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<h3>Display Formats and Diagram Types &nbsp;&nbsp;
<a href="javascript:void(0);" onClick="var e; if (parseInt(navigator.appVersion)>=4){e=event;}  if (parseInt(navigator.appVersion) >=2 && typeof(PopupMenu_Invoke) == 'function') PopupMenu_Invoke(e,'','Display Formats','../../MMI_Reference/Trace/Format/Trace_Format.htm','Measured Quantities','../../MMI_Reference/Trace/Measure/Trace_-_Measure.htm');return false;" title="Related Topics"><img src="../../ss_btn_related_topics.gif" border=0 height=20 width=96 alt="Related Topics"></a>

 </h3>

<p>A display format defines how the set of (complex) measurement points 
 is converted and displayed in a diagram. The display formats in the <span 
 style="font-style: italic;"><I><a HREF="../../MMI_Reference/Trace/Format/Trace_Format.htm"
								id=A1
								style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Trace 
 &#8211; Format</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('A1');
//--></script></I></span> 
 menu use the following basic diagram types: &nbsp;</p>

<ul type="disc" class="whs1">
	
	<li class=kadov-p-CList><p class=List><a href="#Cartesian_Diagrams"
											id=A4
											style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Cartesian</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('A4');
//--></script> 
 (rectangular) diagrams are used for all display formats involving a <a 
 href="Display_Formats_and_Diagram_Types.htm#Conversion"
	id=A10
	style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">conversion</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('A10');
//--></script> 
 of the measurement data into a real (scalar) quantity, i.e. for <span 
 style="font-style: italic;"><I>dB Mag, Phase, Delay, SWR, Lin Mag, Real, 
 Imag </I></span>and <span style="font-style: italic;"><I>Unwrapped Phase. </I></span></p></li>
	
	<li class=kadov-p-CList><p class=List><a href="Display_Formats_and_Diagram_Types.htm#Polar_Diagram"
											id=A5
											style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Polar</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('A5');
//--></script> 
 diagrams are used for the<span style="font-style: italic;"> <I></I></span>display 
 format <span style="font-style: italic;"><I>Polar</I></span> and show a complex 
 quantity as a vector in a single trace. </p></li>
	
	<li class=kadov-p-CList><p class=List><a href="Display_Formats_and_Diagram_Types.htm#Smith_Chart"
											id=A6
											style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Smith</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('A6');
//--></script> 
 charts are used for the display format <span style="font-style: italic;"><I>Smith 
 </I></span>and show a vector like polar diagrams but with grid lines of constant 
 real and imaginary part of the impedance. &nbsp;</p></li>
	
	<li class=kadov-p-CList><p class=List><a href="Display_Formats_and_Diagram_Types.htm#Smith_Chart"
											id=a8
											style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Inverted 
 Smith</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('a8');
//--></script> 
 charts are used for the display format<span style="font-style: italic;"><I> 
 Inverted Smith </I></span>and show a vector like polar diagrams but with grid 
 lines of constant real and imaginary part of the admittance. &nbsp;</p></li>
</ul>

<p class=Note><span style="font-weight: bold;"><B><img src="../../small_images/note.jpg" x-maintain-ratio="TRUE" width="36px" height="37px" border="0" class="img_whs2"></B></span>The 
 analyzer allows arbitrary combinations of display formats and measured 
 quantities (<a href="../../MMI_Reference/Trace/Measure/Trace_-_Measure.htm"
				id=a3
				style="font-style: normal; /*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Trace 
 &#8211; Measure</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('a3');
//--></script>). 
 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 <span style="font-style: italic;"><I><a 
 href="#Display_Formats"
	id=A13
	style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Measured 
 Quantities and Display Formats</a><script type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('A13');
//--></script></I></span>. 
 &nbsp;</p>


<hr align="center" width="50%" class="whs3">


<h4><a name=Cartesian_Diagrams></a>Cartesian Diagrams</h4>

<p>Cartesian diagrams are rectangular diagrams used to display a scalar 
 quantity as a function of the stimulus variable (frequency / power / time). 
 </p>

<ul type="disc" class="whs1">
	
	<li class=kadov-p-CList><p class=List>The stimulus variable appears 
 on the horizontal axis (x-axis), scaled linearly (<a HREF="../../MMI_Reference/Channel/Sweep/Sweep_Type.htm"
														id=A7
														style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">sweep 
 types</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('A7');
//--></script><span 
 style="font-style: italic;"><I> Lin Frequency, Power, Time, CW Mode) </I></span>or 
 logarithmically (sweep type<span style="font-style: italic;"><I> Log Frequency)</I></span>. 
 </p></li>
	
	<li class=kadov-p-CList><p class=List>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 <a href="Display_Formats_and_Diagram_Types.htm#Conversion"
										id=A9
										style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">conversions</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('A9');
//--></script><span 
 style="font-style: italic;"><I>. </I></span></p></li>
</ul>

<p>The following examples show the same trace in Cartesian diagrams with 
 linear and logarithmic x-axis scaling. </p>

<p class=Figure
	style="x-text-underline: normal; line-height: Normal; /*begin!kadov{{*/ text-decoration: underline; /*}}end!kadov*/ "><span 
 style="x-text-underline: off; /*begin!kadov{{*/ text-decoration: none; /*}}end!kadov*/ "><img src="Lin_Cartesian.gif" width="324px" height="194px" border="0" class="img_whs4"> &nbsp;</span><img src="Log_Cartesian.gif" width="323px" height="193px" border="0" class="img_whs5"></p>

<p class=Linkline
	style="x-condition: OLH;"><img src="../../small_images/helpwoval3.gif" x-maintain-ratio="TRUE" width="42px" height="18px" border="0" class="img_whs6"> <a HREF="../../Getting_Started/Basic_Tasks/Scaling_Diagrams.htm"
												id=A11
												style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Scaling 
 diagrams</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('A11');
//--></script></p>


<hr align="center" width="50%" class="whs3">


<h4><a name=Conversion></a> Conversion of Complex into Real Quantities</h4>

<p>The results to be selected in the <span style="font-style: italic;"><I><a 
 HREF="../../MMI_Reference/Trace/Measure/Trace_-_Measure.htm"
	id=A12
	style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Trace 
 &#8211;</I></span> 
 <span style="font-style: italic;"><I>Measure</I></span></a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('A12');
//--></script> 
 menu can be divided into two groups:</p>

<ul type="disc" class="whs1">
	
	<li class=kadov-p-CList><p class=List><span style="font-style: italic;"><I>S-Parameters, 
 Ratios, Wave Quantities, Impedances, Admittances, Z-Parameters </I></span>and<span 
 style="font-style: italic;"><I> Y-Parameters</I></span> are complex. </p></li>
	
	<li class=kadov-p-CList><p class=List><span style="font-style: italic;"><I>Stability 
 Factors</I></span> and <span style="font-style: italic;"><I>DC Input</I></span> values 
 (voltages, PAE) are real. </p></li>
</ul>

<p>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). &nbsp;&nbsp;</p>

<table x-use-null-cells cellspacing="0" width="100%" class="whs7">
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//--></script>
<col class="whs8">
<col class="whs9">
<col class="whs10">

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs12">
<p class=Tabcmu2
	style="font-weight: bold;"><a HREF="../../MMI_Reference/Trace/Format/Trace_Format.htm"
									id=A18
									style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Trace 
 Format</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('A18');
//--></script></td>
<td colspan="1" rowspan="1" width="267px" class="whs13">
<p class=Tabcmu2
	style="font-weight: bold;">Description</td>
<td colspan="1" rowspan="1" width="269px" class="whs14">
<p class=Tabcmu2
	style="font-weight: bold;">Formula</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>dB Mag</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2>Magnitude of z in dB</td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2>|z| = sqrt ( x<span style="font-size: 6pt; vertical-align: Super;"><font size=1 style="font-size:6pt;">2</font></span> 
 + y<span style="font-size: 6pt; vertical-align: Super;"><font size=1 style="font-size:6pt;">2</font></span> ) <br>
dB Mag(z) = 20 * log|z| dB </td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>Lin Mag</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2>Magnitude of z, unconverted</td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2>|z| = sqrt ( x<span style="font-size: 6pt; vertical-align: Super;"><font size=1 style="font-size:6pt;">2</font></span> 
 + y<span style="font-size: 6pt; vertical-align: Super;"><font size=1 style="font-size:6pt;">2</font></span> )</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>Phase</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2>Phase of z</td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2><span style="font-family: Symbol, serif;">f</span> (z) 
 = arctan (y/x)</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>Real</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2>Real part of z</td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2>Re(z) = x</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>Imag</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2>Imaginary part of z</td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2>Im(z) = y</td></tr>

<tr valign="top" class="whs11">
<td width="158px" class="whs15">
<p class=Tabcmu2>SWR</td>
<td width="267px" class="whs16">
<p class=Tabcmu2>(Voltage) Standing Wave Ratio</td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2>SWR = (1 + |z|) / (1 &#8211; 
 |z|)</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs18">
<p class=Tabcmu2>Delay</td>
<td colspan="1" rowspan="1" width="267px" class="whs19">
<p class=Tabcmu2>Group delay, neg. derivative of the phase response</td>
<td colspan="1" rowspan="1" width="269px" class="whs20">
<p class=Tabcmu2>&#8211; 
 d <span style="font-family: Symbol, serif;">f</span> (z) / d<span style="font-family: Symbol, serif;">w</span> 
 &nbsp;(<span 
 style="font-family: Symbol, serif;">w</span> = 2<span style="font-family: Symbol, serif;">p</span> 
 * f)</td></tr>
<script language='JavaScript'><!--
if ((navigator.appName == "Netscape") && (parseInt(navigator.appVersion) == 4)) document.write("</table></table><table>");
//--></script>
</table>

<p><img src="../../small_images/tip.jpg" x-maintain-ratio="TRUE" width="36px" height="37px" border="0" class="img_whs2">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 <a HREF="../../MMI_Reference/Trace/Marker/Marker.htm#Mkr_Format"
								id=A14
								style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">marker 
 format</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('A14');
//--></script>. 
 Marker and trace formats can be selected independently. &nbsp;</p>


<hr align="center" width="50%" class="whs3">


<h4><a name=Polar_Diagram></a> Polar Diagrams</h4>

<p>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. </p>

<ul type="disc" class="whs1">
	
	<li style="/*begin!kadov{{*/ color: #000000; /*}}end!kadov*/ "
		class=kadov-p-CList><p class=List
								style="color: #000000;">The magnitude of 
 the response values corresponds to their distance from the center. Values 
 with the same magnitude are located on circles. </p></li>
	
	<li class=kadov-p-CList><p class=List>The phase of the response values 
 is given by the angle from the positive horizontal axis<span style="font-style: italic;"><I>. 
 </I></span>Values with the same phase on straight lines originating at the 
 center. </p></li>
</ul>

<p>The following example shows a polar diagram with a marker used to display 
 a pair of stimulus and response values. </p>

<p class=Figure><img src="Polar_chart_example.gif" width="327px" height="180px" border="0" class="img_whs21"></p>

<p class=Dropdownline><a name=Polar_Diagram_Example></a><img src="../../small_images/info_red.jpg" x-maintain-ratio="TRUE" width="32px" height="37px" border="0" class="img_whs22"><a class="dropspot" href="javascript:kadovTextPopup(this)" id="a16" style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Example: 
 Reflection coefficients in polar diagrams</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('a16');
//--></script></p>

<div class="droptext" id="POPUP210002800" style="display: 'none'; " x-ss-type="dropdown">
 <p>If the measured quantity is a complex reflection coefficient (S<span 
 style="vertical-align: Sub;">11</span>, S<span style="vertical-align: Sub;">22</span> 
 etc.), then the center of the polar diagram corresponds to a perfect load 
 Z<span style="vertical-align: Sub;">0</span> at the input test port of 
 the DUT (no reflection, matched input), whereas the outer circumference 
 (|S<span style="vertical-align: Sub;">ii</span>| = 1) represents a totally 
 reflected signal. &nbsp;</p>
 <p class=Figure
	style="font-weight: bold;"><img src="Polar_chart.jpg" width="430px" height="220px" border="0" class="img_whs23"></p>
 <p>Examples for definite magnitudes and phase angles: </p>
 <ul type="disc" class="whs1">
  <li class=kadov-p-CList>
  <p class=List>The magnitude of the reflection 
 coefficient of an open circuit (Z = infinity, I = 0) is one, its phase 
 is zero. </p></li>
  <li class=kadov-p-CList>
  <p class=List>The magnitude of the reflection 
 coefficient of a short circuit (Z = 0, U = 0) is one, its phase is &#8211;180<span style="vertical-align: Super;">0</span>. 
 </p></li>
 </ul>
</div>
<p class=Linkline
	style="x-condition: OLH;"><img src="../../small_images/helpwoval3.gif" x-maintain-ratio="TRUE" width="42px" height="18px" border="0" class="img_whs6"> <a href="../../Getting_Started/Basic_Tasks/Scaling_Diagrams.htm#Polar_Diagrams"
												id=a17
												style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Scaling 
 circular diagrams</a><script type="text/javascript"
													language=JavaScript1.2><!--
kadovInitEffects('a17');
//--></script></p>

<p class=Linkline
	style="x-condition: OLH;"><img src="../../small_images/helpwoval3.gif" x-maintain-ratio="TRUE" width="42px" height="18px" border="0" class="img_whs6"> <a href="../../Getting_Started/Reflection_Measurements/Reflection_Measurements.htm"
												id=a25
												style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Reflection 
 measurements</a><script type="text/javascript"
											language=JavaScript1.2><!--
kadovInitEffects('a25');
//--></script></p>


<hr align="center" width="50%" class="whs3">


<h4><a name=Smith_Chart></a> Smith Chart</h4>

<p>The Smith chart is a circular diagram that maps the complex reflection 
 coefficients S<span style="vertical-align: Sub;">ii</span> to normalized 
 impedance values. In contrast to the <a href="../../MMI_Reference/Trace/Format/Trace_Format.htm#Polar"
										id=a20
										style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">polar 
 diagram</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('a20');
//--></script>, 
 the scaling of the diagram is not linear. The grid lines correspond to 
 points of constant resistance and reactance. </p>

<ul type="disc" class="whs1">
	
	<li style="/*begin!kadov{{*/ color: #000000; /*}}end!kadov*/ "
		class=kadov-p-CList><p class=List
								style="color: #000000;">Points with the 
 same resistance are located on circles. </p></li>
	
	<li class=kadov-p-CList><p class=List>Points with the same reactance 
 produce arcs. </p></li>
</ul>

<p>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 <a HREF="../../MMI_Reference/Trace/Marker/Marker.htm#Mkr_Format"
						id=A22
						style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">marker 
 format</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('A22');
//--></script>). </p>

<p class=Figure><img src="Smith_chart.gif" width="327px" height="181px" border="0" class="img_whs24"></p>

<p><img src="../../small_images/tip.jpg" x-maintain-ratio="TRUE" width="36px" height="37px" border="0" class="img_whs2">A comparison of the Smith chart, the <a href="Display_Formats_and_Diagram_Types.htm#Inverted_Smith_Chart"
														id=a35
														style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">inverted 
 Smith</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('a35');
//--></script> 
 chart and the <a href="Display_Formats_and_Diagram_Types.htm#Polar_Diagram"
					id=a24
					style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">polar 
 diagram</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('a24');
//--></script> 
 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 <span style="font-style: italic;"><I>Polar </I></span>to <span style="font-style: italic;"><I>Smith 
 </I></span>or <span style="font-style: italic;"><I>Inverted Smith </I></span>&#8211; the analyzer 
 simply replaces the underlying grid and the default marker format. &nbsp;</p>

<p class=Dropdownline><span class=dropspot><img src="../../small_images/info_red.jpg" x-maintain-ratio="TRUE" width="32px" height="37px" border="0" class="img_whs22"><a class="dropspot" href="javascript:kadovTextPopup(this)" id="a23" style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Smith 
 chart construction</a><script type="text/javascript"
													language=JavaScript1.2><!--
kadovInitEffects('a23');
//--></script></span></p>

<div class="droptext" id="POPUP209754509" style="display: 'none'; " x-ss-type="dropdown">
 <p>In a Smith chart, the impedance plane is reshaped so that the area with 
 positive resistance is mapped into a unit circle. </p>
 <p class=Figure><img src="Smith_chart_construction.jpg" width="428px" height="202px" border="0" class="img_whs25"></p>
 <p>The basic properties of the Smith chart follow from this construction:</p>
 <ul type="disc" class="whs1">
  <li class=kadov-p-CList>
  <p class=List>The central horizontal axis corresponds 
 to zero reactance (real impedance). The center of the diagram represents 
 Z/Z<span style="vertical-align: Sub;">0</span> = 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). </p></li>
  <li class=kadov-p-CList>
  <p class=List>The outer circle corresponds 
 to zero resistance (purely imaginary impedance). Points outside the outer 
 circle indicate an active component. </p></li>
  <li class=kadov-p-CList>
  <p class=List>The upper and lower half of the 
 diagram correspond to positive (inductive) and negative (capacitive) reactive 
 components of the impedance, respectively. </p></li>
 </ul>
</div>
<p class=Dropdownline><a name=Smith_Diagram_Example></a><img src="../../small_images/info_red.jpg" x-maintain-ratio="TRUE" width="32px" height="37px" border="0" class="img_whs22"><a class="dropspot" href="javascript:kadovTextPopup(this)" id="a19" style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Example: 
 Reflection coefficients in the Smith chart</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('a19');
//--></script></p>

<div class="droptext" id="POPUP154676904" style="display: 'none'; " x-ss-type="dropdown">
 <p>If the measured quantity is a complex reflection coefficient <span style="font-family: Symbol, serif;">G</span> 
 (e.g. S<span style="vertical-align: Sub;">11</span>, S<span style="vertical-align: Sub;">22</span>), 
 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) <a HREF="../../MMI_Reference/Trace/Measure/Impedance/Impedance.htm"
									id=a21
									style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">impedances</a>): 
 </p>
 <p class=Figure>Z / Z<span style="vertical-align: Sub;">0</span> = (1 + 
 <span style="font-family: Symbol, serif;">G</span><span style="font-family: Arial, sans-serif;">) 
 / (1 </span>&#8211;<span style="font-family: Symbol, serif;">G</span><span style="font-family: Arial, sans-serif;">) 
 </span></p>
 <p>From this equation it is easy to relate the real and imaginary components 
 of the complex resistance to the real and imaginary parts of <span style="font-family: Symbol, serif;">G:</span></p>
 <p class=Figure><img src="Equation_Re_z.gif" width="362px" height="60px" border="0" class="img_whs26">&nbsp;<span style="font-weight: bold;"><B><img src="Equation_Im_z.gif" width="362px" height="57px" border="0" class="img_whs27"></B></span></p>
 <p>in order to deduce the following properties of the graphical representation 
 in a Smith chart:</p>
 <ul type="disc" class="whs1">
  <li class=kadov-p-CList>
  <p class=List>Real reflection coefficients 
 are mapped to real impedances (resistances).</p></li>
  <li class=kadov-p-CList>
  <p class=List>The center of the <span style="font-family: Symbol, serif;">G</span> 
 plane (<span style="font-family: Symbol, serif;">G</span><span style="font-family: Arial, sans-serif;"> 
 = 0) i</span>s mapped to the reference impedance Z<span style="vertical-align: Sub;">0</span>, 
 whereas the circle with |<span style="font-family: Symbol, serif;">G</span>| 
 = 1 is mapped to the imaginary axis of the Z plane. </p></li>
  <li class=kadov-p-CList>
  <p class=List>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. </p></li>
 </ul>
 <p class=Figure
	style="font-weight: bold;"><img src="Smith_chart_example.jpg" width="415px" height="195px" border="0" class="img_whs28"></p>
 <p>Examples for special points in the Smith chart: </p>
 <ul type="disc" class="whs1">
  <li class=kadov-p-CList>
  <p class=List>The magnitude of the reflection 
 coefficient of an open circuit (Z = infinity, I = 0) is one, its phase 
 is zero. </p></li>
  <li class=kadov-p-CList>
  <p class=List>The magnitude of the reflection 
 coefficient of a short circuit (Z = 0, U = 0) is one, its phase is &#8211;180<span style="vertical-align: Super;">0</span>. 
 </p></li>
 </ul>
</div>
<p class=Linkline
	style="x-condition: OLH;"><img src="../../small_images/helpwoval3.gif" x-maintain-ratio="TRUE" width="42px" height="18px" border="0" class="img_whs6"> <a href="../../Getting_Started/Basic_Tasks/Scaling_Diagrams.htm#Polar_Diagrams"
												id=a15
												style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Scaling 
 circular diagrams</a><script type="text/javascript"
													language=JavaScript1.2><!--
kadovInitEffects('a15');
//--></script></p>

<p class=Linkline
	style="x-condition: OLH;"><img src="../../small_images/helpwoval3.gif" x-maintain-ratio="TRUE" width="42px" height="18px" border="0" class="img_whs6"> <a HREF="../../Getting_Started/Reflection_Measurements/Reflection_Measurements.htm"
												id=A26
												style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Reflection 
 measurements</a><script type="text/javascript"
											language=JavaScript1.2><!--
kadovInitEffects('A26');
//--></script></p>


<hr align="center" width="50%" class="whs3">


<h4><a name=Inverted_Smith_Chart></a> Inverted Smith Chart</h4>

<p>The inverted Smith chart is a circular diagram that maps the complex 
 reflection coefficients S<span style="vertical-align: Sub;">ii</span> 
 to normalized admittance values. In contrast to the <a href="../../MMI_Reference/Trace/Format/Trace_Format.htm#Polar"
														id=a27
														style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">polar 
 diagram</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('a27');
//--></script>, 
 the scaling of the diagram is not linear. The grid lines correspond to 
 points of constant conductance and susceptance. </p>

<ul type="disc" class="whs1">
	
	<li style="/*begin!kadov{{*/ color: #000000; /*}}end!kadov*/ "
		class=kadov-p-CList><p class=List
								style="color: #000000;">Points with the 
 same conductance are located on circles. </p></li>
	
	<li class=kadov-p-CList><p class=List>Points with the same susceptance 
 produce arcs. </p></li>
</ul>

<p>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 <a href="../../MMI_Reference/Trace/Marker/Marker.htm#Mkr_Format"
									id=a28
									style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">marker 
 format</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('a28');
//--></script>). </p>

<p class=nohead><img src="Inverted_Smith_chart_example.gif" width="328px" height="182px" border="0" class="img_whs29"></p>

<p class=nohead><img src="../../small_images/tip.jpg" x-maintain-ratio="TRUE" width="36px" height="37px" border="0" class="img_whs2">A comparison of the inverted Smith chart with 
 the <a href="Display_Formats_and_Diagram_Types.htm#Smith_Chart"
		id=a36
		style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Smith</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('a36');
//--></script> 
 chart and the <a href="Display_Formats_and_Diagram_Types.htm#Polar_Diagram"
					id=a34
					style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">polar 
 diagram</a><script type="text/javascript"
										language=JavaScript1.2><!--
kadovInitEffects('a34');
//--></script> 
 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 <span style="font-style: italic;"><I>Polar </I></span>to <span style="font-style: italic;"><I>Inverted 
 Smith</I></span> or<span style="font-style: italic;"><I> Smith </I></span>&#8211; the analyzer 
 simply replaces the underlying grid and the default marker format. &nbsp;</p>

<p class=Dropdownline><span class=dropspot><img src="../../small_images/info_red.jpg" x-maintain-ratio="TRUE" width="32px" height="37px" border="0" class="img_whs22"><a class="dropspot" href="javascript:kadovTextPopup(this)" id="a29" style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Inverted 
 Smith chart construction</a><script type="text/javascript"
														language=JavaScript1.2><!--
kadovInitEffects('a29');
//--></script></span></p>

<div class="droptext" id="POPUP155399530" style="display: 'none'; " x-ss-type="dropdown">
 <p>The inverted Smith chart is point-symmetric to the Smith chart: </p>
 <p class=Figure><img src="Inverted_Smith_Chart_construction.jpg" width="445px" height="186px" border="0" class="img_whs30"></p>
 <p>The basic properties of the inverted Smith chart follow from this construction: 
 </p>
 <ul type="disc" class="whs1">
  <li class=kadov-p-CList>
  <p class=List>The central horizontal axis corresponds 
 to zero susceptance (real admittance). The center of the diagram represents 
 Y/Y<span style="vertical-align: Sub;">0</span> = 1, where Y<span style="vertical-align: Sub;">0</span> 
 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). </p></li>
  <li class=kadov-p-CList>
  <p class=List>The outer circle corresponds 
 to zero conductance (purely imaginary admittance). Points outside the 
 outer circle indicate an active component. </p></li>
  <li class=kadov-p-CList>
  <p class=List>The upper and lower half of the 
 diagram correspond to negative (inductive) and positive (capacitive) susceptive 
 components of the admittance, respectively. </p></li>
 </ul>
</div>
<p class=Dropdownline><a name=Inverted_Smith_Diagram_Example></a><img src="../../small_images/info_red.jpg" x-maintain-ratio="TRUE" width="32px" height="37px" border="0" class="img_whs22"><a class="dropspot" href="javascript:kadovTextPopup(this)" id="a30" style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Example: 
 Reflection coefficients in the inverted Smith chart</a><script 
 type="text/javascript"
	language=JavaScript1.2><!--
kadovInitEffects('a30');
//--></script></p>

<div class="droptext" id="POPUP77276432" style="display: 'none'; " x-ss-type="dropdown">
 <p>If the measured quantity is a complex reflection coefficient <span style="font-family: Symbol, serif;">G</span> 
 (e.g. S<span style="vertical-align: Sub;">11</span>, S<span style="vertical-align: Sub;">22</span>), 
 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) <a HREF="../../MMI_Reference/Trace/Measure/Admittance/Admittance.htm"
												id=a33
												style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">admittances</a>): 
 </p>
 <p class=Figure>Y / Y<span style="vertical-align: Sub;">0</span> = (1 - 
 <span style="font-family: Symbol, serif;">G</span><span style="font-family: Arial, sans-serif;">) 
 / (1 +</span><span style="font-family: Symbol, serif;">G</span><span 
 style="font-family: Arial, sans-serif;">) </span></p>
 <p>From this equation it is easy to relate the real and imaginary components 
 of the complex admittance to the real and imaginary parts of <span style="font-family: Symbol, serif;">G</span></p>
 <p class=Figure><img src="Equation_Re_y.gif" x-maintain-ratio="TRUE" width="346px" height="58px" border="0" class="img_whs31">&nbsp;&nbsp;<img src="Equation_Im_Y.gif" width="349px" height="56px" border="0" class="img_whs32"></p>
 <p>in order to deduce the following properties of the graphical representation 
 in an inverted Smith chart:</p>
 <ul type="disc" class="whs1">
  <li class=kadov-p-CList>
  <p class=List>Real reflection coefficients 
 are mapped to real admittances (conductances).</p></li>
  <li class=kadov-p-CList>
  <p class=List>The center of the <span style="font-family: Symbol, serif;">G</span> 
 plane (<span style="font-family: Symbol, serif;">G</span><span style="font-family: Arial, sans-serif;"> 
 = 0) i</span>s mapped to the reference admittance Y<span style="vertical-align: Sub;">0</span>, 
 whereas the circle with |<span style="font-family: Symbol, serif;">G</span>| 
 = 1 is mapped to the imaginary axis of the Y plane. </p></li>
  <li class=kadov-p-CList>
  <p class=List>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 (&#8211;1,0)), 
 centered on a straight vertical line. </p></li>
 </ul>
 <p class=Figure
	style="font-weight: bold;"><img src="Inverted_Smith.gif" width="409px" height="197px" border="0" class="img_whs33"></p>
 <p>Examples for special points in the inverted Smith chart: </p>
 <ul type="disc" class="whs1">
  <li class=kadov-p-CList>
  <p class=List>The magnitude of the reflection 
 coefficient of a short circuit (Y = infinity, U = 0) is one, its phase 
 is &#8211;180<span 
 style="vertical-align: Super;">0</span>. </p></li>
  <li class=kadov-p-CList>
  <p class=List>The magnitude of the reflection 
 coefficient of an open circuit (Y = 0, I = 0) is one, its phase is zero. 
 </p></li>
 </ul>
</div>
<p class=Linkline
	style="x-condition: OLH;"><img src="../../small_images/helpwoval3.gif" x-maintain-ratio="TRUE" width="42px" height="18px" border="0" class="img_whs6"> <a href="../../Getting_Started/Basic_Tasks/Scaling_Diagrams.htm#Polar_Diagrams"
												id=a31
												style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Scaling 
 circular diagrams</a><script type="text/javascript"
													language=JavaScript1.2><!--
kadovInitEffects('a31');
//--></script></p>

<p class=Linkline
	style="x-condition: OLH;"><img src="../../small_images/helpwoval3.gif" x-maintain-ratio="TRUE" width="42px" height="18px" border="0" class="img_whs6"> <a href="../../Getting_Started/Reflection_Measurements/Reflection_Measurements.htm"
												id=a32
												style="/*begin!kadov{{*/ position: relative; /*}}end!kadov*/ ">Reflection 
 measurements</a><script type="text/javascript"
											language=JavaScript1.2><!--
kadovInitEffects('a32');
//--></script></p>


<hr align="center" width="50%" class="whs3">


<h4 class=Linkline><a name=Display_Formats></a>Measured Quantities and 
 Display Formats </h4>

<p>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. </p>

<ul type="disc" class="whs1">
	
	<li class=kadov-p-CList><p class=List>All formats are suitable for 
 the analysis of reflection coefficients S<span style="vertical-align: Sub;">ii</span>. 
 The formats <span style="font-style: italic;"><I>SWR, Smith </I></span>and <span 
 style="font-style: italic;"><I>Inverted Smith </I></span>lose their original 
 meaning (standing wave ratio, normalized impedance or admittance) if they 
 are used for transmission <span style="font-style: italic;"><I>S-parameters</I></span>, 
 <span style="font-style: italic;"><I>Ratios</I></span> and other quantities. 
 </p></li>
	
	<li class=kadov-p-CList><p class=List>The complex <span style="font-style: italic;"><I>Impedances, Admittances, 
 Z-parameters, </I></span>and<span style="font-style: italic;"><I> Y-parameters 
 </I></span>are generally displayed in one of the Cartesian diagrams with linear 
 vertical axis scale or in a polar diagram. </p></li>
	
	<li class=kadov-p-CList><p class=List>The real <span style="font-style: italic;"><I>Stability 
 Factors, DC Inputs, </I></span>and the <span style="font-style: italic;"><I>PAE 
 </I></span>is generally displayed in a linear Cartesian diagram <span style="font-style: italic;"><I>(Lin 
 Mag </I></span><span>or </span><span style="font-style: italic;"><I>Real). </I></span>In 
 complex formats, real numbers represent complex numbers with zero imaginary 
 part.<span style="font-style: italic;"> <I>&nbsp;</I></span></p></li>
</ul>

<p>The following table gives an overview of recommended display formats. 
 </p>

<table x-use-null-cells cellspacing="0" width="100%" class="whs7">
<script language='JavaScript'><!--
if ((navigator.appName == "Netscape") && (parseInt(navigator.appVersion) == 4)) document.write("</table><table x-use-null-cells cellspacing='0' width='100%' border='4' bordercolor='gray' bordercolorlight='gray' bordercolordark='gray'>");
//--></script>
<col class="whs8">
<col class="whs9">
<col class="whs10">
<col class="whs34">

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs12">
<p class=Tabcmu2
	style="font-weight: bold;">&nbsp;</td>
<td colspan="1" rowspan="1" width="267px" class="whs13">
<p class=Tabcmu2
	style="font-weight: bold;">Complex dimensionless quantities:</p>
<p class=Tabcmu2
	style="font-weight: bold;">S-parameters and <span>r</span><span>atios</span></td>
<td colspan="1" rowspan="1" width="269px" class="whs14">
<p class=Tabcmu2
	style="font-weight: bold;">Complex quantities with dimensions:</p>
<p class=Tabcmu2
	style="font-weight: bold;">Wave quantities, <span style="font-weight: bold;"><B>Z-parameters, 
 Y-parameters, impedances, admittances</B></span></td>
<td colspan="1" rowspan="1" width="363px" class="whs35">
<p class=Tabcmu2
	style="font-weight: bold;">Real quantities:</p>
<p class=Tabcmu2
	style="font-weight: bold;">Stability Factors, <span style="font-weight: bold;"><B>DC 
 Input 1/2, PAE</B></span></td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>Lin Mag</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"> </td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"> (default for Z-parameters, Y-parameters, 
 impedances, admittances)</td>
<td colspan="1" rowspan="1" width="363px" class="whs37">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"> (default)</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>dB Mag</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"> (default)</td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"> (default for wave quantities)</td>
<td colspan="1" rowspan="1" width="363px" class="whs37">
<p class=Tabcmu2>&#8211; 
 </td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>Phase</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td>
<td colspan="1" rowspan="1" width="363px" class="whs37">
<p class=Tabcmu2>&#8211;</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>Real</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td>
<td colspan="1" rowspan="1" width="363px" class="whs37">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs15">
<p class=Tabcmu2>Imag</td>
<td colspan="1" rowspan="1" width="267px" class="whs16">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td>
<td colspan="1" rowspan="1" width="363px" class="whs37">
<p class=Tabcmu2>&#8211;</td></tr>

<tr valign="top" class="whs11">
<td width="158px" class="whs15">
<p class=Tabcmu2>Unwrapped Phase</td>
<td width="267px" class="whs16">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td>
<td colspan="1" rowspan="1" width="269px" class="whs17">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td>
<td width="363px" class="whs37">
<p class=Tabcmu2>&#8211;</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs18">
<p class=Tabcmu2>Smith</td>
<td colspan="1" rowspan="1" width="267px" class="whs19">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"> (reflection coefficients S<span style="vertical-align: Sub;">ii</span>) 
 </td>
<td colspan="1" rowspan="1" width="269px" class="whs20">
<p class=Tabcmu2>&#8211;</td>
<td colspan="1" rowspan="1" width="363px" class="whs38">
<p class=Tabcmu2>&#8211;</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs18">
<p class=Tabcmu2>Polar</td>
<td colspan="1" rowspan="1" width="267px" class="whs19">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"></td>
<td colspan="1" rowspan="1" width="269px" class="whs20">
<p class=Tabcmu2>&#8211;</td>
<td colspan="1" rowspan="1" width="363px" class="whs38">
<p class=Tabcmu2>&#8211;</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs18">
<p class=Tabcmu2>Inverted Smith</td>
<td colspan="1" rowspan="1" width="267px" class="whs19">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"> (reflection coefficients S<span style="vertical-align: Sub;">ii</span>) 
 </td>
<td colspan="1" rowspan="1" width="269px" class="whs20">
<p class=Tabcmu2>&#8211;</td>
<td colspan="1" rowspan="1" width="363px" class="whs38">
<p class=Tabcmu2>&#8211;</td></tr>

<tr valign="top" class="whs11">
<td colspan="1" rowspan="1" width="158px" class="whs18">
<p class=Tabcmu2>SWR</td>
<td colspan="1" rowspan="1" width="267px" class="whs19">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"> (reflection coefficients S<span style="vertical-align: Sub;">ii</span>) 
 </td>
<td colspan="1" rowspan="1" width="269px" class="whs20">
<p class=Tabcmu2>&#8211;</td>
<td colspan="1" rowspan="1" width="363px" class="whs38">
<p class=Tabcmu2>&#8211;</td></tr>

<tr valign="top" class="whs11">
<td width="158px" class="whs18">
<p class=Tabcmu2>Delay</td>
<td width="267px" class="whs19">
<p class=Tabcmu2><img src="../../small_images/Box_checked.gif" width="16px" height="14px" border="0" class="img_whs36"> (transmission coefficients S<span style="vertical-align: Sub;">ij</span>)</td>
<td colspan="1" rowspan="1" width="269px" class="whs20">
<p class=Tabcmu2>&#8211;</td>
<td width="363px" class="whs38">
<p class=Tabcmu2>&#8211;</td></tr>
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<p>The default formats are activated automatically when the measured quantity 
 is changed. &nbsp;</p>


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