Theory of Operation for RF System Budget Analysis
This section describes the budget analysis process.
S-Parameters for the Cascade of Two, Two-Port, Two-Pin Components
The cascaded two-port, two-pin network signal wave representation used for the network S-parameter derivations can be represented in block diagram form as shown in the following figure.
Signal wave-representations used for network S-parameter derivations
The S-parameters resulting from cascading two two-port, two-pin components, A and B, can be expressed as the following: (see reference 9 in References for RF System Budget Analysis)
The T-parameters resulting from cascading two two-port, two-pin components, A and B, can also be derived and are expressed as follows:
When A is a spectral inverting component, but not B, then their cascade is defined as follows (* in the following represents conjugate):
When B is a spectral inverting component, but not A, then their cascade is defined as follows (* in the following represents conjugate):
When both A and B are spectral inverting components, then their cascade is defined as follows (* in the following represents conjugate):
S-Parameters for a Nonlinear Channel
A nonlinear channel with a cascade connection of a number of nonlinear and linear two-port, two-pin components will have the overall channel S-parameters at the channel input carrier frequency derived as a function of input power using an iterative algorithm (see reference 5 in References for RF System Budget Analysis).
The derivation of these S-parameters only address the carrier frequency throughout the channel and ignore any harmonics generated by the nonlinearities. This is a reasonable assumption because the nonlinearities are characterized with respect to the input to output fundamental carrier with harmonics filtered out. Also, the input signal is assumed to be narrowband.
The S-parameters of each nonlinear two-port, two-pin under large-signal conditions are assumed to be measured as a function of power level incident at only one port; the s11 and s21 parameters are a function of power incident at port 1, and the s12 and s22 parameters are a function of power incident at port 2.
A general nonlinear channel may be composed of alternating linear and nonlinear components as shown in the following figure. In general, the operating point for each nonlinearity is dependent on the operating point of all other nonlinearities.
The S-parameters for each nonlinearity in the channel are interpolated between their given power-dependent values during the iteration process to estimate the intermediate power levels that are incident at the input and output ports of each nonlinear two-port, two-pin component.
Cascade connection of alternately connected linear and nonlinear two-port, two-pin components
In this figure:
- P in is the power incident at the channel input (at a given carrier frequency).
- PO(n) and QO(n) are the initial estimates of the incident power levels at the input and output ports of the n'th nonlinearity obtained from an initial small-signal analysis.
- PK(n) and QK(n) are the incident power levels at the k'th iteration.
- G1K(n) and G2K(n) are the total reflection coefficients looking into the input and output ports of n'th nonlinearity calculated at the k'th iteration.
- The ( k+1 ) terms are derived from the k estimates.
- PT(n) and QT(n) are the operating power levels incident at port 1 and port 2 of the n'th nonlinearity obtained after the final iteration.
The iterative process is continued until the change in PT and QT is below a predetermined threshold. S-parameters for each nonlinearity are then obtained for the PT ( n ) and QT ( n ) values and the overall channel S-parameters are derived as in the linear case.
The greatest advantage of this technique is its ability to incorporate all the interstage mismatches and to handle any number of embedded linear and nonlinear two-port, two-pin components.
Noise Parameters for the Interconnection of Two Components
The cascaded two-port, two-pin network noise wave representation shown in section (b) of the following figure is used for the network noise correlation matrix, [N], derivation (see reference 1 in References for RF System Budget Analysis).
Representations for the connection of two components
This derivation uses the transmission (T) parameters of component A.
Using the definition of the T-matrix for components, that is, b1 = T11 a2; (b2=0), and b1 = T12 b2; (a2=0), the resultant network noise waves an and bn are:
bn = bn1 + TA11 bn2 - TA12 an2
an = an1 - TA21 bn2 + TA22 an2
In matrix form:
Using the definition of [N] and assuming noise from component A is independent and uncorrelated to the noise from component B:
Resulting in (* in the following represents conjugate):
When A is a spectral inverting component, but not B, or when both A and B are spectral inverting components, then:
bn = bn1 + TA11* bn2* - TA12* an2*
an = an1 - TA21* bn2* + TA22* an2*
Resulting in (* in the following represents conjugate):
2nd and 3rd-Order Intercept Definition
The 2nd-order intercept (SOI) and 3rd-order intercept (TOI) of a component or network is a widely accepted system design parameter because they indicate the degree of nonlinearity of a nonlinear component. The volt-out to volt-in relationship for a nonlinear component when S21, SOI, and TOI are specified can typically be described as a polynomial relationship as follows:
where:
X = input voltage
Y = output voltage
S 21 = complex small-signal gain
a 1 = the fundamental small-signal gain magnitude
a 2 = 2nd-order gain coefficient
a 3 = 3rd-order gain coefficient
Given an input signal, V in, with two frequency domain spectral tones (two-tones), ω1 and ω2, such that ω2 > ω1 and (ω2-ω1) << ω1, then a nonlinear component's output intermodulation products will include 2nd-order intermodulation products at (or near) twice ω1 and ω2 (2*ω1, 2*ω2, ω1+ω2), and 3rd-order intermodulation products localized about the two output fundamental tones (2*ω1-ω2, 2*ω1-ω2).
For a plot of the output power versus input power for output fundamental, 2nd-order and 3rd-order tones, see the figure Nonlinear component characterization for power out versus power in.
The small-signal fundamental curve varies with a 1:1 slope. The small-signal 2nd-order and 3rd-order curves vary with a 2:1 and 3:1 slope respectively.
The third-order intercept is that point where the extrapolated small-signal fundamental and 3rd-order curves intersect. At this 3rd-order intercept you may be interested in the input power level, or in the output power level.
Similarly, the 2nd-order intercept is that point where the extrapolated small-signal fundamental and 2nd-order curves intersect.
The following defines the nonlinear amplifier output response for one- and two-tone inputs, and derives the relationship between a 2, a 3 and SOI, TOI.
Nonlinearity Output for One- and Two-Tone Excitation
For one-tone excitation:
The response is:
For two-tone excitation:
The response is:
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For practical nonlinear devices defined by SOI and TOI, there is a maximum input signal level beyond which the device is driven into saturation. The above equations are applicable only below this saturation drive level.
Below saturation, the a 2 term is dependant on SOI (and not TOI) and that the a 3 term is dependant on TOI (and not SOI).
Relating Coefficients a2 and a3 to a1, SOI and TOI
The a 2 and a 3 coefficients are derived from the nonlinear amplifier small-signal gain magnitude, a 1, and output SOI and TOI values.
Given SOI and TOI in dBm power units and given that the amplifier is defined with respect to RefR input and output resistance, they also define the following:
SOI output power level, Watts = po_soi = 10 ( (SOI-30)/10) = vo_soi 2 /(2*RefR)
TOI output power level, Watts = po_toi = 10 ( (TOI-30)/10) = vo_toi 2 /(2*RefR)
SOI input power level, Watts = pi_soi = po_soi/(a1 2 )
TOI input power level, Watts = pi_toi = po_toi/(a1 2 )
SOI and TOI are defined with respect to two-tone inputs with equal amplitude and with small frequency difference (Δω) such that ω1=ω0-Δω and ω2=ω0+Δω.
For SOI, the output tones of interest are at ω1+ω2=2*ω0, and ω2-ω1=2*Δω.
For TOI, the output tones of interest are at
|2*ω1-ω2| = ω0-3*Δω
or
|2*ω2-ω1| = ω0+3*Δω
The two-tone excitation response equation shows the amplitude of the 1st-order product (the fundamental) with a value of
At low power levels, the a i*A 1 term is dominant (the higher-order terms are negligible):
As a result, at low level input power levels (dBm), the 1st, 2nd, and 3rd-order output powers (dBm) vary versus input power (dBm) with ratios 1:1, 2:1, and 3:1 respectively.
By definition, the SOI and TOI points occur where the input and output power levels (dBm) are equal for the extrapolation of the small-signal power levels (dBm) for the fundamental and 2nd-order harmonics (SOI) or 3rd-order harmonics (TOI).
For example, given ω1 = 995 MHz at -30 dBm and ω2 = 1005 MHz at -30 dBm and nonlinear device with S21 = 20 dB, SOI = 50 dBm and TOI = 30 dBm, the 2nd-order and 3rd-order output tones are at (1990 MHz, 2000 MHz, 2010 MHz) and (985 MHz, 1015 MHz) respectively.
For TOI at 30 dBm, the output tones at 1005 MHz and 1015 MHz are at -10 dBm and -90 dBm respectively. TOI is related to the levels at 1005 MHz and 1015 MHz as follows:
TOI = -90 + 3/2*(-10 - (-90)) = 30 dBm
For SOI at 50 dBm, the output tones at 1990 MHz, 2000 MHz and 2010 MHz are at -76 dBm, -70 dBm and -76 dBm respectively. The 2nd-order product at 2000 MHz is the largest. SOI is related to the levels at 1005 MHz and 2000 MHz as follows:
SOI = -70 + 2*(-10 - (-70)) = 50 dBm
For SOI, and from the above equations for two-tone excitation, the maximum 2nd-order output tone of interest occurs at ω1+ω2 for which the relationship between a 2 to a 1 and SOI is as follows:
a 2 = a 1 2 /sqrt(2*RefR*po_soi)
where:
po_soi = 10 ( (SOI-30)/10) with SOI in dBm
For TOI, and from the above equations for two-tone excitation, the 3rd-order tones at |2*ω1-ω2| or |2*ω2-ω1| are the desired 3rd-order intermod tones for which the relationship between a 3 to a 1 and TOI is as follows:
pi_toi = vi_toi 2 /(2*RefR)
a 3 = (4/3)*a 1 3 /(2*RefR*po_toi)
where:
po_toi = 10 ( (TOI-30)/10) with TOI in dBm
2nd and 3rd-Order Intercept for a Cascade Network
When N two-port, two-pin nonlinear components are connected in cascade, the expression for the overall output 2nd- and 3rd-order intercepts (see reference 13 in References for RF System Budget Analysis) are as follows:
where:
vsoi = overall output SOI in volts
vsoi[i] = i'th component output SOI in volts
vg[i] = system voltage gain (magnitude) from the i'th component output to the system output
where:
ptoi = overall output TOI in watts
ptoi[i] = i'th component output TOI in watts
pg[i] = system power gain (magnitude) from the i'th component output to the system output
This expression is typically evaluated (see reference 13 in References for RF System Budget Analysis) as a scalar equation by ignoring each component's reflection coefficients (s11 and s22), transmission phase characteristic (angle of s21), and reverse transmission coefficients (s12).
However, this expression becomes a close approximation to a complete complex nonlinear solution when the pg[i] and vg[i] terms include the effects of each component's reflection coefficients (s11 and s22), transmission phase characteristic (angle of s21), and reverse transmission characteristic (s12).
The preceding expressions for the network 2nd-order and 3rd-order intercepts are based on the small-signal performance of the individual components and on the extrapolated intersection of each components small-signal fundamental and 2nd-order and 3rd-order P out versus P in curves.
This formulation given above is used by the program to derive the network input and output 2nd and 3rd-order intercepts (InSOI, OutSOI, InTOI and OutTOI) and associated measurements utilizing these intercepts (Cmp_OutSOI_dBm, OutIM2_dBm, Cmp_OutTOI_dBm, OutIM3_dBm, OutSFDR_ResBW_dB, OutSFDR_Total_dB, OutS_IM3_dB). These measurements do not require any large-signal analysis, and thus are approximations to the network's actual large-signal performance.
Raw Data Generated for an RF Budget Analysis
The cascaded two-port, two-pin analysis described in the prior sections defined small-signal S-parameter analysis, power dependent S-parameter analysis, and noise parameter analysis. Those analyses result in raw data from which the RF budget measurements are derived. To define this raw data, several cascade system definitions are shown first:
System source definitions
- RefR = system source resistance = 50 (cannot be changed by user)
- Source reflection coefficient, Gs = 0
- TempS = system source temperature in degrees Celsius
- FreqS (FreqPilot) = system source (AGC pilot) frequency
- PwrS, PwrS_dBm (PwrPilot) = system source (AGC pilot) available power in W, dBm
System load definitions
- RefR = system load resistance = 50 (cannot be changed by the user)
- System load resistance, GL = 0
- TempL = system load temperature = -273.15°C (cannot be changed by the user)
Component definitions
- N = number of cascaded components
- n = component index; n = 0, 1, ..., N-1
Simulation setup parameters
- ResBW: user-defined resolution BW for noise measurements
- SimBW, SimFStep: SimBW is the user-defined BW over which noise measurements are swept in frequency with steps defined by SimFStep
Temporary data calculated during cascade analysis
- s11_ss[n], s12_ss[n], s21_ss[n], s22_ss[n], s11_ls[n], s12_ls[n], s21_ls[n], s22_ls[n]
- S-parameters from system input to component n output for small- and large-signal analysis
- b2_ss[n], b2_ls[n]
- System wave out from component n and incident on component n+1 based on small- and large-signal analysis
- b2[N-1] = system wave incident on system load; define b2[-1] = 0
- a2_ss[n], a2_ls[n]
- System wave at output of component n and reflected from component n+1 based on small- and large-signal analysis
- a2_ss[N-1], a2_ls[N-1] = 0 because GL = 0
- Define a2[-1] = reflection from component 0 input
System raw data
With the cascade system definition above, the system raw data is defined here where n represents the n'th component with the index starting at zero for the first component:
- F[n] = frequency (Hz) at component output
- Let F[-1] = FreqS
- F[N-1] = FreqL = system load frequency based on system input FreqS
- G_ss[n], G_ls[n] = reflection coefficient at component input for small- and large-signal analysis
- Let G[N] = GL = 0
- VGI_ss[n], VGI_ls[n] = voltage gain for wave incident on load at component output over system input wave for small- and large-signal analysis input wave
- VGI[n] = b2[n]/as = s21[n]/(1 - s11[n]*Gs - s22[n]*G[n+1] - s12[n]*s21[n]*Gs*G[n+1] + s11[n]*s22[n]*Gs*G[n+1])
- When Gs = 0: VGI[n] = s21[n]/(1 - s22[n]*G[n+1])
- VGR_ss[n], VGR_ls[n] = for wave reflected by load at component output over system input wave for small- and large-signal analysis
- VGR[n] = a2[n]/as = G[n+1] (b2[n]/as)
- When Gs = 0: VGR[n] = G[n+1] s21[n]/(1 - s22[n]*G[n+1])
- P_ss[n], P_ls[n], Q_ss[n], Q_ls[n] = power incident into, reflected from component input for small- and large-signal analysis
- P[n] = PwrS|VGI[n-1]| 2
- Q[n] = PwrS|VGR[n-1]| 2 = |G[n]| 2 P[n]
- PG_ss[n], PG_ls[n] = transducer power gain from system input to power delivered into load at component output for small- and large-signal analysis
- PG[n] = (1-|Gs| 2 )(1-|G[n+1]| 2 )|s21[n]| 2 / |(1 - s11[n]*Gs)(1 - s22[n] G[n+1])- s12[n]*s21[n]*Gs*G[n+1] | 2
- When Gs = 0:
PG[n] = (1-|G[n+1]| 2 ) |s21[n]| 2 / |1 - s22[n]*G[n+1]| 2
PG[n] = |VGI[n]| 2 - |VGR[n]| 2
- NPwr[n] = noise power, dBm, at component output
where:FreqS_NBW[j] = system input frequencies contributing to component j output within its SimBW centered at its primary frequency
j = frequency index of swept system source
k = Boltzmann's constant
NFin[n,j] = NF_RefIn[n] for FreqS_NBW[j]
PGss[n,j] = PG_ss[n] for FreqS_NBW[j] - NBW[n] = noise bandwidth at component output
- NBW[n] = NPwr_W[n]/NPwr0_W[n]
- NPwr0_W[j] = noise power per Hz at the center of the SimulationBW at the node
- Noise figure from system input to component output, NF_RefIn[n]
- Derived from NPwr_NF (W/Hz): defined to be similar as NPwr, but with TempS replaced with 290 K
- NPwr_NF[n] = (k*Ti[n]*Gi[n] + k*Ts*Gi[n])
where:Gi[n] = G1[n] + G2[n] + G3[n] ... = total transducer power gain (ratio) from system input to component n output for system input fundamental and image frequencies
G1[n] = transducer power gain from system input to the component n output at the system input frequency
Ti[n] = T1[n] + T2[n] + T3[n] ... = total noise temperature (K) representing the system noise contribution at system input fundamental and image frequencies
T1[n] = noise temperature representing the system noise contribution from system input to the component n output at the system input frequency
Ts = source noise temperature = T0 = 290 K
NF_RefIn[n] = 10*log10( (T[n]*Gi[n] + T0*Gi[n])/(T0*G1[n]) )The Noise Figure above is from system input to component n output and includes all system input image noise.
Noise Figure from system input to component output
The Noise Figure from system input to component output, but excluding system image noise is not the real system noise figure, but is also available for users:
- NF_RefIn_NoImage[n] = noise figure, dB, from system input to component n output, with exclusion of all image noise
This is available for user reference to compare to their Excel spreadsheet calculations that do not include image noise.
Noise Figure from component input to system output
The Noise Figure from component input to system output available excludes system image noise:
- NF_RefOut_NoImage[n] = noise figure, dB, from component n input to system output, with exclusion of all image noise
The Noise Figure contribution by the component to the overall system noise figure excludes system image noise. - NF_Ctrb_NoImage[n] = component n contribution, in dB, to full system NF
Raw Data Summary for Budget Analysis
With the definitions above, the following table shows the raw data from which all measurements are related by formula.
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