This is the fourth article in a series of blogs , which mainly discusses the importance of GaN HEMT nonlinear model for fast and efficient implementation of power amplifier (PA) design.

introduction
In a simple linear RF / microwave amplifier design, s- parameter matching is generally used to maximize gain and gain flatness. These S- parameter data will also be used to develop matching networks to solve amplifier stability problems. This article discusses the importance of using models to simulate basic S- parameters and stability analysis in the design of gallium nitride (GaN) power amplifiers (PA) . The article introduces the use of models and resistance stability techniques to help avoid device instability, thereby avoiding impact on nonlinear and linear simulations.

In this blog post, we focus on the simple two-port stability analysis used in linear S- parameter calculations. We will use the nonlinear Qorvo GaN power transistor model in the Modelithics Qorvo GaN library , together with the simulation template and Keysight Advanced Design System (ADS) software.

Stability interpretation
Stability refers to the ability of the PA to resist potential stray oscillations. Oscillation may be a full-power large signal problem, or it may be a hidden spectrum problem that has not been properly analyzed and cannot be detected. Even unwanted signals outside the expected frequency range may cause system oscillations and degraded gain performance.

Stability can be divided into two types, and you can use some methods to analyze the stability of PA in your system .

Conditional stability -The system designremains stable when theinput and output exhibit the expected characteristic impedance Z 0 (50Ω or 75Ω), but may be oscillated due to other input or output impedances (input or output ports show negative impedance).
 
Unconditional stability –The system remains stable under any possible positive real impedance in the Smith chart. Note that any system design will oscillate when it encounters negative real impedance (outside of the Smith chart). However, in general, if the system is defined as unconditionally stable, then it can remain stable at all frequencies (the device can gain gain) and all positive real impedances.
 

STABILITY MEASUREMENT
Let's first look at the well-known "K factor " and stability measurement parameter "b" to determine the frequency range that causes instability under a given bias. These derived values calculated from the following equation 1 :

k = {1- |S 11 | 2- |S 22 | 2 + |S 11 *S 22 -S 12 *S 21 | 2 } / {2*|S 12 *S 21 |}

as well as

b = 1 + |S 11 | 2- |S 22 | 2- |S 11 *S 22 -S 12 *S 21 | 2

Unconditional stability is represented by k> 1 and b> 0 .

However, because this standard requires two parameters to check unconditional stability, a more concise formula can be provided, which can be calculated using the "mu-prime" parameter 2 below :

mu_prime = {1-|S 22 | 2 } / {|S 11 -conj(S 22 )*Delta| + |S 21 *S 12 |}

If mu_prime> 1 , it means that it is unconditionally maintained (linearly) stable. 
 

USE MATCHING AND TUNING TO GAIN STABILITY
As mentioned above, S- parameter data can be used to develop a matching network to obtain amplifier stability. Figure 1 shows the single-stage amplifier configuration and the key parameters that affect gain and stability. In the unconditionally stable region, the maximum gain is obtained by setting Γ s and Γ L to achieve simultaneous conjugate matching at two ports. 1

Parameters affecting gain and stability calculations
Figure 1.

Linear stability analysis
STABILITY MEASUREMENT OF UNTUNED TRANSISTORS
Let's look at an example. Figure 2 shows the simulation settings for linear S- parameter analysis of the nonlinear model of Qorvo 's T2G6003028-FS GaN HEMT device ( included in the Modelithics Qorvo GaN model library ) .

Simulation setup for linear S-parameter analysis without stabilization
Figure 2.

Note: The bias conditions for all simulations here are set to Vds = 28 V , Vgs = -3.02 V , which is equivalent to about 200 mA drain current.

In the above schematic diagram, the symbols indicate the parameters that can be calculated using the S parameters of the device , including stability k , b and mu_prime . The "MaxGain1" parameter represents the maximum available gain. The "MaxGain1" parameter calculates the maximum available gain within the frequency range where the device remains unconditionally stable, and displays the value representing the maximum stable gain. In areas where conditions are stable, this value is calculated by simple |S 21 |/|S 12 | .

Figure 3 shows the MaxGain1 parameters, 50Ω gain ( S 21 , unit: dB ) and stability coefficient k , as well as the measured values of b and mu_prime calculated from the schematic diagram of Figure 2 (at m5 ) . This figure shows that the stability measurement value b> 0 and the stability coefficient k> 1 . At about 1.85 GHz (m5) , the stability measurement parameters show a clear turning point. This is the switching frequency between the conditional and unconditional maintaining a stable region. At 3.5 GHz , the maximum gain represented by this simulation parameter is about 18.4 dB (corresponding to the mark m3 in Figure 3 ). Note: At about 10.4 GHz , the maximum usable gain reaches 0 dB ; this frequency is expressed as the maximum frequency or f max analysis from very low frequencies to at least fStability in the max range is a very good practice, which is why in this example, it is set to sweep between 25 MHz and 12 GHz .

According to this analysis, we can draw:

Above 1.85 GHz , the device remains unconditionally stable.
Below 1.85 GHz , the device is conditionally stable.
 

The S parameters derived from the simulation diagram (Figure 2 ) are shown in Figure 4 . S 11 and S 22 are shown in the Smith chart, and the polar chart is used to show S 21 and S 12 .

Note that the gain of 50 Ω input and output ( |S 21 | , unit : dB ) differs greatly from the MaxGain1 value. This is caused by the mismatch associated with S 11 and S 22 in the 50 Ω system .

Gain and stability metrics graphs Figure 3.

S-parameter plots
Figure 4.

Drawing stability circles in the input and output planes can provide more insights. The schematic diagram shown in Figure 2 also includes the symbols "S_StabCircle" and "L_StabCircle" , which correspond to the calculated values ​​of the stability circle in the input and output planes.

The meaning of these circles is as follows. At 25 MHz , the input stability circle is indicated by mark 14 in Figure 5. Each point on the circle represents a Γ s value. According to the following formula, each value can give a Γ out value equal to 1 . .

Γ out = S 22 + S 12 *S 21 *{Γ s / (1-S 11 *Γ s )}

Formula 1

This circle sets the boundary between Γ out <1 and Γ out > 1 , and its meaning is that Γ out > 1 corresponds to the negative impedance of the output port, which may cause oscillations. After that, the question becomes whether there is an unstable (Γ out > 1) area inside or outside the circle . When Γ s = 0 (ie the 50Ω point), perform a quick check. Note that, according to Equation 1 , Gamma] OUT = S 22 is when , for all frequencies less than 1 to analyze. From this, we can conclude that the outside of the circle is a stable area, and the inside of the circle is an unstable area.

The explanation of the output stability circle is basically similar, except that the circle diagram of the Γ L point drawn at this time , Γ in = 1 (according to formula 2 ). Through similar arguments, we can conclude that the inside of the circle diagram shown on the right side of Figure 5 corresponds to an unstable region. Note that reducing the spectrum plan shown in Figure 2 is to reduce the number of circles shown in Figure 5 to make it clearer.

Γ in = S 11 + S 12 *S 21 *{Γ L / (1-S 22 *Γ L )}

Formula 2

Source and load reflection coefficient reference planes
Figure 5.
 

Linear stability analysis
So, what happens when the device cannot meet the requirements for unconditional stability (for example, in our example, the frequency is below 1.85 GHz )?

You can use several matching methods to help stabilize your circuit. In this article, we introduce two methods. One is to use impedance, and the other is to rely on frequency stability.

R impedance - use matching resistors to provide stability
Depend on frequency -use resistance, inductance and capacitance to provide stability
 

IMPEDANCE STABILITY OF MICROWAVE PA DESIGN
In our example, matched resistors can be used to help stabilize high-gain, low-frequency transistors in most microwave applications. These resistors can be connected in series or parallel at the input or output, can be placed in a parallel feedback loop, or included in a bias network. For PA , we want to maximize the output power, so it is best to avoid using resistors in the output network. The feedback amplifier is not in the scope of this article, so we will focus on the series and parallel resistors in the input network.

Figure 6 shows where to add series and parallel resistors in the input network. Adjust these values ​​to achieve unconditional stability over the entire frequency range of 0.025 to 12 GHz . The resulting stability measurements are shown in Figure 7 . These values ​​show that the transistor has unconditional stability over the entire frequency range. However, note that f max will drop from 10.3 GHz to approximately 8.75 GHz . Comparing the maximum gain estimate in Fig. 7 (design frequency is 3.5 GHz [12.3 dB] ) with the value in Fig. 3 ( 18.4 dB , which does not have this stability), we can see that the maximum usable gain is reduced by about 6 dB . This is caused by adding a pure resistance input stabilization network. The S- parameters of the resistance-stabilized device are shown in Fig. 8 , overlapping with the S- parameters of the unstable device . We can see that S 11 and S 12 are affected in the entire frequency range, S 21 is also reduced, S 22 has the smallest change. Fortunately, it can be seen from Figure 9 that after adding the resistance stability network, the stability circle now falls outside the Smith chart in the power and load planes.

Simulation setup for linear S-parameter analysis with stabilization
Figure 6. (Note: The analysis settings are the same as Figure 2 )

Resistively stabilized gain graphs
Figure 7.

S-parameters for resistively stabilized vs non-stabilized device
Figure 8.

Stability circles for resistively stabilized device
Figure 9.

FREQUENCY-DEPENDENT RESISTANCE STABILITY
If the design frequency is higher than 1.85 GHz (for example, 3.5 GHz ), we can implement a frequency-dependent resistance method using a series - parallel stability network. Let's see if we can use this method to reduce the above gain loss.

In Figure 10 , we integrate a resistor (R1) into the modified gate bias network. In addition, place a capacitor (C3) on the series stability resistor (R1) . A serial resistor may be adjusted by adjusting the value of this capacitor (R1) frequency so that the effective short-circuited (it can not " see " ). This can help increase the available gain.

Linear S-parameter analysis simulation setup with frequency-dependent stabilizationFigure 10.

Use inductor (L1) and capacitor (C1) to construct a low-pass filter. This prevents the resistance (R1) from working at higher RF frequencies or lower frequencies in order to achieve stability. For the gain, stability and S- parameter analysis of this solution , please refer to Figure 11 , Figure 12 and Figure 13 . As shown in the figure, the frequency-dependent stability network provides unconditional stability over the entire frequency range while reducing the impact on the maximum available gain at 3.5 GHz . Note that compared with the unstable device, the gain is only reduced by about 1dB at 3.5 GHz , while f max is basically the same as the unstable device (~10.4 GHz) . When looking at the S- parameter comparison results of the unstable device in Figure 12 , we found that, unlike the resistance-stabilized device, the S- parameter has not changed in the entire frequency range, and only changed when the frequency is lower (as needed). It can only be confirmed from Fig. 13 that for an unconditionally stable circuit, the stability circle will not overlap with the Smith chart no matter in the source plane or the load plane.

Frequency-dependent gain stability graphsFigure 11.

S-parameters for stabilized vs non-stabilized device Figure 1 2.

Stability circles for frequency-dependent stabilized device
Figure 1 3.

Main result
So, what are the main findings? As shown in the data below, when using frequency-dependent stability, both stability and gain are optimized

No stability -Themaximum usable gain at3.5 GHzis18.373 dB-Figure3-Above1.85 GHz, unconditional stability-Below1.85 GHz, conditional stability

 
Compliant with resistance stability -Themaximum usable gain at3.5 GHzis12.334 dB-Figure7-Unconditionally stableover the entirefrequency range of0.025to12 GHz-Themaximum usable gain is reduced by6 dB

 
Best result – Complies with frequency - dependent stability – The maximum usable gain at 3.5 GHz is 17.5 dB – Figure 11 – In the entire frequency range of 0.025 to 12 GHz , unconditionally remains stable – The maximum usable gain increases, which is 5.166 dB higher than resistance stability

TO SUM UP
Modeling can help solve common design problems such as stability before actually testing the application. Through accurate modeling and implementation of stability technology, we can perform matching and tuning while maintaining unconditional stability to optimize the performance of s- parameters.

Finally, please note that the stability network discussed here uses ideal lumped elements. In the actual microwave design, you need to include the microstrip interconnection and accurate parasitic models of all RLC components, whether you are doing MMIC design or a board-based hybrid design including lumped components.