Using the Available Power Gain to Design a Bilateral Low-Noise Amplifier

When designing a low-noise amplifier (LNA), we need to consider both gain and noise performance. As we learned in the preceding article, we can use an RF transistor’s constant noise figure (NF) contours to determine the appropriate source termination for a given noise performance. The constant NF contours are plotted in the Γ_S plane; to design an amplifier for noise and gain, we need to plot the transistor’s gain contours in the Γ_S plane as well.

We already covered how to accomplish this in the case of a unilateral device, where the gain of the input and output matching networks are independent of each other. This article explores the design of bilateral LNAs for a specified gain and noise figure, which can be a bit more complicated.

To design a bilateral amplifier for a specified gain other than the maximum, we can use either the operating power gain (G_P) or available power gain (G_A). However, constant G_P circles are plotted in the Γ_L plane, so they can’t be directly used to analyze the amplifier’s gain-noise trade-off. Constant G_A circles, on the other hand, are specified in the Γ_S plane. For that reason, we’ll use the available power gain concept to design our bilateral LNA.

 

Available Power Gain

The available power gain (G_A) is the ratio of the power available from the network (P_AVN) to the power available from the source (P_AVS):

$$G_A ~=~ \frac{P_{AVN}}{P_{AVS}}$$

Equation 1.

 

Figure 1 illustrates how the available power gain of a module is determined.

 

Figure 1. Determining the available power gain from the amplifier (a) and the maximum power available from the source (b).

 

In the G_A equation, P_AVN is normalized to the available power from the source, so our point of reference for this power gain measurement is the input voltage source (V_S). As the RF power travels from the source to the amplifier output, it’s affected by the impedance mismatch at the transistor’s input (Z_S and Z_IN). As a result, P_AVN depends on the source termination (Γ_S).

However, as Figure 1(a) illustrates, we connect the module output to a conjugate-matched load to measure P_AVN. That’s why P_AVN doesn’t depend on the load termination (Γ_L) that we’ll eventually connect to our amplifier. We can verify this by examining the available power gain equation below:

$$G_A ~=~ \frac{1~-~|\Gamma_S|^2}{|1~-~S_{11}\Gamma_S|^2} ~\times~ |S_{21}|^2 ~\times~ \frac{1}{1~-~|\Gamma_{OUT}|^2}$$

Equation 2.

 

where Γ_OUT, the reflection coefficient seen at the output of the transistor, is given by:

$$\Gamma_{OUT}~=~S_{22} ~+~ \frac{S_{12}S_{21} \Gamma_S}{1~-~S_{11}\Gamma_S}$$

Equation 3.

 

Note that G_A isn’t a function of Γ_L, only of Γ_S.

 

Using the Available Power Gain in Amplifier Design

Since Equation 2 is only a function of Γ_S and the S-parameters, we can use it to find the appropriate Γ_S for a given G_A. But does this completely solve our design problem? The actual gain exhibited by an amplifier is its transducer gain, which accounts for both Γ_S and Γ_L. The transducer gain is given by:

$$G_{T} ~=~ \frac{P_L}{P_{AVS}}$$

Equation 4.

 

If we want to design an amplifier using its available power gain, we need to find a relationship between G_A and G_T. Comparing Equations 1 and 4, we see that these two power gains become identical if we have P_L = P_AVN. Therefore, if we set Γ_L equal to Γ^*_OUT, we’ll have G_A = G_T.

To summarize, in order to design for a specific gain through the available power gain concept, we use Equation 2 to find the Γ_S value that produces the desired G_A, then provide a conjugate match at the output so that the actual gain G_T exhibited by the device becomes equal to the chosen G_A.

 

Design Equations for an Unconditionally Stable Device

Now that we understand the overall design procedure, let’s have a look at the required equations for an unconditionally stable device. The values of Γ_S that produce a given G_A lie on the constant G_A circle. The center (c_A) of this circle is given by:

$$c_A ~=~ \frac{g_A c_1^*}{1~+~g_A(|S_{11}|^{2}~-~|\Delta|^2)}$$

Equation 5.

 

and its radius (r_A), by:

$$r_A ~=~ \frac{\Big ( 1~-~2K|S_{12}S_{21}|g_A ~+~ |S_{12}S_{21}|^2g_A^2 \Big )^{\frac{1}{2}}}{|1~+~g_A(|S_{11}|^2~-~|\Delta|^2)|}$$

Equation 6.

The parameters g_A and c₁ in the above equations are defined, respectively, by:

$$g_A ~=~ \frac{G_A}{|S_{21}|^2}$$

Equation 7.

 

and:

$$c_1~=~S_{11}~-~\Delta S_{22}^*$$

Equation 8.

 

K is Rollet’s stability factor:

$$K ~=~ \frac{1~-~|S_{11}|^2 ~-~ |S_{22}|^2 ~+~ |\Delta|^2}{2|S_{12}S_{21}|}$$

Equation 9.

 

and Δ is the determinant of the S-parameters matrix:

$$\Delta ~=~ S_{11}S_{22}~-~S_{12}S_{21}$$

Equation 10.

 

For an unconditionally stable device, the maximum of G_A is given by:

$$G_{A, max} ~=~ \frac{|S_{21}|}{|S_{12}|} ~\times~ (K~-~\sqrt{K^{2}~-~1})$$

Equation 11.

 

As an aside, the equation for G_A,max is also the equation for G_P,max and G_T,max.

 

Example 1: Plotting G_A Circles for an RF Transistor

Now that we have the necessary equations, we’ll work through some examples. To start with, let’s plot some constant G_A circles for a transistor with the S-parameters in Table 1 at f = 1.4 GHz.

 

Table 1. S-parameters for an example transistor. Z₀ = 50 Ω.

f (GHz)	S11	S21	S12	S22
0.8	0.440 ∠ –157.6 degrees	4.725 ∠ 84.3 degrees	0.06 ∠ 55.4 degrees	0.339 ∠ –51.8 degrees
1.4	0.533 ∠ 176.6 degrees	2.800 ∠ 64.5 degrees	0.06 ∠ 58.4 degrees	0.604 ∠ –58.3 degrees
2.0	0.439 ∠ 159.6 degrees	2.057 ∠ 49.2 degrees	0.17 ∠ 58.1 degrees	0.294 ∠ –68.1 degrees

 

Properly, we should verify that this device is unconditionally stable at all three of the frequencies above. For the sake of brevity, though, I’m including a K-factor test only for f = 1.4 GHz.

$$K ~=~ \frac{1~-~|S_{11}|^2 ~-~ |S_{22}|^2 ~+~ |\Delta|^2}{2|S_{12}S_{21}|} ~=~ 1.12$$

Equation 12.

 

Since  K is greater than unity and |Δ| < 1, the device is unconditionally stable at our chosen frequency. We can therefore use the design procedure outlined above to find the appropriate source and load terminations. First, applying Equation 11, we find the maximum of G_A:

$$G_{A, max} ~=~ \frac{|S_{21}|}{|S_{12}|} ~\times~ (K-\sqrt{K^{2}-1})~=~28.73$$

Equation 13.

 

This converts to G_A,max = 14.58 dB.

I’ve arbitrarily chosen to plot the constant gain circles of G_A = 11 dB, 12 dB, 13 dB, and 14 dB. Using the data and equations above, we know the following:

|S₂₁|² = 7.84

K = 1.12

Δ = 0.16 ∠ 113.32 degrees

c₁ = 0.44 ∠ 177.66 degrees.

We can therefore apply Equations 5 and 6 to find the center and radius of the constant G_A circles. Table 2 provides a summary of the calculations.

Note that the conversion of G_A from dB to linear terms is the first set of calculations we have to do. We need to use Equation 7 to determine g_A before we can calculate either c_A or r_A, and Equation 7 requires G_A to be in linear terms.

 

Table 2. Summary of the calculations necessary to plot constant G_A circles for an example transistor.

GA (dB)	GA (linear terms)	gA	cA	rA
11 dB	12.59	1.61	0.50 ∠ –177.66 degrees	0.48
12 dB	15.85	2.00	0.58 ∠ –177.66 degrees	0.39
13 dB	19.95	2.55	0.67 ∠ –177.66 degrees	0.29
14 dB	25.12	3.20	0.77 ∠ –177.66 degrees	0.16

 

These constant G_A circles are plotted in Figure 2.

 

Figure 2. Constant G_A circles for an example transistor at G_A = 11 dB (pink circle), 12 dB (red circle), 13 dB (green circle), and 14 dB (purple circle).

 

The centers of the constant G_A circles are always on the line connecting c^*₁ to the origin of the Smith chart. To find the origin, see Equation 5.

 

Example 2: Designing an Amplifier With a Perfectly Matched Output for a Specific Gain

Let’s use the transistor from the previous example design an amplifier with a gain of 13 dB at f = 1.4 GHz, and ensure a perfect match on the output side of the transistor when we do so.

Since we want a perfect match on the output side, we’ll use the available power gain method. Any source termination residing on the green constant G_A circle in Figure 2 can be used to achieve a 13 dB available gain—I’ve arbitrarily chosen Γ_S = 0.38 ∠ –177.66 degrees, marked as point A in the above figure. With Γ_S determined, we can find the reflection coefficient seen at the output of the transistor:

$$\Gamma_{OUT}~=~S_{22} ~+~ \frac{S_{12}S_{21} \Gamma_S}{1~-~S_{11}\Gamma_S}~=~0.68 ~\angle~ -57.92~\text{degrees}$$

Equation 14.

 

Now we only need to provide a complex conjugate match on the output side to have G_T = G_A = 13 dB:

$$\Gamma_L ~=~ \Gamma_{OUT}^{*}~=~0.68 ~\angle~ -57.92~\text{degrees}$$

Equation 15.

 

This choice of Γ_L also ensures a perfect match at the output (VSWR = 1).

If we substitute Γ_S = 0.38 ∠ –177.66 degrees and Γ_L = 0.68 ∠ 57.92 degrees into the transducer gain equation, we obtain:

$$G_{T}~=~\frac{1~-~|\Gamma_{S}|^{2}}{|1~-~S_{11} \Gamma_{S}|^{2}}~\times~|S_{21}|^{2}~\times~\frac{1~-~| \Gamma_{L}|^{2}}{|1~-~\Gamma_{OUT} \Gamma_{L}|^{2}}~=~19.82$$

Equation 16.

 

Converted from linear terms, this becomes 12.97 dB. This is very close to the target value of G_T = 13 dB.

We can now use a Z Smith chart to design the input and output matching networks. To design the input matching network, we first locate Γ_S on the Smith chart and find its associated normalized admittance (y_S) through a 180 degree rotation along the constant |Γ_S| circle (Figure 3).

 

Figure 3. Smith chart for the design of an example RF amplifier's input matching network.

 

We interpret the Smith chart as a Y Smith chart from now on. We want a circuit that takes us from the center of the chart (the 50 Ω termination) to y_S. The intersection point of the constant |Γ_S| circle with the 1 + jb circle, marked as point A, has a susceptance of j0.84.

To create a susceptance of j0.84, we add a parallel open-circuited stub of length l₁ = 0.109λ to the 50 Ω termination. We then add a series line of length l₂ = 0.091λ to travel along the constant |Γ_S| circle to y_S. That takes care of the input matching section; the output matching section can be designed in a similar way, as we see in Figure 4.

 

Figure 4. Smith chart for the design of the example amplifier's output matching network.

 

We can see that the output matching network needs an open-circuited stub of length l₃ = 0.171λ and a series line of length l₄ = 0.236λ.

Figure 5 shows the AC schematic of the final design, including both input and output matching networks.

 

Figure 5. The final design of our example amplifier.

 

Figure 6 shows the simulated gain of the amplifier, which is very close to the desired value G_T = 13 dB.

 

Figure 6. Simulated gain of the amplifier we designed above. At 1400 MHz, gain is 12.96 dB.

 

Figure 7 shows the input and output reflection coefficients of the amplifier. We can see from this figure that using the available power gain approach produces amplifiers with well-matched output and mismatched input.

 

Figure 7. Input (blue) and output (red) reflection coefficients of the example amplifier over the 800 MHz to 2000 MHz frequency range.

 

Example 3: Designing a Bilateral Amplifier for a Specific Gain and Noise Performance

Assume that, at f = 1.4 GHz, the noise parameters of the transistor we’ve been investigating are:

F_min = 1.6 dB

Γ_opt = 0.62 ∠ 100 degrees

R_N = 20 Ω.

Using this transistor, let’s design an amplifier with a noise figure of 3 dB and the maximum gain that’s compatible with that noise figure.

First, we find the center and radius of the NF = 3 dB constant noise circle. Table 3 provides a summary of the required calculations. Note that we’re plugging the noise factor, not the noise figure, into our equations.

 

Table 3. Summarized calculations for the NF = 3 dB constant noise circle.

NF (Noise Figure)	F (Noise Factor)	N	cF	rF
3 dB	2	0.4	0.44 ∠ 100 degrees	0.46

 

Figure 8 plots the constant noise circle of NF = 3 dB alongside the constant G_A circles from Example 1.

 

Figure 8. Example 1’s constant G_A circles plotted alongside the 3 dB constant NF circle.

 

The NF = 3 dB noise circle intersects the G_A = 13 dB gain circle at point A, which corresponds to Γ_S = 0.46 ∠ 161.4 degrees. With Γ_S determined, we can now find the reflection coefficient seen at the output of the transistor:

$$\Gamma_{OUT}~=~S_{22} ~+~ \frac{S_{12}S_{21} \Gamma_S}{1~-~S_{11}\Gamma_S}~=~0.7 ~\angle~ -61.65~\text{degrees}$$

Equation 17.

 

Finally, to produce G_T = G_A = 13 dB, we provide a complex conjugate match on the output side:

$$\Gamma_L ~=~ \Gamma_{OUT}^{*}~=~0.7 ~\angle~ 61.65~\text{degrees}$$

Equation 18.

 

We’ll once again use a Z Smith chart to design the matching networks. Figures 9 and 10 show the design details for the input and output matching networks, respectively.

 

Figure 9. Smith chart for the design of a new example amplifier’s input matching network.

 

Figure 10. Smith chart for the design of the amplifier’s output matching network.

 

Figure 11 shows the final AC schematic.

 

Figure 11. AC schematic for a bilateral LNA intended to have 13 dB of gain and 3 dB of noise at f = 1.4 GHz.

 

A computer analysis of the above design gives a gain of 12.94 dB and noise figure of 3 dB at f = 1.4 GHz.

 

Wrapping Up

RF LNAs are a critical component of our increasingly connected world. I hope that this article, taken together with the previous one, has helped improve your understanding of low-noise amplifier design. If there is more about LNA design that you’d like me to cover, please consider leaving a note for me in the comments section.

 

Featured image used courtesy of Adobe Stock; all other images used courtesy of Steve Arar