RF Cascade Workbook

Copyright

1996 -
2016

Webmaster:

Kirt
Blattenberger,

BSEE - KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The Internet was still largely an unknown entity at the time and not much was available in the form of WYSIWYG ...

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to Find What You Need.

There are 1,000s of Pages Indexed on RF Cafe !

Here is a reprint of an article I (Kirt Blattenberger) had published in Wireless Design & Development magazine
in 1995. Some of the references are a bit dated, but the info is all still very useful. Waypoint Software is now RF
Cafe, and TxRx Designer is now Shareware by the name of RF Workbench.

With the advent of high speed personal computers, a very insightful graphical method of determining inband mixer spurious products has been largely forgotten. The Spur Web^{1} chart rapidly identifies both inband and out-of-band spurs, affording a pictorial view of
where conversion system frequencies lie with respect to all spur products. A comparison will be presented between
the Spur Web chart method and the common numerical method.

Contemporary efforts at identifying inband mixer spurious products consist of employing a computer program, into which a range of frequencies is defined for the input, output, and local oscillator (LO). A frequency step size or number of frequency steps determines the resolution of the computation. If too few steps are specified, then stepping past an inband spur is possible. If too many steps are specified, then computation time becomes excessive and a long list of spur product frequencies results. The number of computations for a given range of frequencies is governed by Equation 1

Mixer spurious products exist due to nonlinear characteristics of diodes used in mixer construction, mathematically resulting in the sum and difference of integer value harmonics of input and LO frequencies. Theoretically the harmonics run to infinity in a simple diode mixer; however, in practice harmonics greater than 9th order often generate spur products that are below the noise floor and can be ignored. Double balanced mixers tend to suppress odd order spur products to a greater degree than even order products, while triple balanced mixers tend to suppress even order spur products to a greater degree than odd order products. A spur product defined by 3*I – 5*L is of the eighth order and is considered even (although both harmonics are odd). In general, spur products take on the form given in Equation 2.

A 60 MHz computer can complete the necessary calculations in less than a second for even large numbers of steps. Interpreting the data can take much longer, however. To show how quickly the number of output frequencies can grow, Table 1 uses Equation 1 to predict the number of calculations for both fixed and variable input and LO frequencies.

Typically, the first range of frequencies tested is defined by input and output filter passbands since the spurs generated within those bands cannot be filtered. The next step would be to test a range of input frequencies below the lower input band edge and a range of frequencies above the upper input band edge. Sufficient steps in the input frequency must be used to guarantee that all possible spur products are generated in the calculations. If a variable LO is used, then additional iterations must be run to cover multiple values of LO frequency. Considerable amounts of time can be spent sifting through all of the data generated by the aforementioned method.

Enter the Spur Web chart method of spurious mixer product detection. For the record, I make no claim to have originated the method (my first encounter with the chart was in an October, 1989, article in MSN magazine); I simply hope to resurrect a seemingly lost art and to introduce a helpful standard chart. Figure 1 makes clear why I dubbed the chart a Spur Web.

Choosing axis scales to represent ratios that are obtained by dividing the input and output frequencies by the LO frequency allows straight lines to be drawn which are defined by all spur products. The slope of a particular spur line is determined by the harmonic of the input frequency (I) while the intercept of the spur line with the vertical axis is determined by the harmonic of the LO frequency (L).

Similarly, the position within the Spur Web chart of a conversion system frequency plan can be represented by scaling the input and output range of frequencies by the LO frequency. Here, the input and output filter band edges constitute a good choice for the frequency ranges. The calculated ratio values describe a rectangular area within the Spur Web chart. All spur lines that pass through the rectangle are representative of inband spurs at the output - again, those which cannot be filtered.

Both inband and out-of-band spur products are apparent using the Spur Web method. If an intolerable spur product is looming just outside your filter band, then its corresponding input and/or output frequency at any point can be inferred by multiplying its x and/or y coordinate, respectively, by the LO frequency at the point of interest. A major advantage is that the sample step size is infinitesimally small so stepping over a spur product is impossible.

Evaluating a frequency conversion system with a variable LO is equally uninvolved. Simply construct two rectangles within the Spur Web - one by scaling the input and output frequencies using the lowest LO frequency (the larger rectangle) and one using the highest LO frequency (the smaller rectangle). Connect the four corners of the two rectangles with straight lines to create a wire frame object. Spur lines that intersect the frame represent inband spurs at the output. Again, multiplying any point on the chart by the particular LO frequency recovers the corresponding input and output frequencies.

Referring back to Figure 1, the effects of using a high-side^{2} LO versus
a low-side LO for either an upconversion or a downconversion becomes apparent from a spurious product perspective.
The associated spur products are vastly different depending upon the position of the rectangle within the Spur Web
chart. A particular mixer model might provide superior suppression for a given set of spur products in one region
of the chart compared to another region, making a choice of high-side LO versus low-side LO an obvious decision. Other
factors, such as expense, might outweigh the spur product considerations when selecting an LO.

As would be expected, the ±1*I ±1*L spur lines run diagonally from corner to corner through each rectangular area, indicating that the fundamental conversion frequencies define the rectangles. Although in the example all of the spur lines are labeled, a simple set of equations defines each line and can be applied to identify or create any spur line on the Spur Web chart. Using Equation 3 and Equation 4, the values for the input harmonic and the LO harmonic can be calculated.

Two unencumbered Spur Web charts are provided with this article. One chart includes all input and LO harmonics from 0 through 5, with all spur lines labeled. The other Spur Web chart includes input and LO harmonics from 0 through 9 when either of the harmonics is equal to or greater than 6. Not all spur lines are labeled in the second chart; however, Equation 3 and Equation 4 can be used to identify any unlabeled spur line. Including all harmonics from 0 through 9 on a single chart would produce an annoyingly cluttered chart.

Waypoint Software offers a unique frequency conversion CAE package named TxRx Designer v3.0, that incorporates the Spur Web chart as a feature in the program. A dynamic cursor readout provides corresponding input and output frequencies based on the LO reference frequency. Per the foregoing discussion, the band edges of the input and output filters are used to describe the rectangle boundaries within the chart. Figure 2 illustrates how the high-side downconversion example of Figure 1 is presented by TxRx Designer v3.0. To demonstrate the full power of the method, a variable LO is specified to cover a range of 130 to 150 (frequency units will be omitted in the discussion).

All of the detected spur products for the 0 through 5th harmonics are listed in the INBAND SPURS table to the left of the Spur Web chart. Note that the cursor is positioned on the 2*LO – 3*RF spur line that passes through the upper left corner of the backmost rectangle. With a reference LO frequency of 150.00, the indicated IF Freq is 29.7520 (actually 30.0000, but limited by pixel resolution in this case) while the indicated RF Freq is 90.0000. The CALCULATE SPUR LEVEL window is open wherein the frequency of the spur product is verified. Additionally, the mixer spur suppression for the spur product (derived from the mixer definition file) is displayed along with the power level of the spur product at the system output (inclusive of system gain and filter rejection).

Immediately apparent is that there are nearby spur lines that are not included in the detected inband spur table. Simply moving the cursor to a point on a spur line will reveal the input frequency that generated the spur product. The CALCULATE SPUR LEVEL window can be used to determine the power level of the out-of-band spur product at the system output, thereby ascertaining whether the filtering is sufficient or whether a different mixer model will be required. Filter specifications can be easily modified by clicking the mouse on a filter icon to open an edit window. Rerunning the Spur Web calculation takes only a few seconds at most.

After a final configuration is achieved, simply print the screen for a permanent record of the frequency conversion plan and the system components that are responsible for it. For a record of the filter responses, switch to the graph screen and print the system amplitude versus frequency.

Rather than spending valuable time running through multiple iterations of input, output, and LO frequencies to analyze long lists of spur product frequencies, recommend learning to use the Spur Web chart is highly recommended. The venerable Smith Chart^{3} revolutionized the solution of impedance computations in the pre-computer era
and has been carried over into the computer era via manifestations in many programs. Hopefully, this reintroduction
of the Spur Web chart and a demonstration of its viability in a computer program will prompt inclusion of the Spur
Web chart into system analysis programs. A great service will hence be provided to designers at a time when work overload
is the rule rather than the exception.

1. Spur Web is a trademark of Waypoint Software.^{™} is a registered trademark of Analog Instruments Company.

With the advent of high speed personal computers, a very insightful graphical method of determining inband mixer spurious products has been largely forgotten. The Spur Web

Contemporary efforts at identifying inband mixer spurious products consist of employing a computer program, into which a range of frequencies is defined for the input, output, and local oscillator (LO). A frequency step size or number of frequency steps determines the resolution of the computation. If too few steps are specified, then stepping past an inband spur is possible. If too many steps are specified, then computation time becomes excessive and a long list of spur product frequencies results. The number of computations for a given range of frequencies is governed by Equation 1

_{
}

Equation 1

Mixer spurious products exist due to nonlinear characteristics of diodes used in mixer construction, mathematically resulting in the sum and difference of integer value harmonics of input and LO frequencies. Theoretically the harmonics run to infinity in a simple diode mixer; however, in practice harmonics greater than 9th order often generate spur products that are below the noise floor and can be ignored. Double balanced mixers tend to suppress odd order spur products to a greater degree than even order products, while triple balanced mixers tend to suppress even order spur products to a greater degree than odd order products. A spur product defined by 3*I – 5*L is of the eighth order and is considered even (although both harmonics are odd). In general, spur products take on the form given in Equation 2.

_{
}

Equation 2

A 60 MHz computer can complete the necessary calculations in less than a second for even large numbers of steps. Interpreting the data can take much longer, however. To show how quickly the number of output frequencies can grow, Table 1 uses Equation 1 to predict the number of calculations for both fixed and variable input and LO frequencies.

# Input Steps | # LO Steps | I | L | # Spur Products |

0 | 0 | 6 | 6 | 147 |

50 | 0 | 6 | 6 | 7,497 |

0 | 10 | 6 | 6 | 1,617 |

50 | 10 | 6 | 6 | 82,467 |

Table 1

Typically, the first range of frequencies tested is defined by input and output filter passbands since the spurs generated within those bands cannot be filtered. The next step would be to test a range of input frequencies below the lower input band edge and a range of frequencies above the upper input band edge. Sufficient steps in the input frequency must be used to guarantee that all possible spur products are generated in the calculations. If a variable LO is used, then additional iterations must be run to cover multiple values of LO frequency. Considerable amounts of time can be spent sifting through all of the data generated by the aforementioned method.

Enter the Spur Web chart method of spurious mixer product detection. For the record, I make no claim to have originated the method (my first encounter with the chart was in an October, 1989, article in MSN magazine); I simply hope to resurrect a seemingly lost art and to introduce a helpful standard chart. Figure 1 makes clear why I dubbed the chart a Spur Web.

Choosing axis scales to represent ratios that are obtained by dividing the input and output frequencies by the LO frequency allows straight lines to be drawn which are defined by all spur products. The slope of a particular spur line is determined by the harmonic of the input frequency (I) while the intercept of the spur line with the vertical axis is determined by the harmonic of the LO frequency (L).

Similarly, the position within the Spur Web chart of a conversion system frequency plan can be represented by scaling the input and output range of frequencies by the LO frequency. Here, the input and output filter band edges constitute a good choice for the frequency ranges. The calculated ratio values describe a rectangular area within the Spur Web chart. All spur lines that pass through the rectangle are representative of inband spurs at the output - again, those which cannot be filtered.

Figure 1

Both inband and out-of-band spur products are apparent using the Spur Web method. If an intolerable spur product is looming just outside your filter band, then its corresponding input and/or output frequency at any point can be inferred by multiplying its x and/or y coordinate, respectively, by the LO frequency at the point of interest. A major advantage is that the sample step size is infinitesimally small so stepping over a spur product is impossible.

Evaluating a frequency conversion system with a variable LO is equally uninvolved. Simply construct two rectangles within the Spur Web - one by scaling the input and output frequencies using the lowest LO frequency (the larger rectangle) and one using the highest LO frequency (the smaller rectangle). Connect the four corners of the two rectangles with straight lines to create a wire frame object. Spur lines that intersect the frame represent inband spurs at the output. Again, multiplying any point on the chart by the particular LO frequency recovers the corresponding input and output frequencies.

Referring back to Figure 1, the effects of using a high-side

As would be expected, the ±1*I ±1*L spur lines run diagonally from corner to corner through each rectangular area, indicating that the fundamental conversion frequencies define the rectangles. Although in the example all of the spur lines are labeled, a simple set of equations defines each line and can be applied to identify or create any spur line on the Spur Web chart. Using Equation 3 and Equation 4, the values for the input harmonic and the LO harmonic can be calculated.

Equation 3 |
Equation 4 |

Two unencumbered Spur Web charts are provided with this article. One chart includes all input and LO harmonics from 0 through 5, with all spur lines labeled. The other Spur Web chart includes input and LO harmonics from 0 through 9 when either of the harmonics is equal to or greater than 6. Not all spur lines are labeled in the second chart; however, Equation 3 and Equation 4 can be used to identify any unlabeled spur line. Including all harmonics from 0 through 9 on a single chart would produce an annoyingly cluttered chart.

Waypoint Software offers a unique frequency conversion CAE package named TxRx Designer v3.0, that incorporates the Spur Web chart as a feature in the program. A dynamic cursor readout provides corresponding input and output frequencies based on the LO reference frequency. Per the foregoing discussion, the band edges of the input and output filters are used to describe the rectangle boundaries within the chart. Figure 2 illustrates how the high-side downconversion example of Figure 1 is presented by TxRx Designer v3.0. To demonstrate the full power of the method, a variable LO is specified to cover a range of 130 to 150 (frequency units will be omitted in the discussion).

All of the detected spur products for the 0 through 5th harmonics are listed in the INBAND SPURS table to the left of the Spur Web chart. Note that the cursor is positioned on the 2*LO – 3*RF spur line that passes through the upper left corner of the backmost rectangle. With a reference LO frequency of 150.00, the indicated IF Freq is 29.7520 (actually 30.0000, but limited by pixel resolution in this case) while the indicated RF Freq is 90.0000. The CALCULATE SPUR LEVEL window is open wherein the frequency of the spur product is verified. Additionally, the mixer spur suppression for the spur product (derived from the mixer definition file) is displayed along with the power level of the spur product at the system output (inclusive of system gain and filter rejection).

Figure 2

Immediately apparent is that there are nearby spur lines that are not included in the detected inband spur table. Simply moving the cursor to a point on a spur line will reveal the input frequency that generated the spur product. The CALCULATE SPUR LEVEL window can be used to determine the power level of the out-of-band spur product at the system output, thereby ascertaining whether the filtering is sufficient or whether a different mixer model will be required. Filter specifications can be easily modified by clicking the mouse on a filter icon to open an edit window. Rerunning the Spur Web calculation takes only a few seconds at most.

After a final configuration is achieved, simply print the screen for a permanent record of the frequency conversion plan and the system components that are responsible for it. For a record of the filter responses, switch to the graph screen and print the system amplitude versus frequency.

Rather than spending valuable time running through multiple iterations of input, output, and LO frequencies to analyze long lists of spur product frequencies, recommend learning to use the Spur Web chart is highly recommended. The venerable Smith Chart

Figure 3

Figure 4

1. Spur Web is a trademark of Waypoint Software.

2. A high-side LO refers to an LO frequency higher than the input range of frequencies for a

downconversion, and an LO frequency higher than the output range of frequencies for an

upconversion.
The converse is true for a low-side LO.