# Learn About Designing Unilateral Low-Noise Amplifiers

## In this article, we learn about noise parameters and use Smith charts to design a unilateral low-noise amplifier (LNA) for a specified gain.

In receiver applications, the first amplifier in the signal chain has a dominant effect on the noise performance of the overall system. This amplifier should exhibit as low a noise figure as possible while providing an acceptably high power gain. The design procedure for this low-noise amplifier (LNA) should therefore account for both gain and noise performance.

In this article, we’ll learn about how to design a unilateral LNA based on these requirements. We’ll start off by exploring how the noise parameters of a two-port network are specified in RF applications, then work through the process of designing a unilateral amplifier that achieves both a specific gain and a specific level of noise. Finally, we’ll test our design using the RF design software introduced in the previous article of this series.

### Noise Parameters of a Two-Port Network

As my article on the noise figure metric discussed in detail, the output noise of a circuit depends greatly on its source impedance. Meanwhile, the noise factor (*F*) of a transistor connected to the source admittance *Y _{S}* =

*G*+

_{S}*jB*is given by the following equation:

_{S}$$F = F_{min}~+~\frac{R_N}{G_S}|Y_S~-~Y_{opt}|^2$$

*Equation 1.*

*Equation 1.*

where:

*F _{min}* is the minimum noise factor of the device

*R _{N}* is the equivalent noise resistance of the device

*Y _{opt}* is the optimum source admittance

*G _{S}* is the real part of the source admittance,

*Y*

_{S}.We can see from this equation how *F* changes with the source admittance (*Y _{S}*). Observe that for

*Y*=

_{S}*Y*, the noise factor reduces to its minimum value,

_{opt}*F*.

_{min}The quantities *F _{min}*,

*R*, and

_{N}*Y*are called the noise parameters of the transistor. We don’t calculate these—instead, they're either given by the manufacturer or obtained through measurement.

_{opt}*F*, which is sometimes given in dB as

_{min}*NF*, changes with the transistor’s bias point, temperature, and frequency of operation. The

_{min}*R*parameter is a sensitivity factor, showing how fast the noise factor increases as the source admittance moves away from

_{N}*Y*.

_{opt}At low frequencies *Y _{opt}* is real, but it becomes a complex value above 50 to 100 MHz for most active devices. For any given two-port network, we can find a value for

*Y*that minimizes the noise factor. Note that the S-parameters aren’t present in Equation 1. In fact, the S-parameters of a device don’t provide us with any information about its noise performance.

_{opt}*F*, as stated previously, is the noise factor. It’s expressed in linear terms. The noise *figure*, abbreviated as *NF*, is the noise factor converted to dB. The relationship between *F *and *NF *can therefore be expressed as follows:

$$NF~=~10 \log_{10}(F)$$

**Equation** **2.**

**Equation**

**2.**

In practice, determining the *NF* dependence on the source impedance requires specialized noise measurement equipment. This equipment uses stub tuners to apply a range of complex impedances to the device, and these measurements are then analyzed to produce the contours of constant *NF* on the Γ_{S} plane.

Figure 1 shows the constant *NF* contours for a hypothetical device. As we’ll shortly discuss in greater detail, these contours are circular in form.

**Figure 1.** Smith chart displaying NF* contours of a hypothetical device, demonstrating the impact of driving-point impedance on the noise figure. Image used courtesy of D. Boyd*

**Figure 1.**Smith chart displaying

Note that common noise figure analyzers and network analyzers are not capable of producing these *NF* contours.

### Alternative Form of the Noise Factor Equation

The *R _{N}* parameter introduced above can also be specified as a conductance term, \(G_{N}~=~\frac{1}{R_{N}}\). Also, instead of specifying the optimum admittance, it’s possible to solve the equation by specifying either the equivalent optimum source impedance (\(Z_{opt}~=~\frac{1}{Y_{opt}}\)) or its associated optimum source reflection coefficient (Γ

_{opt}). The parameters

*Y*and Γ

_{opt}_{opt}are related by the following equation:

$$Y_{opt}~=~\frac{1}{Z_0}~ \times~ \frac{1~-~\Gamma_{opt}}{1~+~\Gamma_{opt}}$$

**Equation 3.**

**Equation 3.**

Using the Γ_{opt} parameter, Equation 1 can also be expressed as:

$$F~=~F_{min}~ + ~\frac{4R_N}{Z_0} ~\times~ \frac{|\Gamma_S ~-~ \Gamma_{opt}|^2}{(1~-~|\Gamma_S|^2)\times|1~+~\Gamma_{opt}|^2}$$

**Equation 4.**

**Equation 4.**

Note that the amplifier’s load reflection coefficient (Γ_{L}) doesn’t appear in Equation 4. We can see from this that the output matching doesn’t have any effect on the noise factor. However, a matched output can provide more gain and reduce the effect of the noise of the following stages.

There’s generally a trade-off between the gain and noise performance of an amplifier—the minimum noise can’t be achieved at the maximum gain.

### Drawing Constant *NF* Circles

In order to draw the constant *NF* circles for a given noise factor (*F*), we first find the noise figure parameter (*N*). This is given by:

$$N~=~\frac{F~-~F_{min}}{4R_N/Z_0}|1~+~\Gamma_{opt}|^2$$

**Equation 5.**

**Equation 5.**

The center (*c _{F}*) of the constant

*NF*circle is given by:

$$c_F ~=~\frac{\Gamma_{opt}}{N~+~1}$$

**Equation 6.**

**Equation 6.**

and its radius ( *r _{F}*), by:

$$r_F ~=~\frac{\sqrt{N(N~+~1~-~ |\Gamma_{opt}|^2})}{N~+~1}$$

**Equation 7.**

**Equation 7.**

To cement these concepts, let’s work through an example.

#### Example 1: Plotting Constant *NF* Circles

Assume that *Z*_{0} = 50 Ω and *f *= 1.4 GHz for a transistor with the following S-parameters:

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

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

f (GHz) |
S_{11} |
S_{21} |
S_{12} |
S_{22} |

1.4 | 0.533 ∠ 176.6 degrees | 2.8 ∠ 64.5 degrees | 0.02 ∠ 58.4 degrees | 0.604 ∠ –58.3 degrees |

The noise parameters of the device are:

*NF _{min}* = 1.6 dB

Γ_{opt} = 0.5 ∠ 130 degrees

*R _{N}* = 20 Ω.

Let’s plot constant *NF* circles for this transistor at *NF* = 2 dB, 2.5, dB, and 3 dB. Table 2 summarizes the required calculations. Note that our equations use *F*, not *NF*, so we can’t plug the noise figure values into the equations directly. Instead, we have to convert them from decibel measurements to the linear terms in which the noise factor is expressed.

**Table 2.** These calculation results allow us to plot constant NF* circles for our example transistor.*

**Table 2.**These calculation results allow us to plot constant

NF |
F |
N |
c_{F} |
r_{F} |

2.0 dB | 1.58 | 0.05 | 0.47 ∠ 130 degrees | 0.20 |

2.5 dB | 1.78 | 0.13 | 0.44 ∠ 130 degrees | 0.30 |

3.0 dB | 2.00 | 0.21 | 0.41 ∠ 130 degrees | 0.37 |

These constant *NF* circles are plotted in Figure 2.

**Figure 2.** Constant NF* circles for an example transistor. Image used courtesy of Steve Arar*

**Figure 2.**Constant

Note that the centers of the constant noise circles lie on the line drawn from the center of the Smith chart to the point Γ_{opt} (see Equation 6). At Γ_{opt}, where we obtain *NF _{min}* = 1.6 dB, the noise circle transforms to a single point. As we increase the noise figure, the center of the circle moves closer to the origin, and its radius becomes larger.

### Designing Unilateral RF Amplifiers for Both Gain and Noise

The constant *NF* circles are plotted in the Γ_{S} plane and can be used to find the appropriate source termination for a given noise figure. To account for both noise and gain, we need to plot the gain contours in the Γ_{S} plane as well. This is straightforward in the case of a unilateral device, where the gain of the input and output matching sections are independent of each other. We’ll save the design of bilateral LNAs for the next article.

#### Example 2: Designing a Unilateral LNA for a Specific Gain and Noise Performance

Using the transistor from the previous example, let’s design an amplifier that has a 2.5 dB noise figure with the maximum possible gain.

The transistor has a small *S*_{12}, suggesting that it might be considered unilateral. Applying the unilateral figure of merit (*U*), we obtain:

$$U~=~\frac{|S_{12}||S_{21}||S_{11}||S_{22}|}{(1~-~|S_{11}|^2)(1~-~|S_{22}|^2)}~=~0.04$$

**Equation 8.**

**Equation 8.**

Since *U* is less than 0.1, we immediately know that the error of the unilateral method is less than ±1 dB. The unilateral approach can therefore be applied. We can also calculate the exact value of the error bound for the unilateral approximation. This works out to be:

$$-0.34~\text{dB}~ < ~G_T ~-~ G_{TU} ~<~0.35~\text{dB}$$

**Equation 9.**

**Equation 9.**

This means that we should expect less than about ±0.35 dB error in the actual gain of our final design.

Next, we determine *G _{S,max}*, the maximum possible gain for the input matching section of our unilateral device:

$$G_{S, max}~=~\frac{1}{1~-~|S_{11}|^2}~=~1.4$$

**Equation 10.**

**Equation 10.**

which converts to 1.46 dB.

This allows us to choose appropriate values for our constant gain circles. In this example, I arbitrarily chose to plot the *G _{S}* = 0.5, 1, 1.28, and 1.4 dB circles. The centers and radii of these constant G

_{S}circles are given in Table 3.

**Table 3.** Centers and radii of constant G_{S}* circles.*

**Table 3.**Centers and radii of constant

Gain | Normalized Gain | Center | Radius |

G = 0.50 dB_{S} |
g = 0.80_{S} |
c_{S}_{1} = 0.45 ∠ –176.6 degrees |
r_{S}_{1} = 0.34 |

G = 1.00 dB_{S} |
g = 0.90_{S} |
c_{S}_{2} = 0.49 ∠ –176.6 degrees |
r = 0.23_{S2} |

G = 1.28 dB_{S} |
g = 0.96_{S} |
c_{S}_{3} = 0.52 ∠ –176.6 degrees |
r_{S}_{3} = 0.15 |

G = 1.40 dB_{S} |
g = 0.99_{S} |
c_{S}_{4} = 0.53 ∠ –176.6 degrees |
r_{S}_{4} = 0.07 |

Figure 3 plots these circles and the *NF* = 2.5 dB circle in the Γ_{S} plane.

**Figure 3.** Constant G_{S}* and *NF* circles in the *Γ_{S}* plane. Image used courtesy of Steve Arar*

**Figure 3.**Constant

The *G _{S}* = 1.28 dB gain circle only intersects the noise circle of

*NF*= 2.5 dB at Γ

_{S}= 0.45 ∠ 169.17 degrees. Any higher value of

*G*will move us away from Γ

_{S}_{opt}, resulting in a larger noise figure.

For the output section, we choose a conjugate match to maximize the gain. This results in:

$$\Gamma_L ~=~ S_{22}^* ~=~ 0.604~ \angle ~58.3~\text{degrees}$$

**Equation 11.**

**Equation 11.**

and:

$$G_L ~=~\frac{1}{1~-~|S_{22}|^2}~=~1.57$$

**Equation 12.**

**Equation 12.**

which converts to 1.96 dB.

The total gain is calculated as:

$$G_{TU}~=~G_{S}~+~G_0~+~G_{L}~=~ 1.28~\text{dB}~+~8.94~\text{dB}~+~1.96~\text{dB}~=~12.18~\text{dB}$$

**Equation 13.**

**Equation 13.**

*G*_{0} =|*S*_{21}|^{2} in the above equation. This is the basic *Z*_{0}-based transducer power gain of the transistor.

Next, we use a *Z* Smith chart to design the input and output matching networks. For the input matching section, we locate Γ_{S} on the Smith chart in Figure 4 and find its associated normalized admittance (*y _{S}*) through a 180 degree rotation along the constant |Γ

_{S}| circle.

**Figure 4.** The constant |Γ_{S}|* circle. Important points on the circle are marked in blue. Image used courtesy of Steve Arar*

**Figure 4.**The constant

From now on, we interpret the Smith chart as a *Y* Smith chart. We want a circuit that takes us from the center of the chart, located at the 50 Ω termination, to *y _{S}*. The intersection point of the constant |Γ

_{S}| circle with the 1 +

*jb*circle is marked as point

*A*, which has a susceptance of approximately

*j*1.

When designing the input matching section of the two-port network, we add a parallel open-circuited stub of length *l*_{1} = 0.125λ to the 50 Ω termination to create a susceptance of *j*1. We then add a series line of length *l*_{2} = 0.103λ to travel along the constant |Γ_{S}| circle to *y _{S}*.

The output matching section can be designed in a similar way. As Figure 5 shows, the output matching network needs an open-circuited stub of length *l*_{3} = 0.157λ and a series line of length *l*_{4} = 0.243λ.

**Figure 5.** Constant |Γ_{L}|* circle. Note the values of l*_{3}* and l*_{4}*. Image used courtesy of Steve Arar*

**Figure 5.**Constant

Figure 6 shows the AC schematic of the final amplifier design.

**Figure 6.** Final design of the example LNA. Image used courtesy of Steve Arar

**Figure 6.**Final design of the example LNA. Image used courtesy of Steve Arar

Now we can use design software to verify the performance of the circuit.

### Adding Noise Parameters to a Touchstone File

As we learned in the recent article about RF amplifier stabilization techniques, the Touchstone (.s2p) file format is commonly used in RF design software to specify the S-parameters of a two-port network. Table 4 shows the .s2p file for the S-parameters of the amplifier in Figure 6. The noise parameters are also included at the end of the file, though this is optional.

**Table 4. **S-parameter and noise parameters of the amplifier in Figure 6, saved as a Touchstone file.

**Table 4.**S-parameter and noise parameters of the amplifier in Figure 6, saved as a Touchstone file.

Recall that the options line, which starts with the # mark, contains the header information. This header information specifies the frequency unit and the data format for the S-parameters. The term “R 50” in the options line indicates that the load termination resistor for the S-parameters is 50 Ω. The lines that start with the ! symbol are comment lines.

As you can see, there isn’t a separate options line for noise parameters. In order for the simulator to be able to distinguish where the S-parameters data ends and the noise data begins, the first frequency of the noise parameters must be less than or equal to the highest frequency of the S-parameters.

The data format for the noise information is as follows:

- The first column specifies the frequency (1400 MHz).
- The second column gives the minimum noise figure in dB (1.6 dB).
- The next two columns give the magnitude and phase of the optimum reflection coefficient (Γ
_{opt}= 0.5 ∠ 130 degrees). - The last column is the effective noise resistance (
*R*= 20 Ω) normalized to the system impedance we defined in the options line._{N}

Linking the above .s2p file to a s2p component in Pathwave ADS, we can analyze the gain and noise performance of the system. The Pathwave ADS schematic we produce is shown in Figure 7.

**Figure 7.** Pathwave ADS schematic of an example amplifier. Image used courtesy of Steve Arar

**Figure 7.**Pathwave ADS schematic of an example amplifier. Image used courtesy of Steve Arar

Note that the simulation temperature is set to 16.85 °C, ensuring that the noise figure measurement is consistent with the IEEE definition of the noise figure. The computer analysis shows that the circuit we’ve designed has a gain of 12.466 dB and noise figure of 2.522 dB. These numbers are acceptably close to our design specifications.

### Wrapping Up

The latter part of this article focused on working through examples. Here’s a brief summary of the concepts introduced earlier on, should you want to review them:

- The noise performance of a two-port network with given operating conditions can be fully characterized by its noise parameters:
*F*(or_{min}*NF*), Γ_{min}_{opt,}and R_{N}. - The constant
*NF*circles are plotted in the Γ_{S}plane and can be used to find the appropriate source termination for a given noise figure. - To account for both noise and gain, we need to plot the gain contours in the Γ
_{S}plane as well. This is straightforward if the device is unilateral.

Note that it’s less straightforward to plot gain contours on the Γ_{S} plane if the device is bilateral. We previously used operating power gain (*G _{P}*) circles to design a bilateral amplifier for a specific gain, but

*G*circles are in the Γ

_{P}_{L}plane and don’t directly specify the usable source terminations.

Fortunately, there’s a method based on the available power gain (*G _{A}*) concept that allows us to plot the gain contours of a bilateral device in the Γ

_{S}plane. We’ll discuss how to use constant

*G*circles to design bilateral amplifiers for both gain and noise performance in the next article.

_{A}

*Featured image used courtesy of Adobe Stock*

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