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# Z-Parameters Amplifier

Model nonlinear amplifier using Z-parameters

## Library

Amplifiers sublibrary of the Physical library

## Description

The Z-Parameters Amplifier block models the nonlinear amplifier described in the block dialog box, in terms of its frequency-dependent Z-parameters, the frequencies of the Z-parameters, noise data, and nonlinearity data

### Network Parameters

In the Z-Parameters field of the block dialog box, provide the Z-parameters for each of M frequencies as a 2-by-2-by-M array. In the Frequency field, specify the frequencies for the Z-parameters as an M-element vector. The elements of the frequencies vector must be in the same order as the Z-parameters. All frequencies must be positive. For example, the following figure shows the correspondence between the Z-parameters array and the vector of frequencies.

The Z-Parameters Amplifier block uses the RF Toolbox™ `z2s` function to convert the Z-parameters to S-parameters, and then interpolates the resulting S-parameters to determine their values at the modeling frequencies. See SimRF Equivalent Baseband Algorithms for more details.

### Nonlinearity

You can introduce nonlinearities into your model by specifying parameters in the Nonlinearity Data tab of the Z-Parameters Amplifier block dialog box. Depending on which of these parameters you specify, the block computes up to four of the coefficients ${c}_{1}$, ${c}_{3}$, ${c}_{5}$, and ${c}_{7}$ of the polynomial

`${F}_{AM/AM}\left(s\right)={c}_{1}s+{c}_{3}{|s|}^{2}s+{c}_{5}{|s|}^{4}s+{c}_{7}{|s|}^{6}s$`

that determines the AM/AM conversion for the input signal $s$. The block automatically calculates ${c}_{1}$, the linear gain term. If you do not specify additional nonlinearity data, the block operates as a linear amplifier. If you do, the block calculates one or more of the remaining coefficients as the solution to a system of linear equations, determined by the following method.

1. The block checks whether you have specified a value other than `Inf` for:

• The third-order intercept point ($OIP3$ or $IIP3$).

• The output power at the 1-dB compression point (${P}_{1dB,out}$).

• The output power at saturation (${P}_{sat,out}$).

In addition, if you have specified ${P}_{sat,out}$, the block uses the value for the gain compression at saturation ($G{C}_{sat}$). Otherwise, $G{C}_{sat}$ is not used. You define each of these parameters in the block dialog box, on the Nonlinearity Data tab.

2. The block calculates a corresponding input or output value for the parameters you have specified. In units of dB and dBm,

`$\begin{array}{c}{P}_{sat,out}+G{C}_{sat}={P}_{sat,in}+{G}_{lin}\\ {P}_{1dB,out}+1={P}_{1dB,in}+{G}_{lin}\\ OIP3=IIP3+{G}_{lin}\end{array}$`

where ${G}_{lin}$ is ${c}_{1}$ in units of dB.

3. The block formulates the coefficients ${c}_{3}$, ${c}_{5}$, and ${c}_{7}$, where applicable, as the solutions to a system of one, two, or three linear equations. The number of equations used is equal to the number of parameters you provide. For example, if you specify all three parameters, the block formulates the coefficients according to the following equations:

`$\begin{array}{c}\sqrt{{P}_{sat,out}}={c}_{1}\sqrt{{P}_{sat,in}}+{c}_{3}{\left(\sqrt{{P}_{sat,in}}\right)}^{3}+{c}_{5}{\left(\sqrt{{P}_{sat,in}}\right)}^{5}+{c}_{7}{\left(\sqrt{{P}_{sat,in}}\right)}^{7}\\ \sqrt{{P}_{1dB,out}}={c}_{1}\sqrt{{P}_{1dB,in}}+{c}_{3}{\left(\sqrt{{P}_{1dB,in}}\right)}^{3}+{c}_{5}{\left(\sqrt{{P}_{1dB,in}}\right)}^{5}+{c}_{7}{\left(\sqrt{{P}_{1dB,in}}\right)}^{7}\\ 0=\frac{{c}_{1}}{IIP3}+{c}_{3}\end{array}$`

The first two equations are the evaluation of the polynomial ${F}_{AM/AM}\left(s\right)$ at the points $\left(\sqrt{{P}_{sat,in}},\sqrt{{P}_{sat,out}}\right)$ and $\left(\sqrt{{P}_{1dB,in}},\sqrt{{P}_{1dB,out}}\right)$, expressed in linear units (such as W or mW) and normalized to a 1-Ω impedance. The third equation is the definition of the third-order intercept point.

The calculation omits higher-order terms according to the available degrees of freedom of the system. If you specify only two of the three parameters, the block does not use the equation involving the parameter you did not specify, and eliminates any ${c}_{7}$ terms from the remaining equations. Similarly, if you provide only one of the parameters, the block uses only the solution to the equation involving that parameter and omits any ${c}_{5}$ or ${c}_{7}$ terms.

If you provide vectors of nonlinearity and frequency data, the block calculates the polynomial coefficients using values for the parameters interpolated at the center frequency.

### Active Noise

You can specify active block noise in one of the following ways:

• Spot noise data in the Z-Parameters Amplifier block dialog box.

• Noise figure, noise factor, or noise temperature value in the Z-Parameters Amplifier block dialog box.

If you specify block noise as spot noise data, the block uses the data to calculate noise figure. The block first interpolates the noise data for the modeling frequencies, using the specified Interpolation method. It then calculates the noise figure using the resulting values.

## Parameters

### Main Tab

Z-Parameters

Z-parameters for a nonlinear amplifier in a 2-by-2-by-M array. M is the number of Z-parameters.

Frequency (Hz)

Frequencies of the Z-parameters as an M-element vector. The order of the frequencies must correspond to the order of the Z-parameters in Z-Parameters. All frequencies must be positive.

Interpolation method

The method used to interpolate the network parameters. The following table lists the available methods describes each one.

MethodDescription
`Linear` (default)Linear interpolation
`Spline`Cubic spline interpolation
`Cubic`Piecewise cubic Hermite interpolation

### Noise Data Tab

Noise type

Type of noise data. The value can be `Noise figure`, ```Spot noise data```, `Noise factor`, or ```Noise temperature```. This parameter is disabled if the data source contains noise data.

Noise figure (dB)

Scalar ratio or vector of ratios, in decibels, of the available signal-to-noise power ratio at the input to the available signal-to-noise power ratio at the output, (Si/Ni)/(So/No). This parameter is enabled if Noise type is set to `Noise figure`.

Minimum noise figure (dB)

Minimum scalar ratio or vector of minimum ratios of the available signal-to-noise power ratio at the input to the available signal-to-noise power ratio at the output, (Si/Ni)/(So/No). This parameter is enabled if Noise type is set to `Spot noise data`.

Optimal reflection coefficient

Optimal amplifier source impedance. This parameter is enabled if Noise type is set to ```Spot noise data```. The value can be a scalar or vector.

Equivalent normalized resistance

Resistance or vector of resistances normalized to the resistance value or values used to take the noise measurement. This parameter is enabled if Noise type is set to ```Spot noise data```.

Noise factor

Scalar ratio or vector of ratios of the available signal-to-noise power ratio at the input to the available signal-to-noise power ratio at the output, (Si/Ni)/(So/No). This parameter is enabled if Noise type is set to `Noise factor`.

Noise temperature (K)

Equivalent temperature or vector of temperatures that produce the same amount of noise power as the amplifier. This parameter is enabled if Noise type is set to ```Noise temperature```.

Frequency (Hz)

Scalar value or vector corresponding to the domain of frequencies over which you are specifying the noise data. If you provide a scalar value for your noise data, the block ignores the Frequency (Hz) parameter and uses the noise data for all frequencies. If you provide a vector of values for your noise data, it must be the same size as the vector of frequencies. The block uses the Interpolation method specified in the Main tab to interpolate noise data.

### Nonlinearity Data Tab

IP3 type

Type of third-order intercept point. The value can be `IIP3` (input intercept point) or `OIP3` (output intercept point). This parameter is disabled if the data source contains power data or IP3 data.

IP3 (dBm)

Value of third-order intercept point. This parameter is disabled if the data source contains power data or IP3 data. Use the default value, `Inf`, if you do not know the IP3 value. This parameter can be a scalar (to specify frequency-independent nonlinearity data) or a vector (to specify frequency-dependent nonlinearity data).

1 dB gain compression power (dBm)

Output power value (${P}_{1dB,out}$) at which gain has decreased by 1 dB. This parameter is disabled if the data source contains power data or 1-dB compression point data. Use the default value, `Inf`, if you do not know the 1-dB compression point. This parameter can be a scalar (to specify frequency-independent nonlinearity data) or a vector (to specify frequency-dependent nonlinearity data).

Output saturation power (dBm)

Output power value (${P}_{sat,out}$) that the amplifier produces when fully saturated. This parameter is disabled if the data source contains output saturation power data. Use the default value, `Inf`, if you do not know the saturation power. If you specify this parameter, you must also specify the Gain compression at saturation (dB). This parameter can be a scalar (to specify frequency-independent nonlinearity data) or a vector (to specify frequency-dependent nonlinearity data).

Gain compression at saturation (dB)

Decrease in gain ($G{C}_{sat}$) when the power is fully saturated. The block ignores this parameter if you do not specify the Output saturation power (dBm). This parameter can be a scalar (to specify frequency-independent nonlinearity data) or a vector (to specify frequency-dependent nonlinearity data).

Frequency (Hz)

Scalar or vector value of frequency points corresponding to the third-order intercept and power data. This parameter is disabled if the data source contains power data or IP3 data. If you use a scalar value, the IP3 (dBm), 1 dB gain compression power (dBm), and Output saturation power (dBm) parameters must all be scalars. If you use a vector value, one or more of the IP3 (dBm), 1 dB gain compression power (dBm), and Output saturation power (dBm) parameters must also be a vector.

### Visualization Tab

For information about plotting, see Create Plots.

## Examples

### Plotting Parameters with the Z-Parameters Amplifier Block

The following example specifies Z-parameters [12.60+3.80i, 3.77-0.17i; 80.02+54.68i, 26.02+3.84i] and [15.12+3.55i, 4.14-0.92i; 92.10+23.67i, 27.59+2.71i] at frequencies 2.0 GHz and 2.1 GHz respectively. It uses the MATLAB® `cat` function to create the 2-by-2-by-2 Z-parameters array.

```cat(3,... [12.60+3.80i, 3.77-0.17i; 80.02+54.68i, 26.02+3.84i],... [15.12+3.55i, 4.14-0.92i; 92.10+23.67i, 27.59+2.71i]) ```
1. Type the following command at the MATLAB prompt to create a variable called `zparams` that stores the values of the Z-parameters.

```zparams = cat(3,... [12.60+3.80i, 3.77-0.17i; 80.02+54.68i, 26.02+3.84i],... [15.12+3.55i, 4.14-0.92i; 92.10+23.67i, 27.59+2.71i])```

2. Set the Z-Parameters Amplifier block parameters on the Main tab as follows:

• Set the Z-Parameters parameter to `zparams`.

• Set the Frequency (Hz) parameter to `[2.0e9,2.1e9]`.

Click Apply. This action applies the specified settings.

3. Set the Z-Parameters Amplifier block parameters on the Visualization tab as follows:

• In the Source of frequency data list, select `User-specified`.

• Set the Frequency data (Hz) parameter to `[1.9e9:1.0e8:2.2e9]`.

• In the Plot type list, select ```X-Y plane```.

• In the Y parameter1 list, select `S11`.

Click Plot. This action creates an X-Y Plane plot of the S11 parameters in the frequency range 1.9 to 2.2 GHz.

## See Also

`z2s` (RF Toolbox)

`interp1` (MATLAB)

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