Amplifier designers have so far learned about the use of the unilateral gain design approach as well as a method for achieving simultaneous conjugate matching of a stable transistor's input and output ports to its source and load. This month, the article series focuses on a method corresponding to the *operating gain* design approach in which particular load impedance is specified for an amplifier design.

For the 50-Ω loaded amplifier, for the unilateral amplifier designs, and for the simultaneous conjugate match design, the *transducer gain* (GT) definition was applied. For a network described by S-parameters, two other commonly used gain definitions are used (p. 213 of ref. 1): the *operating gain *(GP) and the *available gain* (GA), which are special cases derived from the transducer gain:

where:

and

and

The three gain definitions (Fig. 1) can be expressed in terms of the S-parameters and the reflection coefficients of the network, source, and load.^{1,2}

The transducer gain is provided by

or

The operating gain can be calculated by

The available gain can be determined by

where:

and

The operating gain design is used with a matched source and optional load. The design is exact, and S_{12} is not neglected. The operating gain procedure consists of selecting a Γ_{L} value from constant gain circles (to be presented shortly) and then finding the corresponding input match, Γ_{S}.

The operating gain method is particularly useful in the case of power amplifiers, for which a specific load impedance for the transistor is often required for maximum power output. The load, Z_{L}, is typically specified by the transistor manufacturer as that which was found empirically to yield the highest power output for the device at a given frequency. Other applications for this method arise in which the Z_{L} is predetermined, perhaps by the input impedance of a subsequent filter.

Unlike the unilateral gain and simultaneous conjugate match designs, the mathematics of the operating gain method do not require an unconditionally stable network. However, if the amplifier is not unconditionally stable, care must be taken that the chosen Z_{S} and Z_{L} values do not lie within or too near the impedances that cause potential instability. Furthermore, if the design is applied with a potentially unstable network, there will be no safeguard against the network oscillating when the designed Z_{S} and/or Z_{L} may be absent, as might occur when the intended source or load is disconnected. Therefore, while the operating gain method can be used with potentially unstable networks, good engineering practice suggests that one first make the network unconditionally stable if this can be done while satisfying the amplifier's performance requirements.

The operating gain can be rewritten as^{1 }

where:

and

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These relations yield circles in the Γ_{L} plane (the Smith Chart) having constant gain. The circles are described by

where the center of the circle C_{P} is located at

and the radius of the circle is

The procedure to design to a given transducer power gain, G_{T} = G_{P}. is:

- For a given G
_{P}, plot the center and radius of the gain circle on the Smith Chart. - Select a desired Γ
_{L}(or the corresponding Z_{L}). - For the particular Γ
_{L}, maximum power is obtained by matching the input according to

where:

Using the equations above, these steps can be performed manually. Alternatively, they can be accomplished using the network simulator. Constant gain circles, obtained using the Genesys3 software simulation suite from Eagleware Corp. (Norcross, GA) in increments of −1, −2, −3, −4, −5, and −6 dB below the optimum load, Γ_{M} (which gives the simultaneous conjugate match), as shown in Fig 2. These are calculated for the 2N6679A transistor alone (described in Part 1)—without any stabilizing circuit elements.

Notice that the load instability circle (the shaded circle) lies partially within the |Γ| ≤ 1 circle of the Smith chart. The network simulator does this to indicate that these are unstable load impedances to be avoided in the selection of Γ_{L}. Suppose a point on the −1-dB circle is selected corresponding to the normalized impedance z_{L} = 1+ j1.6. This means that we must load the transistor with an unnormalized impedance of 50 +j80 Ω. This choice is convenient, since the 50-Ω resistive part is already included in the matched termination. An inductor provides the reactance:

L_{1} = 80 Ω/(6.28 Ω/nH) = 12.74 nH

The resulting circuit is shown in Fig. 3.

Next, the input impedance, Z_{IN}, required for the − 1-dB gain must be determined, using the network simulator for the calculation (Table 1). The required source impedance is the complex conjugate of Z_{IN}, thus

Z_{S} = Z_{IN}*

Z_{S} = Z_{IN}* = 4.133 Ω + j1.448 Ω.

A quarter-wave impedance inverter^{4} can be used to obtain the real part. Its characteristic impedance, Z_{T}, is

Z_{T} = ^{0.5} = 14.38 Ω

The required reactive part of Z_{S} can be obtained with a series inductor, L_{2}, is

L_{2} = 1. 448 Ω/(6.28 Ω/nH) = 0.23 nH

The complete circuit and performance is shown in Fig. 4. The gain at 1 GHz is 22.4 dB, higher than that obtained when input and output were matched with the *stabilized* 2N6679A circuit. As expected, the input circuit is matched, |S_{11}| ≤ −20 dB, but the output is not matched,, and this mismatch results in a 23-percent power loss or about 1.1 dB. However, to obtain this additional gain would require selection of a load quite near those that cause potential instability.

This operating gain result might appear to be preferable to the simultaneous conjugate match and unilateral designs performed earlier. It has more gain at 1 GHz, and also has far more gain below 1 GHz, should that prove desirable. However, this design did not provide unconditional stability, as was provided by the earlier designs.

Rechecking the K and B_{1} values (Table 2), it is found that the amplifier that was designed is potentially unstable from 100 to 1500 MHz. Checking the instability circles (Fig. 5) confirms this fact. However, the amplifier will be stable if 50-Ω source and loads are used, since none of the input and output instability circles includes the origin of the Smith Chart.

For low-noise amplifiers, optimum performance generally consists of achieving the lowest noise figure possible for the highest possible gain. This requires specifying a particular source impedance for the transistor, using a method called the *available gain* design approach, which will be detailed next month in Part 6 of this amplifier design series.

REFERENCES

- Guillermo Gonzalez,
*Microwave Transistor Amplifiers, Analysis and Design*, Second Edition, Prentice-Hall, Upper Saddle River, NJ, 1984. An excellent engineering reference for transistor amplifier design. - Theodore Grosch,
*Small Signal Microwave Amplifier Design*, Noble Publishing Corp., Norcross, GA, 1999. The author presents a thorough derivation of all of the important formulas for transistor amplifier design and evaluation, based on the S parameters of the transistor and the circuit, which surrounds it. - Genesys 7 Reference Manual, Eagleware, Norcross, GA, 1986-2000. This is the reference manual for the CAD program used as a network simulator for the examples of this text. Note that early versions of this manual have an error in the equation for B1 on page 324. Use the corrected expression (10.3-5) given in this text.
- Joseph F. White,
*High Frequency Techniques, An Introduction to RF and Microwave Engineering*, Wiley, New York, 2004.