This section presents all of the
formulas used in the VBA functions to calculate system cascade
parameters. In the following equations, the subscript “N” refers
to the current stage device parameter, while “N1” refers to
the cascaded system parameters up through and including the
previous stage. Use Figure 7 as a reference when interpreting
the formulas. For example, Gain_{N} refers to the gain
of the current stage’s device. Gain_{N1} refers to
the cascaded gain of all stages preceding the current stage.
So referring to Figure 7, for N=3, DeviceGain_{N} =
0.15 dB, and NomGain_{N1} = 0.5 dB. Finally, upper
case names refer to decibel values (e.g., G = 20 dB or P = 20
dBm), while lower case names refer to linear values (e.g., g
= 100 or p = 100 mW).

Min/max formulas are constructed
using combinations most likely to occur based on the available
combinations of parameters, rather than just those combinations
that yield absolutely the largest and smallest values. For instance,
the maximum IMD3 value uses the cascaded values of MaxP[sig]
and MaxIP3 because both use MaxGain. You may easily modify
the Min/Max formula variables to suit
your particular needs.


Figure 7 Formula
Reference Designations


Figure 8 Function
(fx) Dialog Box

Convert between decibels and non
decibels as follows:

P(dB) = 10*log_{10} (p)

At the bottom of the calculated
OIP2 and OIP3 columns, there is a row titled “Input Reference
(dBm)” that contains the equivalent system input referenced
values for the Nom, Min and
Max cases.


5.1
Gain (dB)

Gain values expressed in decibels
add arithmetically for the linear operating region (possible
output compression is ignored). If the “Use VSWR” cell is set
to “Y,” then the interstage VSWR mismatch values are added to
the total gain for Min and Max values. Mismatch loss is not
included in the calculation due to the complexity involved in
making a full accounting of the effects – reverse isolation
of preceding components, multiple reflection cycles and other
phenomenon would have to be accommodated, and is therefore way
beyond the scope of this workbook.

 Nom  
Nominal cascaded
gain uses the sum of the nominal gains for each of the stages. 

 Max  
Maximum cascaded
gain uses the sum of the maximum gains of each of the stages. 

 Min  
Minimum cascaded
gain uses the sum of the minimum gains for each of the stages. 

5.2
Noise Figure, NF (dB) 
Noise figure is
effectively the reduction of signal to noise ratio from the
cascade input to the output. Noise figure must be calculated
using the non decibel (linear) forms of gain and noise figure,
and then converted into decibels. These equations deal with
power, so the “10*log10 (v)” decibel form is used. 
Image Noise 
Image noise, which
can and often does contribute to the total noise power in systems,
is not accounted for in RF Cascade Workbook. Image noise is
added to the system when frequency mixing is performed and the
image frequency band at the input of the mixer is not sufficiently
filtered. The image frequency is defined as 2*LO1*RF. Unless
the mixer is an imagereject model specifically designed to
attenuate the image frequency, it will translate the image band
content (even if it is only noise) into the IF band just as
it will the intended signals. The result is increased noise
power that, worst case, could add 3 dB to the noise floor. If
your system can tolerate the hit in sensitivity, then there
is no problem. If you must maintain the highest level of sensitivity,
then consider placing a filter before the mixer that will know
the image band content down sufficiently to be negligible at
the mixer output. 10 dB of attenuation is a good minimum target
for just thermal noise, but more could be required if there
is also signal content in the image band. 

 Nom  
Nominal cascaded
noise figure uses the nominal gains and noise figures of each
of the stages. 


 Max  
Maximum cascaded
noise figure uses the minimum gains and maximum noise figures
of each of the stages. 


 Min  
Minimum cascaded
noise figure uses the maximum gains and minimum noise figures
of each of the stages. 


5.3
2Tone, 2ndOrder Intercept Point, OIP2 (dBm) 
IP2 is the theoretical
power at which the 2^{nd}order intermodulation products
would intersect the power of the original tones (CW) when input/output
power slopes are plotted. In the linear region of operation,
the original tones plot on a 1:1 (normalized) slope, while the
2^{nd}order products plot on a 2:1 (normalized) slope.
Therefore, the product tones increase at twice the rate of the
original tones, and the lines cross at the IP2 point. See Figure
9 for visualization. 2ndorder intermodulation products are
an important consideration in direct conversion and near zero
IF systems. 

Figure 9 Intercept
Points & Saturated Power

 Nom  
Nominal cascaded
2^{nd}order intercept point uses the nominal gains
and IP2s of each of the stages. 



 Max  
Maximum cascaded
2^{nd}order intercept point uses the maximum gains
and maximum IP2s of each of the stages. 



 Min  
Minimum cascaded
2^{nd}order intercept point uses the minimum gains
and minimum IP2s of each of the stages. 



5.4
2Tone, 3rdOrder Intercept Point, IP3 (dBm) 
IP3 is the theoretical
power at which the 3^{rd}order intermodulation products
would intersect the power of the original tones (CW) when input/output
power slopes are plotted. In the linear region of operation,
the original tones plot on a 1:1 (normalized) slope, while the
3^{rd}order products plot on a 3:1 (normalized) slope.
Therefore, the product tones increase at three times the rate
of the original tones, and the lines cross at the IP3 point.
See Figure 9 for visualization. 
 Nom  
Nominal cascaded 3^{rd}order
intercept point uses the nominal gains and IP3s of each of the
stages.




 Max  
Maximum cascaded
3^{rd}order intercept point uses the maximum gains
and maximum IP3s of each of the stages. 



 Min  
Minimum cascaded
3^{rd}order intercept point uses the minimum gains
and minimum IP3s of each of the stages. 



5.5
Saturated Power, P[sat] (dBm) 
P[sat] is the output
power at which no further increase in the input power will result
in an increase at the output. This, along with the 1 dB compression
point (P1dB) is a very nonlinear region of operation and can
only be modeled by sophisticated transfer functions that are
unique to each component. See Figure 9 for visualization. Therefore,
no attempt is made to model it here. Instead, the P[sat] value
is used as a monitor for the power level in the system to alert
the user to a potential problem. A check is made to determine
whether the power level at the component input, plus the linear
gain of the component, results in a power level equal to or
greater than the P[sat] of the component. If so, then the output
power is limited to the component’s P[sat] power level. No tolerance
input parameter is provided for P[sat] because it normally is
not an intentional design parameter. Note: The cascaded signal
power level displayed in the worksheet is calculated as if there
is no saturation limit. 
 Nom 

Nominal gain and P[sat] values are used per the following
equation.


 Max 

Not used. 
 Min  
Not used.

5.6
Signal Power, P[sig] (dBm) 
P[sig] is the power
of the signal as it propagates through the cascade, and is increased
or decreased by the linear gain of each stage. Note that it
is possible for the calculated value to exceed the P[sat] value,
because no adjustment is made. This is done to prevent the annoying
case where all of the other power dependant values are thrown
off by an adjusted output power value. There is an indication
of a saturated condition given in the component parameter input
area column labeled “!!!.” 
 Nom  

 Max 


 Min 


5.7
Noise Bandwidth, NBW (Freq Units) 
Cascaded noise
bandwidth merely checks the NBW of the current component, and
sets the system NBW to the lesser of either the component NBW
or the system’s previous NBW. Only a nominal value is calculated.
Frequency units are set on the “FilterMixer” worksheet. This
value is used with calculations that include noise power levels,
like dynamic range and spurious free dynamic range. 

 Max 

Not used. 
 Min  
Not used.

5.8
Noise Power, P[n] (dBm) 
P[n] is the power
of the noise as it propagates through the cascade, and is increased
or decreased by the gain, noise figure and NBW of each stage.
Since the system temperature is given in Celsius degrees, 273.15
is added to get equivalent Kelvin degrees. NWB is given in whatever
Frequency Units are specified on the “FilterMixer” worksheet,
so a multiplication by the appropriate factor is done. Note
that MaxGain is used with MaxNF and MinGain is used with MinNF.
This is because in most systems the noise figure is set near
the cascade input where the least amount of gain has accumulated.
Your specific application might warrant a different combination
of Min/Max values. See section 5.2 for discussion on image noise
contribution to noise power. In the following equations, k is
Boltzmann's constant. 
 Nom  

 Max 


 Min 


, where:

5.9
Signal to Noise Ratio, SNR (dB) 
SNR is the difference
between the noise power level and the signal power level. 
 Nom  

 Max 


 Min 


5.10
Saturated Dynamic Range, SDR (dB) 
SDR is the difference
between the saturated power level and the noise power level,
minus the minimum system SNR, as specified in the system parameter
area (Min SNR). This is different than the traditional dynamic
range (DR), which references the 1 dB compression point (P1dB);
RF Cascade Workbook 2004 does not calculate P1dB. Since P[sat]
is typically about 2 or 3 dB above P1dB, the traditional dynamic
range will be about 2 or 3 dB lower. 
 Nom  

 Max 


 Min 


5.11
2ndOrder SpuriousFree Dynamic Range, SFDR2 (dB) 
SFDR2 is the difference
between the IP2 power and the theoretical power of two tones
at the system input that would generate 2^{nd} order
products at the output with a power just equal to the noise
power at the output. 
 Nom  

 Max 


 Min 


5.12
2nd Order Intermodulation Product Power, IMD2 (dBm) 
Calculations of
IMD2 in RF Cascade Workbook 2004 assume intermod products are
caused by the nonlinear mixing of two input tones of equal
amplitude. It is essentially the same process as in a mixer
for frequency conversion, where an infinite series is produced
that consists of every possible frequency according to
±j*Tone1
±k*Tone2. 2nd order products
are more likely to fall inband for the direct conversion system
popular these days. A smaller (minimum) IMD2 is better. 

 Max 


 Min 


5.13
Delta 2nd Order Intermodulation Products, ΔIMD2 (dB) 
Delta IMD2 intermod
is the difference between the IMD2 product power (IMD2) and
the signal power, P[sig]. 
 Nom  

 Max 


 Min 


5.14
3rdOrder SpuriousFree Dynamic Range, SFDR3 (dB) 
SFDR3 is the difference
between the IP3 power and the theoretical power of two tones
at the system input that would generate 3rd order products at
the output with a power just equal to the noise power at the
output. 
 Nom  

 Max 


 Min 


5.15
3rd Order Intermodulation Product Power, IMD3 (dBm) 
Calculations of
IMD3 in RF Cascade Workbook 2004 assume intermod products are
caused by the nonlinear mixing of two input tones of equal
amplitude. It is essentially the same process as in a mixer
for frequency conversion, where an infinite series is produced
that consists of every possible frequency according to
±j*Tone1
±k*Tone2. The 3rd order
products that most likely fall inband at the output are
±2*Tone1
±Tone2 and
±Tone1
±2*Tone2. In reality, the
powers of most products are below the noise power. A smaller
(minimum) IMD3 is better. 
 Nom  

 Max 


 Min 


5.16
Delta 3rd Order Intermodulation Products, ΔIMD3 (dB) 
Delta IMD3 intermod
is the difference between the IMD3 product power (IMD3) and
the signal power, P[sig]. 
 Nom  

 Max 


 Min 


5.17
Interstage VSWR Mismatch Error (dB) 
VSWR mismatch errors
are caused by constructive and destructive interference of the
voltage standing waves at component interfaces due to impedance
mismatches. This is not the sum of all VSWR errors (see VSWR
Mismatch columns to the right cumulative). It is assumed that
there is infinite isolation between the component’s input and
output ports. These equations deal with voltage, so the “20*log10
(v)” decibel form is used. Only amplitude errors (not phase)
are used. 
 Neg  

 Pos 


5.17
Cumulative VSWR Mismatch Error (dB) 
This is the sum
of the Pos and Neg Interstage VSWR Mismatch Errors for all stages.
If a “y” or “Y” is entered in the “Use VSWR” cell in the input
parameters area, then these values are included in the Min/Max
Gain calculations, and consequently ripple through all the calculations
that depend on the min/max gain values. If “n” or “N” is entered,
then all of the values are reported as zeros. 
Chapter 1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13

Version 1.11 by Kirt Blattenberger
RF Cafe Website (www.rfcafe.com)

Chapter 5
