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

## Learn how the available power gain concept can help us solve the problem of designing a bilateral RF amplifier for a specified gain and noise figure.

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*). However, constant

_{A}*G*circles are plotted in the Γ

_{P}_{L}plane, so they can’t be directly used to analyze the amplifier’s gain-noise trade-off. Constant

*G*circles, on the other hand, are specified in the Γ

_{A}_{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*) to the power available from the source (

_{AVN}*P*):

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

**Equation 1.**

**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).

**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*is normalized to the available power from the source, so our point of reference for this power gain measurement is the input voltage source (

_{AVN}*V*). As the RF power travels from the source to the amplifier output, it’s affected by the impedance mismatch at the transistor’s input (

_{S}*Z*and

_{S}*Z*). As a result,

_{IN}*P*depends on the source termination (Γ

_{AVN}_{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*doesn’t depend on the load termination (Γ

_{AVN}_{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.**

**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.**

**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.**

**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*. Comparing Equations 1 and 4, we see that these two power gains become identical if we have

_{T}*P*=

_{L}*P*. Therefore, if we set Γ

_{AVN}_{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*exhibited by the device becomes equal to the chosen

_{T}*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*circle. The center (

_{A}*c*) of this circle is given by:

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

**Equation 5.**

**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.**

**Equation 6.**

The parameters *g _{A}* and

*c*

_{1}in the above equations are defined, respectively, by:

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

**Equation 7.**

**Equation 7.**

and:

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

**Equation 8.**

**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.**

**Equation 9.**

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

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

**Equation 10.**

**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.**

**Equation 11.**

As an aside, the equation for *G _{A,max}* is also the equation for

*G*and

_{P,max}*G*.

_{T,max}

#### Example 1: Plotting *G*_{A} Circles for an RF Transistor

_{A}

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. Z_{0}* = 50 Ω.*

**Table 1.**S-parameters for an example transistor.

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.**

**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.**

**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:

|*S*_{21}|^{2} = 7.84

*K* = 1.12

Δ = 0.16 ∠ 113.32 degrees

*c*_{1} = 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*before we can calculate either

_{A}*c*or

_{A}*r*, and Equation 7 requires

_{A}*G*to be in linear terms.

_{A}

*Table 2.** Summary of the calculations necessary to plot constant *G_{A}* circles for an example transistor.*

*Table 2.*

G(dB)_{A} |
G(linear terms)_{A} |
g_{A} |
c_{A} |
r_{A} |

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).*

**Figure 2.**Constant

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.**

**Equation 14.**

Now we only need to provide a complex conjugate match on the output side to have *G _{T}* =

*G*= 13 dB:

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

**Equation 15.**

**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.**

**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.

**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

*j*0.84.

To create a susceptance of *j*0.84, we add a parallel open-circuited stub of length *l*_{1} = 0.109λ to the 50 Ω termination. We then add a series line of length *l*_{2} = 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.

**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 *l*_{3} = 0.171λ and a series line of length *l*_{4} = 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 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 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.

**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.*

**Table 3.**Summarized calculations for the

NF (Noise Figure) |
F (Noise Factor) |
N |
c_{F} |
r_{F} |

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.*

**Figure 8.**Example 1’s constant

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.**

**Equation 17.**

Finally, to produce *G _{T}* =

*G*= 13 dB, we provide a complex conjugate match on the output side:

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

**Equation 18.**

**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 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 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.*

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

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*

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