Reference Measurements and Models for RF System Budget Analysis
This section provides the details about the measurements and component models used by the Budget controller:
RF Budget Cascade Measurements
You can select Budget Cascade Measurements of interest from the Budget controller's Measurements tab. These measurements produce a measurement value at each system node. For example, a system with five components will produce five values for each cascade measurement. These measurements are grouped as follows, and are described in the following sections:
- Component Measurements are for individual components. The Cmp_Ctrb type measurements are for the overall system without contributions of the individual component. The other Cmp type measurements are without contributions from the other system components.
- Noise Figure Measurements are system noise figure measurements that are calculated either from the system input to each component output (RefIn), or from each component input to the system output (RefOut).
- System Measurements at Component Inputs are for the subsystems defined from the component input to the system output.
- System Measurements at Component Outputs are for the subsystems defined from the system input to the component output.
In the following, power measurements use transducer power gain. Transducer power gain is dependant on all S-parameters of a network (s11, s12, s21, s22) along with the network source reflection coefficient (gs) and load reflection coefficient (gl). The standard expression for transducer power gain, g0, is
Measurements that involve power use source and load coefficients along with the S-parameters of the subnetwork being measured. Thus, power delivered into the system load at component n input is based on the transducer power gain for the subnetwork defined from the system input through component n-1, the system source reflection coefficient, the system load reflection coefficient into component n , and the system input power. The system load reflection coefficient at component n input is based on the subnetwork defined from component n to the system output and on the system output load reflection coefficient. The system load reflection coefficient at component n output is based on the subnetwork defined from component n+1 to the system output and on the system output load reflection coefficient.
Component Measurements
The formulas in the following table reference the raw data defined in Raw Data Generated for an RF Budget Analysis.
Component Measurements
Noise Figure Measurements
The formulas in the following table reference the raw data defined in Raw Data Generated for an RF Budget Analysis.
Noise Figure Measurements
System Measurements at Component Inputs
The formulas in the following table reference the raw data defined in Raw Data Generated for an RF Budget Analysis.
System Measurements at Component Inputs
System Measurements at Component Outputs
The formulas in the following table reference the raw data defined in Raw Data Generated for an RF Budget Analysis.
System Measurements at Component Outputs
RF Budget Summary Measurements
The System Summary Measurements define overall system performance from input to output.
The formulas in the following table reference the raw data defined in Raw Data Generated for an RF Budget Analysis.
System Summary Measurements
RF Budget Analysis Component Models
This section describes the component models used for RF budget analysis.
Two-Port, Two-Pin Small-Signal S-Parameter Definitions
Scattering parameters (S-parameters) are used to define the signal properties of a two-port, two-pin circuit component at a single frequency. The S-parameters over a range of frequencies define the component's performance for all defined spectral tones.
S-parameter definitions can be found in any textbook on circuit theory. The following discussion is for a two-port, two-pin component.
A two-port, two-pin circuit component signal wave representation can be shown in block diagram form. See the following figure.
Two-port, two-pin signal wave
where:
a 1 = wave into port 1
b 1 = wave out of port 1
a 2 = wave into port 2
b 2 = wave out of port 2
The S-parameters for this conventional component are defined in standard microwave text books as follows:
b 1 = a 1 s 11 + a 2 s 12
b 2 = a 1 s 21 + a 2 s 22
where:
s 11 = port 1 reflection coefficient: s 11 = b 1 /a 1 ; a 2 = 0
s 22 = port 2 reflection coefficient: s 22 = b 2 /a 2 ; a 1 = 0
s 21 = forward transmission coefficient: s 21 = b 2 /a 1 ; a 2 = 0
s 12 = reverse transmission coefficient: s 12 = b 1 /a 2 ; a 1 = 0
S-parameters are defined with respect to a reference impedance that is typically 50 ohms. For 50-ohm S-parameters, and with the two-port, two-pin component terminated with 50 ohms at each port, the s 21 parameter is simply the voltage gain of the component from port 1 to port 2.
These equations can be solved for b 1 and a 1 in terms of a 2 and b 2 to yield the transmission (T) parameters as follows:
b 1 = a 2 t 11 + b 2 t 12
a 1 = a 2 t 21 + b 2 t 22
The T-parameters are related to the S-parameters as follows:
S-Parameter Definitions for Components with Spectral Inversions
A spectral inversion (SI) component is a component whose output signal is derived from the conjugate phase of the input signal. This typically occurs for down converting mixers with an LO frequency greater than the input RF frequency.
The frequency inversion of signals through a spectral inverting component brings about a conjugate transformation to the transmitted wave. This transformation makes use of the property of the mixer which can be modeled as a multiplier of the input and local oscillator waveforms:
V in(t) = cos(wi • t + Phi)
V LO(t) = cos(wlo • t); assume wlo > wi
V o(t) = Vin * VLO
V o(t) = 0.5 cos((wlo-wi) t - Phi) + 0.5 cos((wlo+wi) t + Phi)
As shown, the lower sideband component, wlo-wi, has a phase component which is the conjugate of the input phase.
The S-parameter definitions for a spectral inverting component must account for the spectral inversion that occurs at the output. Therefore, the S-parameters for a spectral inverting component are slightly different than those of a conventional component (see references 10, 11, and 12 in References for RF System Budget Analysis).
(* in the following represents conjugate):
s 11 is the port 1 reflection coefficient: s 11 = b 1 /a 1 ; a 2 = 0
s 22 is the port 2 reflection coefficient: s 22 = b 2 /a 2 ; a 1 = 0
s 21 is the forward transmission coefficient: s 21 = b 2 /a 1 *; a 2 = 0
s 12 is the reverse transmission coefficient: s 12 = b 1 /a 2 *; a 1 = 0
Note that s 21 and s 12 account for the conjugate of the incident wave. The definitions for s 11 and s 12 above are slightly different from the convention used in reference 10 (in References for RF System Budget Analysis) in which s 11 = b 1 */a 1 * and s 12 = b 1 */a 2.
The reverse transmission wave, b 1, and forward transmission wave, b 2, are as follows:
b 1 = a 1 s 11 + a 2 * s 12
b 2 = a 1 * s 21 + a 2 s 22
Two-Port, Two-Pin Noise Parameter Definitions
Noise parameters are used to define the noise properties of a circuit component at a single frequency. The noise parameters over a range of frequencies define the component's performance for all noise power spectral density spectral tones defining an incident noise.
Noise parameter definitions can be found in any textbook on circuit theory. Noise wave parameters are used by the program to define the noise properties of any circuit component. The following discussion is for a two-port, two-pin component.
The two-port, two-pin component noise wave representation may use two noise waves at the component input (see section (b) in the next figure; see reference 1 in References for RF System Budget Analysis). Otherwise, it may use one noise wave at the component input and one at the component output (see section (c) in the next figure; see reference 2 in References for RF System Budget Analysis).
In the following noise discussions, the spot noise in a bandwidth of 1 Hz is assumed.
A two-port, two-pin component noise wave representation with two noise waves at the component input is shown in section (b) in the following figure; (also see reference 1 in References for RF System Budget Analysis).
The noise correlation matrix, [N], is defined as follows:
where the noise is considered to be within a 1 Hz bandwidth and * represents the complex conjugate.
A two-port, two-pin component noise wave representation with one noise wave at the component input and at the component output is shown in section (c) in the following figure (also see reference 2 in References for RF System Budget Analysis).
Component signal and noise wave representations
The noise correlation matrix, [A], is defined as follows:
where the noise is considered to be within a 1 Hz bandwidth.
These noise waves, bn1 and bn2, are related to the first pair of noise waves, an and bn, as follows:
bn1 = an s11 + bn an = bn2/s21 bn2 = an s21 bn = bn1 - bn2 s11/s21
This results in the following relationship between the [A] and [N] noise correlation parameters:
A11 = N22 |s11|2 + N21 s11 + N12 s11* + N11
A12 = N22 s11 s21* + N12 s21*
A21 = N22 s11* s21 + N21 s21
A22 = N22 |s21|2
N11 = A11 + A22 |s11|2/|s21|2 - A12 s11*/s21* - A12* s11/s21
N12 = A12/s21* - A22 s11/|s21|2
N21 = A21/s21 - A22 s11*/|s21|2
N22 = A22/|s21|2
Linear Component Noise Models
A two-port, two-pin linear circuit component has a mathematical model defined by a 2x2 S-parameter matrix and a 2x2 noise wave parameter matrix. The linear component may be passive or active.
A passive component has S-parameters that satisfy the energy conservation requirement for port index i from 1 to N:
The noise wave parameters for a linear passive component are derived from the component S-parameters and its physical temperature as follows:
For an n-port passive component the noise correlation matrix is given by reference 3 in References for RF System Budget Analysis:
[A] = k • Tphys • {[I] - [s][s*] T}
where:
k = Boltzmann's constant
Tphys = physical temperature in K
[I] = the identity matrix
[s] = the component's S-parameter matrix
[s*] T = the transpose of the conjugate of the [s] matrix
For a two-port, two-pin component:
The [N] noise correlation matrix may be derived from this [A] matrix.
All active linear components within the program are two-port, two-pin components with noise wave parameters that are related to the more common noise parameters of NFmin, Gopt, and Rn (minimum noise figure (dB)), optimum source reflection coefficient for NFmin, equivalent input normalized noise resistance, respectively) as follows (see reference 1 in References for RF System Budget Analysis):
N11 = k Tb T0
N12 = k Tc T0 (cos(phi) + j sin(phi))
N21 = k Tc T0 (cos(phi) - j sin(phi))
N22 = k Ta T0
k = Boltzmann's constant
T0 = reference temperature = 290 K
Ta = Fmin - 1 + Td |Gopt| 2
Tb = Td - Fmin + 1
Tc = Td |Gopt|
phi = PI - angle(Gopt)
Td = 4 Rn/ | 1 + Gopt | 2
Fmin = 10 (NFmin / 10)
angle(Gopt) = angle of Gopt
Z0 = reference resistance
These noise correlation parameters, Nij, can be converted back to the standard noise parameters as follows:
k Td = 0.5 {(N22+N11) + sqrt((N22+N11) 2 - 4 |N12| 2 )}
NFmin = 10 log10(Td + 1 - N11/k)
|Gopt| = |N12/k|/(Td)
angle(Gopt) = PI - angle(N12/k)
Rn = Td/4 | 1+Gopt| 2
The component noise is dependent on the source reflection coefficient, Gams, as follows (see reference 1 in References for RF System Budget Analysis):
NF = 10 log10(nf)
where:
There is a physical realizability requirement for the noise parameters of an active two-port, two-pin component. This requirement may be expressed with respect to the common noise parameters of NFmin, Gopt, and Rn (minimum noise in dB, optimum source reflection coefficient for NFmin, equivalent input normalized noise resistance, respectively) as follows:
This is based on the requirement that the component's combined noise wave power at port 1 not be negative.
Any active component has its noise parameters checked against this physical requirement. If the noise parameters supplied by the user in any component are such that the Rn value supplied is less than this minimum, then the specific component model will either error out and quit with error message to the user, or will proceed by setting the Rn value to this limit value.
Nonlinear Component Models
All nonlinear circuit components are two-port, two-pin components with a mathematical model defined by a 2x2 S-parameter matrix versus input power at port 1 and port 2, and a 2x2 noise wave parameter matrix derived at small-signal conditions.
In general, all S-parameters (s11, s12, s21, s22) vary as a function of input power. The parameters s11 and s21 are defined as a function of power incident at port 1 with no power incident at port 2; whereas s12 and s22 are defined as a function of power incident at port 2 with no power incident at port 1 (see reference 5 in References for RF System Budget Analysis). The data set for this nonlinear model is readily measured for a nonlinear RF two-port, two-pin in a hardware measurement lab.
This data set has been accepted as a convenient means of characterizing nonlinear devices by their large-signal S-parameters and have been successfully used for designing power amplifiers, oscillators, etc. (see references 6, 7, and 8 in References for RF System Budget Analysis).
This general model for a circuit component nonlinearity is derived during a system simulation for a nonlinear component. For details about the P2D data file format, see P2D Format.
The noise wave parameters for an electrical nonlinearity are the same as those defined for an active linear component in Linear Component Noise Models.
Characterization of Component Nonlinearities
In general, nonlinear two-port, two-pin amplifiers have an output power versus input power characteristic as shown in the following figure.
Nonlinear component characterization for power out versus power in
As shown, the nonlinear characteristic includes:
- Fundamental small-signal linear gain
- 2nd-order products
- 3rd-order products
- 5th and higher order products
- 3rd-order intercept point (IP3, also called TOI)
- 2nd-order intercept (IP2, also called SOI)
- 1 dB gain compression point (1DBC)
- Power saturation point (PSat)
- Gain compression at saturation (Gcs)
For budget analysis, nonlinear two-port, two-pin amplifiers are modeled using one of the following modeling techniques:
- Nonlinear models with S-parameters versus power used for all nonlinear measurements except for SOI and TOI measurements.
- Nonlinear models with defined SOI used for SOI measurements.
- Nonlinear models with defined TOI used for TOI measurements.
- Nonlinear models with no explicitly defined TOI used for TOI measurements.
All nonlinear models represent zero memory nonlinearities.
Nonlinear models with S-parameters versus power
This nonlinear model is used for all nonlinear measurements except for SOI and TOI measurements. It is based on the large-signal S-parameter (P2D) data set for each nonlinear component. This data set is obtained for each nonlinear component during budget analysis set-up. In this data set, all S-parameters (S11, S21, S12, S22) can vary as a function of frequency and power. This data set is linearly interpolated and used for budget analysis.
The P2D data for a nonlinear component is collected at the component input frequency and for a power range that is dependant on whether the user has selected any budget measurements that require the 1 dB power compression point (P1dB), SOI, or TOI for each component.
When no P1dB, SOI, and TOI measurements are selected The maximum component input power used to collect the P2D data for a nonlinear component is equal to the component large-signal incident power (P_LS_inc) plus 5 dB (P_LS_inc+5dB). The large-signal incident power is determined by analyzing the system to determine the system large-signal gain from the system source output to the evaluated nonlinear component input. This maximum power level (P_LS_inc+5dB) must be less than or equal to the CmpMaxPin value (Component maximum input power in dBm). You can set CmpMaxPin in the Budget controller's setup dialog box, on the Setup tab.
When any P1dB, SOI, or TOI measurement is selected The maximum component input power used to collect the P2D data for a nonlinear component is equal to the component small-signal incident power (P_SS_inc) that places the nonlinear component into at least 5 dB gain compression (P_SS_inc_5dB). The small-signal incident power is determined by analyzing the system to determine the system small-signal gain from the system source output to the evaluated nonlinear component input. This maximum power level (P_LS_inc_5dB) must be less than or equal to the CmpMaxPin value (Component maximum input power in dBm). You can set CmpMaxPin in the Budget controller's setup dialog box, on the Setup tab. The P1dB measurements are any of the following: Cmp_OutP1dB_dBm, InP1dB_dBm, OutCDR_ResBW_dB, OutCDR_Total_dB, OutP1dB_dBm.
Given the component maximum input power from the above conditions (P_LS_inc+5dB or P_SS_inc_5dB), the P2D data (S11, S12, S21, S22 versus power) is obtained for the nonlinear component over a 100 dB range below this maximum input power in 1 dB steps.
During budget analysis, the P2D data for each nonlinear component is linearly interpolated in an iterative process to obtain the overall system operating points at each nonlinear component input and output. All P2D S-parameters (S11, S12, S21, S22) are used in this analysis.
Nonlinear models with defined SOI used for SOI measurements
This nonlinear model is used only for components Amplifier, Amplifier2, and AGC_Amp, and only when these models are at the top level of the RF System design being analyzed. If these models are within a nonlinear subnetwork design, then the subnetwork is considered to not have any defined SOI and the nonlinear subnetwork is modeled as described in the section Nonlinear models with S-parameters versus power.
For components Amplifier, Amplifier2 and AGC_Amp, SOI can only be used if TOI is also specified.
The SOI value for these nonlinear amplifiers is used directly in the Budget SOI measurements Cmp_OutSOI_dBm, InSOI_dBm, OutIM2_dBm, OutSOI_dBm.
For a definition of these measurements, see sections Component Measurements, System Measurements at Component Inputs and System Measurements at Component Outputs.
Nonlinear models with defined TOI used for TOI measurements
This nonlinear model is used only for components Amplifier, Amplifier2 and AGC_Amp and only when these models are at the top level of the RF system design being analyzed. If these models are within a nonlinear subnetwork design, then the nonlinear subnetwork is modeled as described in the section for Nonlinear model with no explicitly defined TOI used for TOI measurements.
The TOI value for these nonlinear amplifiers is used directly in the Budget TOI measurements Cmp_OutTOI_dBm, InTOI_dBm, OutIM3_dBm, OutTOI_dBm, OutSFDR_ResBW_dB, OutSFDR_Total_dB, OutS_IM3_dB.
For definition of these measurements, see sections Component Measurements, System Measurements at Component Inputs, and System Measurements at Component Outputs.
Nonlinear model with no explicitly defined TOI used for TOI measurements
This nonlinear model is used for any nonlinear component with no defined TOI or for any nonlinear subnetwork for which TOI measurements are to be made.
The TOI value for these nonlinear amplifiers is derived from their P2D dataset for S21 versus power, and then these models are treated the same as for those described in Nonlinear models with defined TOI used for TOI measurements.
The TOI value is obtained by curve fitting the component S21 data versus power to a third order polynomial expression evaluated under low power conditions. It is presumed that the nonlinear model does not contain any SOI characteristic.
The 3rd-order polynomial expression relates output RF voltage (V out ) to input RF voltage (V in).
where:
V in = input signal voltage
V out = output signal voltage
a 1 = small-signal gain
a 3 = 3rd-order gain coefficient
The S21 data versus power (V out versus V in) is used to find the input signal level where 0.2 dB gain compression occurs (Vin_0.2dB). It is presumed that only the 3rd-order nonlinearity dominates at this level of gain compression and that all higher order nonlinear polynomial terms are negligible.
Given the small-signal gain (a 1) and the value for Vin_0.2dB, the a 3 coefficient (and thus the TOI value) is derived.
For details on how a 3 and TOI are related, see the section 2nd and 3rd-Order Intercept Definition.
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