A display format defines how the set of (complex) measurement points is converted and displayed in a diagram. The display formats in the Trace – Format menu use the following basic diagram types:
Cartesian (rectangular) diagrams are used for all display formats involving a conversion of the measurement data into a real (scalar) quantity, i.e. for dB Mag, Phase, Delay, SWR, Lin Mag, Real, Imag and Unwrapped Phase.
Polar diagrams are used for the display format Polar and show a complex quantity as a vector in a single trace.
Smith charts are used for the display format Smith. They show a complex quantity like polar diagrams but with grid lines of constant real and imaginary part of the impedance.
Inverted Smith charts are used for the display format Inverted Smith. They show a complex quantity like polar diagrams but with grid lines of constant real and imaginary part of the admittance.
The
analyzer allows arbitrary combinations of display formats and measured
quantities (Trace
– Measure).
Nevertheless, in order to extract useful information from the data, it
is important to select a display format which is appropriate to the analysis
of a particular measured quantity; see Measured
Quantities and Display Formats.
Cartesian diagrams are rectangular diagrams used to display a scalar quantity as a function of the stimulus variable (frequency / power / time).
The stimulus variable appears on the horizontal axis (x-axis), scaled linearly (sweep types Lin Frequency, Power, Time, CW Mode) or logarithmically (sweep type Log Frequency).
The measured data (response values) appears on the vertical axis (y-axis). The scale of the y-axis is linear with equidistant grid lines although the y-axis values may be obtained from the measured data by non-linear conversions.
The following examples show the same trace in Cartesian diagrams with linear and logarithmic x-axis scaling.
The results in the Trace – Measure menu can be divided into two groups:
S-Parameters, Ratios, Wave Quantities, Impedances, Admittances, Z-Parameters, and Y-Parameters are complex.
Stability Factors and DC Input values (voltages, PAE) are real.
The following table shows how the response values in the different Cartesian diagrams are calculated from the complex measurement values z = x + jy (where x, y, z are functions of the sweep variable). The formulas also hold for real results, which are treated as complex values with zero imaginary part (y = 0).
|
Description |
Formula | |
|
dB Mag |
Magnitude of z in dB |
|z| = sqrt ( x2
+ y2 ) |
|
Lin Mag |
Magnitude of z, unconverted |
|z| = sqrt ( x2 + y2 ) |
|
Phase |
Phase of z |
φ (z) = arctan (y/x) |
|
Real |
Real part of z |
Re(z) = x |
|
Imag |
Imaginary part of z |
Im(z) = y |
|
SWR |
(Voltage) Standing Wave Ratio |
SWR = (1 + |z|) / (1 – |z|) |
|
Delay |
Group delay, neg. derivative of the phase response |
– d φ (z) / dΩ (Ω = 2p * f) |
An extended range of formats and conversion formulas is
available for markers. To convert any point on a trace, create a marker
and select the appropriate marker
format.
Marker and trace formats can be selected independently.
Polar diagrams show the measured data (response values) in the complex plane with a horizontal real axis and a vertical imaginary axis. The grid lines correspond to points of equal magnitude and phase.
The magnitude of the response values corresponds to their distance from the center. Values with the same magnitude are located on circles.
The phase of the response values is given by the angle from the positive horizontal axis. Values with the same phase are on straight lines originating at the center.
The following example shows a polar diagram with a marker used to display a pair of stimulus and response values.
Example:
Reflection coefficients in polar diagrams
If the measured quantity is a complex reflection coefficient (S11, S22 etc.), then the center of the polar diagram corresponds to a perfect load Z0 at the input test port of the DUT (no reflection, matched input), whereas the outer circumference (|Sii| = 1) represents a totally reflected signal.
Examples for definite magnitudes and phase angles:
The magnitude of the reflection coefficient of an open circuit (Z = infinity, I = 0) is one, its phase is zero.
The magnitude of the reflection coefficient of a short circuit (Z = 0, U = 0) is one, its phase is –180 deg.
The Smith chart is a circular diagram that maps the complex reflection coefficients Sii to normalized impedance values. In contrast to the polar diagram, the scaling of the diagram is not linear. The grid lines correspond to points of constant resistance and reactance.
Points with the same resistance are located on circles.
Points with the same reactance produce arcs.
The following example shows a Smith chart with a marker used to display the stimulus value, the complex impedance Z = R + j X and the equivalent inductance L (see marker format).
A comparison of the Smith chart, the inverted
Smith
chart and the polar
diagram
reveals many similarities between the two representations. In fact the
shape of a trace does not change at all if the display format is switched
from Polar to Smith
or Inverted Smith – the analyzer
simply replaces the underlying grid and the default marker format.
In a Smith chart, the impedance plane is reshaped so that the area with positive resistance is mapped into a unit circle.
The basic properties of the Smith chart follow from this construction:
The central horizontal axis corresponds to zero reactance (real impedance). The center of the diagram represents Z/Z0 = 1 which is the reference impedance of the system (zero reflection). At the left and right intersection points between the horizontal axis and the outer circle, the impedance is zero (short) and infinity (open).
The outer circle corresponds to zero resistance (purely imaginary impedance). Points outside the outer circle indicate an active component.
The upper and lower half of the diagram correspond to positive (inductive) and negative (capacitive) reactive components of the impedance, respectively.
Example:
Reflection coefficients in the Smith chart
If the measured quantity is a complex reflection coefficient Γ (e.g. S11, S22), then the unit Smith chart can be used to read the normalized impedance of the DUT. The coordinates in the normalized impedance plane and in the reflection coefficient plane are related as follows (see also: definition of matched-circuit (converted) impedances):
Z / Z0 = (1 + Γ) / (1 – Γ)
From this equation it is easy to relate the real and imaginary components of the complex resistance to the real and imaginary parts of G:
in order to deduce the following properties of the graphical representation in a Smith chart:
Real reflection coefficients are mapped to real impedances (resistances).
The center of the Γ plane (Γ = 0) is mapped to the reference impedance Z0, whereas the circle with |Γ| = 1 is mapped to the imaginary axis of the Z plane.
The circles for the points of equal resistance are centered on the real axis and intersect at Z = infinity. The arcs for the points of equal reactance also belong to circles intersecting at Z = infinity (open circuit point (1,0)), centered on a straight vertical line.
Examples for special points in the Smith chart:
The magnitude of the reflection coefficient of an open circuit (Z = infinity, I = 0) is one, its phase is zero.
The magnitude of the reflection coefficient of a short circuit (Z = 0, U = 0) is one, its phase is –180 deg.
The inverted Smith chart is a circular diagram that maps the complex reflection coefficients Sii to normalized admittance values. In contrast to the polar diagram, the scaling of the diagram is not linear. The grid lines correspond to points of constant conductance and susceptance.
Points with the same conductance are located on circles.
Points with the same susceptance produce arcs.
The following example shows an inverted Smith chart with a marker used to display the stimulus value, the complex admittance Y = G + j B and the equivalent inductance L (see marker format).
A comparison of the inverted Smith chart with
the Smith
chart and the polar
diagram
reveals many similarities between the different representations. In fact
the shape of a trace does not change at all if the display format is switched
from Polar to Inverted
Smith or Smith – the analyzer
simply replaces the underlying grid and the default marker format.
Inverted
Smith chart construction
The inverted Smith chart is point-symmetric to the Smith chart:
The basic properties of the inverted Smith chart follow from this construction:
The central horizontal axis corresponds to zero susceptance (real admittance). The center of the diagram represents Y/Y0 = 1, where Y0 is the reference admittance of the system (zero reflection). At the left and right intersection points between the horizontal axis and the outer circle, the admittance is infinity (short) and zero (open).
The outer circle corresponds to zero conductance (purely imaginary admittance). Points outside the outer circle indicate an active component.
The upper and lower half of the diagram correspond to negative (inductive) and positive (capacitive) susceptive components of the admittance, respectively.
Example:
Reflection coefficients in the inverted Smith chart
If the measured quantity is a complex reflection coefficient Γ (e.g. S11, S22), then the unit inverted Smith chart can be used to read the normalized admittance of the DUT. The coordinates in the normalized admittance plane and in the reflection coefficient plane are related as follows (see also: definition of matched-circuit (converted) admittances):
Y / Y0 = (1 - Γ) / (1 + Γ)
From this equation it is easy to relate the real and imaginary components of the complex admittance to the real and imaginary parts of Γ
in order to deduce the following properties of the graphical representation in an inverted Smith chart:
Real reflection coefficients are mapped to real admittances (conductances).
The center of the Γ plane (Γ = 0) is mapped to the reference admittance Y0, whereas the circle with |Γ| = 1 is mapped to the imaginary axis of the Y plane.
The circles for the points of equal conductance are centered on the real axis and intersect at Y = infinity. The arcs for the points of equal susceptance also belong to circles intersecting at Y = infinity (short circuit point (–1,0)), centered on a straight vertical line.
Examples for special points in the inverted Smith chart:
The magnitude of the reflection coefficient of a short circuit (Y = infinity, U = 0) is one, its phase is –1800.
The magnitude of the reflection coefficient of an open circuit (Y = 0, I = 0) is one, its phase is zero.
The analyzer allows any combination of a display format and a measured quantity. The following rules can help to avoid inappropriate formats and find the format that is ideally suited to the measurement task.
All formats are suitable for the analysis of reflection coefficients Sii. The formats SWR, Smith and Inverted Smith lose their original meaning (standing wave ratio, normalized impedance or admittance) if they are used for transmission S-parameters, Ratios and other quantities.
The complex Impedances, Admittances, Z-parameters, and Y-parameters are generally displayed in one of the Cartesian diagrams with linear vertical axis scale or in a polar diagram.
The real Stability Factors, DC Inputs, and the PAE is generally displayed in a linear Cartesian diagram (Lin Mag or Real). In complex formats, real numbers represent complex numbers with zero imaginary part.
The following table gives an overview of recommended display formats.
|
|
Complex dimensionless quantities: S-parameters and ratios |
Complex quantities with dimensions: Wave quantities, Z-parameters, Y-parameters, impedances, admittances |
Real quantities: Stability Factors, DC Input 1/2, PAE |
|
Lin Mag |
|
|
|
|
dB Mag |
|
|
– |
|
Phase |
|
|
– |
|
Real |
|
|
|
|
Imag |
|
|
– |
|
Unwrapped Phase |
|
|
– |
|
Smith |
|
– |
– |
|
Polar |
|
– |
– |
|
Inverted Smith |
|
– |
– |
|
SWR |
|
– |
– |
|
Delay |
|
– |
– |
The default formats are activated automatically when the measured quantity is changed.
