Technical Article

# Using the Operating Power Gain to Design a Bilateral RF Amplifier

December 03, 2023 by Dr. Steve Arar

## For newbie RF designers, the field's variety of power gain definitions can cause confusion. This article explains how, why, and when to use the operating power gain definition in RF amplifier design.

The two previous articles in this series covered, respectively, how to design unilateral amplifiers for a specified gain and how to design bilateral amplifiers for maximum gain. In both cases, we used the transducer power gain definition in our calculations. However, this power gain definition isn’t very helpful when designing a bilateral amplifier for a gain other than the maximum. For this design problem, the operating power gain provides a more convenient solution.

In this article, we’ll explore the use of the operating power gain definition in the design of a bilateral RF amplifier—not just how to use it, but also why it’s the best choice for this type of design problem. To help us understand this, we’ll start with a brief overview of key concepts, then move to a more detailed exploration of the transducer and operating power gain definitions. After providing the required design equations, we’ll take a look at an example to clarify the concepts we’ve discussed.

The equations might look a little intimidating—keep in mind that our main goal is to gain insight into the equations so that we can use them confidently.

### A Variety of Power Gain Definitions

Consider the basic single-stage RF amplifier in Figure 1.

##### Figure 1. Basic single-stage RF amplifier.

The three common types of power gain we might use for this circuit are:

• Available power gain (GA).
• Transducer power gain (GT).
• Operating power gain (GP), which is usually what’s meant by “power gain” without a modifier.

This article is mostly concerned with the transducer power gain and operating power gain definitions. However, the concept of available power from a source (PAVS) is important when discussing RF power gain definitions, so we’ll take a moment to review available power gain before we move on.

#### Available Power From a Source

PAVS refers to the maximum power that a source can provide. The maximum power transfer occurs when the load is conjugately matched to the source impedance (Figure 2).

##### Figure 2. Conditions for maximum power transfer.

PAVS is given by the following equation:

$$P_{AVS}~=~\frac{|V_S|^2}{8Z_0}~\times~ \frac{|1~-~\Gamma_S|^2}{1~-~|\Gamma_S|^2}$$

##### Equation 1.

where VS is the peak value of the voltage source and Z0 is the reference characteristic impedance of the system.

Note that PAVS is a function only of the source parameters. It can therefore be used to characterize the input voltage source.

### Transducer Power Gain

For the circuit in Figure 1, the transducer power gain is defined as the ratio of the power delivered to the load (PL) to the power available from the source (PAVS):

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

##### Equation 2.

Note that the load power is normalized to the available power from the source. Because of this, our point of reference for the power gain measurement is the input voltage source (VS). As the RF power travels from the source to the load, it’s affected by the impedance mismatch (between ΓS and ΓIN) at the input of the transistor, then by the gain of the transistor itself, and finally by the impedance mismatch on the output side of the transistor (between ΓL and ΓOUT).

In short, GT accounts for both the input mismatch and the output mismatch. The GT equation below verifies this:

$$G_{T} ~=~ \frac{1~-~|\Gamma_S|^2}{|1~-~\Gamma_S \Gamma_{IN}|^2} ~\times~ |S_{21}|^2 ~\times~ \frac{1~-~|\Gamma_L|^2}{|1~-~S_{22} \Gamma_{L}|^2}$$

##### Equation 3.

ΓIN is the reflection coefficient seen at the input of the transistor, and is given by:

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

##### Equation 4.

Equation 3 shows that GT depends on both ΓS and ΓL. Since it takes both source and load mismatch into account, GT can be considered a full characterization of the amplifier’s gain. That’s why we commonly use the transducer power gain in amplifier design.

#### Using the Transducer Gain in Amplifier Design

The design of unilateral amplifiers is one type of design problem to which the transducer gain lends itself. In this case, we have S12 = 0, and Equation 3 simplifies to the product of three independent gain terms:

$$G_{T}~=~ \frac{1~-~|\Gamma_{S} |^{2}}{| 1~-~S_{11}\Gamma_{S}|^{2}}~\times~|S_{21}|^{2}~\times~ \frac{1~-~|\Gamma_{L}|^{2} }{|1~-~S_{22} \Gamma_{L}|^{2}}$$

##### Equation 5.

Observe that the first fractional term is only related to the input parameters (ΓS and S11) whereas the last term is only dependent on the output parameters (ΓL and S22). Since the gain terms are independent of each other, we can easily split up the overall desired gain between the gain terms and find the appropriate ΓS and ΓL. For example, if the desired overall gain is 12 dB, and |S21|2 = 9 dB, we might decide to allocate 2 dB to the input gain term and 1 dB to the output term.

Another design problem that can be solved through the transducer gain is the design of bilateral amplifiers for maximum gain. In this case, we need to provide a simultaneous conjugate match condition:

$$\Gamma_{S}~=~ \Gamma_{IN}^{*}~~\text{ and }~~\Gamma_{L}~=~\Gamma_{OUT}^{*}$$

##### Equation 6.

where ΓOUT is the reflection coefficient seen at the output of the transistor:

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

##### Equation 7.

The two constraints defined in Equation 6 are coupled, so the two equations need to be solved simultaneously. However, the equations give us the exact value of ΓS and ΓL for maximum gain.

#### When the Transducer Gain isn’t a Good Choice

But what if we’re designing a bilateral amplifier for a specific gain that isn’t the maximum? Can we split up the overall desired gain between different gain terms in Equation 3, like we did in the unilateral approach? Let’s explore this idea a little bit.

First, we’ll rewrite Equation 3 as:

$$G_{T} ~=~G_S ~\times~ G_0 ~\times~ G_L$$

##### Equation 8.

where:

$$G_S ~=~ \frac{1~-~|\Gamma_S|^2}{|1~-~ \Gamma_{IN} \Gamma_S|^2}$$

##### Equation 9.

$$G_0 ~=~ |S_{21}|^2$$

##### Equation 10.

$$G_L~=~ \frac{1~-~|\Gamma_L|^2}{|1~-~S_{22} \Gamma_{L}|^2}$$

##### Equation 11.

Assume that, to produce a given overall gain of GT, the GS and GL terms are supposed to be equal to arbitrary values G1 and G2, respectively. The output term GL has the same form as the unilateral case. We then take the following steps:

1. Plot the GL = G2 circles to find the appropriate ΓL.
2. Once we have ΓL, calculate ΓIN from Equation 4.
3. With ΓIN determined, plot constant GS circles for a given value (G1, in this case).

However, what if the maximum value of GS is less than G1, which we initially allocated to this gain term? When we begin our design, we don’t know the range of gain values that GS can produce—we can only determine that after calculating ΓIN, which we can only do after choosing ΓL. Therefore, the values of GS might not be satisfactory for the desired GT.

This will require trying a new ΓL and repeating the whole process until we find the appropriate terminations. This procedure is not recommended in practice. Note that the root cause of this difficulty is the bilateral behavior of the transistor that makes ΓIN dependent on ΓL.

The above discussion also shows that if we treat the gain terms of a bilateral amplifier independently, we cannot design for maximum gain. As an independent gain term, the maximum of GL occurs at ΓL = S*22, but this value won’t maximize GT. From Equation 6, the maximum of GT occurs under the simultaneous conjugate match condition (ΓL = Γ*OUT). To address these issues, we can use the operating power gain concept.

### Operating Power Gain

Referring back to Figure 1, the operating power gain is defined as the ratio of the power delivered to the load (PL) divided by the power delivered to the input of the transistor (PIN):

$$G_P ~=~ \frac{P_L}{P_{IN}}$$

##### Equation 12.

The difference between GT and GP is in the denominator of the power gain equations. GT is defined with respect to the input voltage source. On the other hand, the reference power for GP is the power that enters the transistor. The input voltage source and the input matching network obviously affects the amount of power that enters the transistor. However, since our reference point is the transistor’s input, we only need to know the amount of power that gets into the transistor, not how it gets there!

GP is given by the following equation:

$$G_{P} ~=~ \frac{1}{(1~-~|\Gamma_{IN}|^2)}~\times~|S_{21}|^2~\times~ \frac{1~-~|\Gamma_L|^2}{|1~-~S_{22} \Gamma_{L}|^2}$$

##### Equation 13.

Observe that ΓS doesn’t appear in the GP equation.

#### Using Operating Power Gain in Amplifier Design

Since Equation 13 is only a function of ΓL and the S-parameters, it can easily be used to find the appropriate ΓL for a given GP. But does this completely solve the problem of designing an amplifier for a specific gain?

The actual gain that an amplifier exhibits is its transducer gain, which accounts for both ΓS and ΓL. We need to find a relationship between the operating power gain and GT. Comparing Equation 2 and Equation 12, we observe that these two power gains become identical if we have PIN = PAVS. For these two power quantities to be equal, we only need to set ΓS = Γ*IN (see Figure 2). This leads to GP = GT.

To summarize, in order to design for a specific gain, we use Equation 13 to find the ΓL value that produces the desired operating power gain, then provide a conjugate match at the input so that the actual gain exhibited by the device becomes equal to the chosen GP.

#### 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 ΓL that produce a given GP lie on a circle known as a constant GP circle. The center (CP) of this circle is given by:

$$C_P ~=~ \frac{g_P C_2^*}{1~+~g_P(|S_{22}|^2~-~|\Delta|^2)}$$

##### Equation 14.

and the circle’s radius (rP), by:

$$r_P ~=~ \frac{\Big ( 1~-~2K|S_{12}S_{21}|g_P ~+~ |S_{12}S_{21}|^2g_P^2 \Big )^{\frac{1}{2}}}{|1~+~g_P(|S_{22}|^2~-~|\Delta|^2)|}$$

##### Equation 15.

where the parameters gP and C2 are defined by:

$$g_P ~=~ \frac{G_P}{|S_{21}|^2}$$

##### Equation 16.

and:

$$C_2~=~S_{22}~-~\Delta S_{11}^*$$

##### Equation 17.

K is Rollet’s stability factor:

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

##### Equation 18.

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

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

#### Minimum and Maximum Values of GP

When designing amplifiers, it’s important to know the range of values that GP can take. The maximum of GP is given by:

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

##### Equation 20.

Interestingly GP,max is the same as GT,max, which we discussed in the previous article, “How to Design a Bilateral RF Amplifier for Maximum Gain.” The termination ΓL,max that produces this gain is also equal to the ΓL value that produces GT,max. With ΓL,max, the constant GP gain circle transforms to a single point. ΓL,max can therefore be found by substituting GP,max into Equation 14.

Also, GP has a minimum of zero that occurs at |ΓL| = 1 (see Equation 13). When that’s the case, all of the output power is reflected from the load.

#### Example: Finding ΓS and ΓL for a desired gain

In the abovementioned article, we found the maximum transducer gain at f = 1.4 GHz for a transistor with Z0 = 50 Ω and the S-parameters in Table 1.

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

For that frequency, we found GT,max = 28.73, which translates to 14.58 dB. Now, for the same transistor, let’s find the appropriate ΓS and ΓL values for a gain of 11 dB at f = 1.4 GHz. We’ll need to ensure a perfect match on the transistor’s input side.

We can verify that the transistor is unconditionally stable at the three frequency points listed in the table. For example, at f = 1.4 GHz, |Δ| < 1 and K greater than unity:

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

##### Equation 21.

Since the device is unconditionally stable, the equations presented in the previous sections are applicable. The desired gain (GP = 11 dB) is equal to 12.59 in linear terms. With |S21|2 = 7.84, we obtain gP = 1.61.

Next, we apply Equations 14 and 15 to find the center CP and radius rP of the GP = 12.59 circle. Table 2 provides a summary of the calculations.

##### Table 2. Summary of calculations for the constant gain circle.
 gP K Δ C2 CP rP 1.61 1.12 0.16 ∠ 113.32 degrees 0.52 ∠ –57.51 degrees 0.54 ∠ 57.51 degrees 0.44

The constant gain circle of GP = 12.59 is shown in Figure 3. GP is labeled with its value in decibels.

##### Figure 3. Constant gain circle for GP = 12.59.

Depending on your design goals, you can choose ΓL to be anywhere on the circle. In the above figure, I arbitrarily chose it to be at the intersection of the constant gain circle with the constant-resistance circle of r = 1.

Reading from the Smith chart, we obtain ΓL = 0.11 ∠ 90 degrees. With ΓL determined, Equation 4 produces ΓIN = 0.55 ∠ 177.87 degrees. To have GT = GP, we should provide a conjugate match at the input:

$$\Gamma_S~=~\Gamma_{IN}^{*}~=~0.55~ \angle~ -177.87~\text{degrees}$$

##### Equation 22.

This choice of ΓS also ensures a perfect match at the input (VSWR = 1). If we substitute ΓL = 0.11 ∠ 90 degrees and ΓS = 0.55 ∠ –177.87 degrees into Equation 3, we obtain GT = 12.43. This value is in linear terms—converted to dB, it becomes GT = 10.94 dB. This is very close to the desired value of 11 dB.

The matching networks can easily be determined using a Z Smith chart. For the input matching section, we locate ΓS on the Smith chart and find its associated normalized admittance (yS) through a 180 degree rotation along the constant |ΓS| circle. The point yS has a normalized admittance of approximately 3.4 + j0.2, as shown in Figure 4.

##### Figure 4. Smith chart showing the constant |ΓS| circle and normalized admittance of an example transistor.

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 (or the 50 Ω termination) to yS. The intersection point of the constant |ΓS| circle with the 1 + jb circle is marked as point A, which has a susceptance of j1.33.

We add a parallel open-circuited stub of length l1 = 0.147λ to the 50 Ω termination to create a susceptance of j1.33. We then add a series line of length l2 = 0.075λ to travel along the constant |ΓS| circle to yS. The final input matching section is shown in Figure 6.

The output matching section can be designed in a similar way, as the Smith chart in Figure 5 illustrates.

##### Figure 5. Smith chart showing the constant |ΓL| circle and load admittance of an example transistor.

We can see that the output matching network needs an open-circuited stub of length l3 = 0.032λ and a series line of length l4 = 0.25λ. The AC schematic of the final design is shown in Figure 6.

##### Figure 6. Schematic of the amplifier’s final design.

Figure 7 plots the simulated gain of the amplifier, which is very close to the desired value of GT = 11 dB.

##### Figure 7. The amplifier’s simulated gain is 10.94 dB.

The amplifier’s input reflection coefficient is shown in Figure 8.

##### Figure 8. Input reflection coefficient for the example amplifier.

We can see that the input is well matched to the 50 Ω source impedance.

Though the design procedure above produces an input VSWR of unity, the output VSWR might be unacceptably high. If that’s the case, we can try other values of ΓL to see if we can achieve a better output VSWR. We can also introduce a specific mismatch at the input to improve the output VSWR.