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

## 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.

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

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

- Available power gain (
*G*)._{A} - Transducer power gain (
*G*)._{T} - Operating power gain (
*G*), which is usually what’s meant by “power gain” without a modifier._{P}

This article is mostly concerned with the transducer power gain and operating power gain definitions. However, the concept of available power from a source (*P _{AVS}*) 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

*P _{AVS}* 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.

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

*P _{AVS} *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.**

**Equation 1.**

where *V _{S}* is the peak value of the voltage source and

*Z*is the reference characteristic impedance of the system.

_{0}Note that *P _{AVS}* 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 (*P _{L}*) to the power available from the source (

*P*):

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

**Equation 2.**

**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 (*V _{S}*). As the RF power travels from the source to the load, it’s affected by the impedance mismatch (between Γ

_{S}and Γ

_{IN}) 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 Γ

_{L}and Γ

_{OUT}).

In short, *G _{T}* accounts for both the input mismatch and the output mismatch. The

*G*equation below verifies this:

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

**Equation 3.**

**Equation 3.**

Γ_{IN} 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 4.**

Equation 3 shows that *G _{T}* depends on both Γ

_{S}and Γ

_{L}. Since it takes both source and load mismatch into account,

*G*can be considered a full characterization of the amplifier’s gain. That’s why we commonly use the transducer power gain in amplifier design.

_{T}

#### 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 *S _{12}* = 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.**

**Equation 5.**

Observe that the first fractional term is only related to the input parameters (Γ_{S} and S_{11}) whereas the last term is only dependent on the output parameters (Γ_{L} and *S _{22}*). 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 Γ

_{S}and Γ

_{L}. For example, if the desired overall gain is 12 dB, and |

*S*|

_{21}^{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.**

**Equation 6.**

where Γ_{OUT} 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.**

**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 Γ_{S} and Γ_{L} 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.**

**Equation 8.**

where:

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

**Equation 9.**

**Equation 9.**

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

**Equation 10.**

**Equation 10.**

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

**Equation 11.**

**Equation 11.**

Assume that, to produce a given overall gain of *G _{T}*, the

*G*and

_{S}*G*terms are supposed to be equal to arbitrary values

_{L}*G*and

_{1}*G*, respectively. The output term

_{2}*G*has the same form as the unilateral case. We then take the following steps:

_{L}- Plot the
*G*=_{L}*G*circles to find the appropriate Γ_{2}_{L}. - Once we have Γ
_{L}, calculate Γ_{IN}from Equation 4. - With Γ
_{IN}determined, plot constant*G*circles for a given value (_{S}*G*, in this case)._{1}

However, what if the maximum value of *G _{S}* is less than

*G*, which we initially allocated to this gain term? When we begin our design, we don’t know the range of gain values that

_{1}*G*can produce—we can only determine that after calculating Γ

_{S}_{IN}, which we can only do after choosing Γ

_{L}. Therefore, the values of

*G*might not be satisfactory for the desired

_{S}*G*.

_{T}This will require trying a new Γ_{L} 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 Γ_{IN} dependent on Γ_{L}.

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 *G _{L}* occurs at Γ

_{L}=

*S*, but this value won’t maximize

^{*}_{22}*G*. From Equation 6, the maximum of

_{T}*G*occurs under the simultaneous conjugate match condition (Γ

_{T}_{L}= 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 (*P _{L}*) divided by the power delivered to the input of the transistor (

*P*):

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

**Equation 12.**

**Equation 12.**

The difference between *G _{T}* and

*G*is in the denominator of the power gain equations.

_{P}*G*is defined with respect to the input voltage source. On the other hand, the reference power for

_{T}*G*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!

_{P}*G _{P}* 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.**

**Equation 13.**

Observe that Γ_{S} doesn’t appear in the *G _{P}* equation.

#### Using Operating Power Gain in Amplifier Design

Since Equation 13 is only a function of Γ_{L} and the S-parameters, it can easily be used to find the appropriate Γ_{L} for a given *G _{P}*. 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 Γ_{S} and Γ_{L}. We need to find a relationship between the operating power gain and *G _{T}*. Comparing Equation 2 and Equation 12, we observe that these two power gains become identical if we have

*P*= P

_{IN}_{AVS}. For these two power quantities to be equal, we only need to set Γ

_{S}= Γ

^{*}

_{IN}(see Figure 2). This leads to

*G*=

_{P}*G*.

_{T}To summarize, in order to design for a specific gain, we use Equation 13 to find the Γ_{L} 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 *G _{P}*.

#### 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_{L} that produce a given *G _{P}* lie on a circle known as a constant

*G*circle. The center (

_{P}*C*) of this circle is given by:

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

**Equation 14.**

**Equation 14.**

and the circle’s radius (*r _{P}*), 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.**

**Equation 15.**

where the parameters *g _{P}* and

*C*are defined by:

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

**Equation 16.**

**Equation 16.**

and:

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

**Equation 17.**

**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.**

**Equation 18.**

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

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

**Equation 19.**

**Equation 19.**

#### Minimum and Maximum Values of *G*_{P}

_{P}

When designing amplifiers, it’s important to know the range of values that *G _{P}* can take. The maximum of

*G*is given by:

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

**Equation 20.**

**Equation 20.**

Interestingly *G _{P,max}* is the same as

*G*, which we discussed in the previous article, “How to Design a Bilateral RF Amplifier for Maximum Gain.” The termination Γ

_{T,max}_{L,max}that produces this gain is also equal to the Γ

_{L}value that produces

*G*. With Γ

_{T,max}_{L,max}, the constant

*G*gain circle transforms to a single point. Γ

_{P}_{L,max }can therefore be found by substituting G

_{P,max}into Equation 14.

Also, *G _{P}* has a minimum of zero that occurs at |Γ

_{L}| = 1 (see Equation 13). When that’s the case, all of the output power is reflected from the load.

#### Example: Finding Γ_{S} and Γ_{L} for a desired gain

In the abovementioned article, we found the maximum transducer gain at *f *= 1.4 GHz for a transistor with *Z _{0}* = 50 Ω and the S-parameters in Table 1.

**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} |

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 *G _{T,max}* = 28.73, which translates to 14.58 dB. Now, for the same transistor, let’s find the appropriate Γ

_{S}and Γ

_{L}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.**

**Equation 21.**

Since the device is unconditionally stable, the equations presented in the previous sections are applicable. The desired gain (*G _{P}* = 11 dB) is equal to 12.59 in linear terms. With |

*S*|

_{21}^{2}= 7.84, we obtain

*g*= 1.61.

_{P}Next, we apply Equations 14 and 15 to find the center *C _{P}* and radius

*r*of the

_{P}*G*= 12.59 circle. Table 2 provides a summary of the calculations.

_{P}

**Table 2.** Summary of calculations for the constant gain circle.

**Table 2.**Summary of calculations for the constant gain circle.

g_{P} |
K |
Δ | C_{2} |
C_{P} |
r_{P} |

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 *G _{P}* = 12.59 is shown in Figure 3.

*G*is labeled with its value in decibels.

_{P}

**Figure 3.** Constant gain circle for G_{P}* = 12.59.*

**Figure 3.**Constant gain circle for

Depending on your design goals, you can choose Γ_{L} 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 Γ_{L} = 0.11 ∠ 90 degrees. With Γ_{L} determined, Equation 4 produces Γ_{IN} = 0.55 ∠ 177.87 degrees. To have *G _{T}* =

*G*, we should provide a conjugate match at the input:

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

**Equation 22.**

**Equation 22.**

This choice of Γ_{S} also ensures a perfect match at the input (*VSWR* = 1). If we substitute Γ_{L} = 0.11 ∠ 90 degrees and Γ_{S} = 0.55 ∠ –177.87 degrees into Equation 3, we obtain *G _{T}* = 12.43. This value is in linear terms—converted to dB, it becomes

*G*= 10.94 dB. This is very close to the desired value of 11 dB.

_{T}The matching networks can easily be determined using a *Z* Smith chart. For the input matching section, we 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. The point

*y*has a normalized admittance of approximately 3.4 +

_{S}*j*0.2, as shown in Figure 4.

**Figure 4.** Smith chart showing the constant |Γ_{S}|* circle and normalized admittance of an example transistor.*

**Figure 4.**Smith chart showing 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 (or 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

*j*1.33.

We add a parallel open-circuited stub of length *l _{1}* = 0.147λ to the 50 Ω termination to create a susceptance of

*j*1.33. We then add a series line of length

*l*= 0.075λ to travel along the constant |Γ

_{2}_{S}| circle to

*y*. The final input matching section is shown in Figure 6.

_{S}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 |Γ_{L}|* circle and load admittance of an example transistor.*

**Figure 5.**Smith chart showing the constant

We can see that the output matching network needs an open-circuited stub of length *l _{3}* = 0.032λ and a series line of length

*l*= 0.25λ. The AC schematic of the final design is shown in Figure 6.

_{4}

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

**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 *G _{T}* = 11 dB.

**Figure 7.** The amplifier’s simulated gain is 10.94 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.

**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 Γ_{L} 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*.

### Suggestions for Further Reading

It’s worth mentioning that the available power gain concept can also be used to design bilateral amplifiers for a specific gain. Also, in this article, we only provided the equations for the unconditionally stable case. Now that you’re familiar with the basic concepts, I recommend you take a look at G. Gonzalez’s well-known book “Microwave Transistor Amplifiers: Analysis and Design” to learn about the design of potentially unstable devices as well.

*Featured image used courtesy of Keysight; featured image background used courtesy of Adobe Stock; all other images used courtesy of Steve Arar*

0 Comments