection Coefficient P" is equivalent to the "Return Loss, dB."

Note 3: On a Smith Chart the "Reflection Coefficient, P" is a radially scaled parameter.

Reflection Coefficient, X or Y Component

In a waveguide, the in-phase or quadrature-phase rectangular component, respectively, of the "Reflection Coefficients E or Il" represented on a Smith Chart as a rectangular-coordinate overlay. (See Chap. 8.)

Reflection Loss, dB

A nondissipative loss introduced at a discontinuity along a uniform waveguide, such as at a mismatched termination. "Reflection Loss, dB" can be expressed as a ratio, in dB, of the reflected to the absorbed power at the discontinuity and/or at all other points along a uniform waveguide toward the generator therefrom.

Note 1: If the input impedance of a lossless waveguide is matched to the internal impedance of the generator, a compensating gain will occur at the generator end of the waveguide. Any difference between the "Reflection Loss, dB" at each end of a waveguide corresponds to the increase in attenuation in a waveguide due to reflected power from the load.

Note 2: On a Smith Chart "Reflection Loss, dB" is a radially scaled parameter.

Return Gain, dB

In a waveguide terminated in an impedance or admittance with a negative real part, the ratio in dB of the power in the reflected and incident waves.

Note 1: On a Smith Chart whose impedance or admittance coordinates are designated with negative real parts this is a radially scaled parameter.

Return Loss, dB

In a waveguide, the ratio in dB of the power in the incident and reflected waves. The term "Return Loss, dB" is synonymous with "Power Reflection Coefficient" when the latter is expressed in dB.

Note: On a Smith Chart this is a radially scaled parameter.

Smith Chart

A circular reflection chart composed of two families of mutually orthogonal circular coordinate curves representing rectangular components of impedance or admittance, normalized with respect to the characteristic impedance and/or characteristic admittance of a waveguide. Peripheral scales completely surrounding the coordinates include a set of linear waveguide position and phase angle reference scales. The Smith Chart also includes a set of radial scales representing mutually related radially scaled parameters.

Note: The Smith Chart is commonly used for the graphical representation and analysis of the electrical properties of waveguides or circuits .

Standing Wave Loss Coefficient (Factor)

The ratio of combined dissipation and radiation losses in a waveguide when mismatch-terminated and when match-terminated.

Note 1: A specific value of this coefficient applies to the transmission losses integrated over plus or minus one-half wavelengths from the point of observation, as compared to the attenuation in the same length of waveguide. Thus, spatially repetitive variations in transmission loss within each standing half wavelength are smoothed.

Note 2: On a Smith Chart this is a radially scaled parameter.

Standing Wave Peak, Const. P

The ratio of the maximum amplitude of the standing voltage or current wave along a mismatch-terminated waveguide to the amplitude of the corresponding wave along a match-terminated waveguide when conducting the same power to the load.

Note: On a Smith Chart this is a radially scaled parameter.

Standing Wave Ratio (dBS)

In a waveguide, twenty times the logarithm to the base 10 of the standing wave ratio (S).

Note: On a Smith Chart this is a radially scaled parameter.

Standing Wave Ratio (SWR)

The ratio of the maximum to the minimum amplitudes of the voltage (or current) along a waveguide.

Note 1: For a given termination, and in a given region along a waveguide the SWR is identical for voltage or current. If the waveguide is lossy the SWR will diminish as the point of observation is moved toward the generator.

Note 2: On a Smith Chart the SWR is a radially scaled parameter.

Transmission Coefficient E or I

At a specified point along a waveguide the ratio of the amplitude of the transmitted voltage (or current) wave to the amplitude of the corresponding incident wave.

Note 1: The "transmitted voltage (or current) wave" is the complex resultant of the incident and reflected voltage (or current wave at the point.

Note 2: On the Smith Chart the "Transmission Coefficient E or I" is a linear scale equal in length to the diameter of the impedance or admittance coordinates and pivoted from their origin. As so plotted, this scale applies to voltage in relation to impedance coordinates or to current in relation to admittance coordinates.

Transmission Coefficient P

In a waveguide, the ratio of the transmitted to the incident power. The "Transmission Coefficient P" is equal to unity minus the "Reflection Coefficient P."

Note: On a Smith Chart the "transmission Coefficient P" is a radially scaled parameter.

Transmission Coefficient, X and Y Components

In a waveguide, the in-phase and quadrature-phase components of the "Transmission Coefficient E or I" represented on a Smith Chart as a rectangular coordinate overlay. (See Chap. 8.)

Transmission Loss Coefficient

In a waveguide, the "Standing Wave Loss Coefficient (Factor)."

Note: On Smith Chart this is a radially scaled parameter.

Transmission Loss, 1-dB Steps

A term used on Smith Charts to indicate the total losses due to dissipation of power within a waveguide and/or the radiation of power therefrom when the waveguide is match-terminated. "Transmission Loss" is expressed as a ratio in dB of the relative powers in the forward-traveling waves at two separated reference points along the waveguide.

Note: As above defined and used on earlier Smith Charts, the term is synonymous with the term "Attenuation (1 dB Maj. Div.)" which is used on more recent Smith Charts. The change in designation was made to avoid possible misunderstanding of the foregoing Smith Chart usage of the term which in other usage frequently means the total losses when the waveguide is mismatch-terminated. (See "Standing Wave Loss Coefficient (Factor).")

Wavelengths Toward Generator (or Toward Load)

In a waveguide, the relative distances and directions between any two reference points, represented on Smith Charts by the two outermost peripheral scales expressing electrical lengths in wavelengths from an arbitrarily selected radial reference locus on the impedance or admittance coordinates.

Note: On Smith Charts with fixed scales, the zero points of these scales are arbitrarily referenced to the position of a voltage standing wave minimum on the impedance coordinates and/or a current standing wave minimum or admittance coordinates. On a Smith Chart instrument, in which these scales are rotatable with respect to the impedance or admittance coordinates, the zero point may be aligned with any other reference position on the coordinates.

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