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Microwave Amplifiers
Design of Microwave Transistor Amplifiers Using S Parameters
Microwave amplifiers combine active elements with passive transmission line circuits to provide functions critical to microwave systems and instruments. The history of microwave amplifiers begins with electron devices using resonant or slow-wave structures to match wave velocity to electron beam velocity.
The design techniques used for BJT and FET amplifiers employ the full range of concepts we have developed in the study of microwave transmission lines, two-port networks and Smith chart presentation.
The development of S-parameter matrix concepts grew from the need to characterize active devices and amplifiers in a form that recognized the need for matched termination rather than short- or open-circuit termination. Much of the initial work was performed at the Hewlett-Packard Company in connection with the development of instruments to measure device and amplifier parameters.
We'll begin by considering microwave amplifiers that are
- Small signal so that superposition applies, and
- Built with microwave bipolar junction or field-effect transistors
References
The following books and notes are references for this material:
Pozar^1 , D. M., Microwave Engineering
Gonzalez 2 , G., Microwave Transistor Amplifiers
Vendelin, Pavio & Rohde^3 , Microwave Circuit Design Using Linear and Nonlinear Techniques
Review of Transmission Lines
(^1) Pozar, D., Microwave Engineering , 2nd Edition, J. Wiley, 1998, pg. 600- (^2) Gonzalez, G., Microwave Transistor Amplifiers, (^3) Vendelin, Pavio & Rohde, Microwave Circuit Design Using Linear and Nonlinear Techniques , J. Wiley, 1990
For the purpose of characterizing microwave amplifiers, key transmission line concepts are
Traveling waves in both directions, V+^ and V-
Characteristic impedance Zo and propagation constant jß
Reflection coefficient Γ=
ZL - Zo ZL + Zo for complex load ZL
- Standing waves resulting from Γ≠ 0
- Transformation of ZL through line of Zo and length ßl
- Description of Γ and Z on the Smith chart (polar graph of Γ)
Review of Scattering Matrix
- Normalization with respect to Zo of wave amplitudes:
a =
V+
Zo
and b =
V-
Zo
- Relationship of bi and ai: b (^) i = Γi ai
- Expressions for b 1 and b 2 at reference planes: b1 = S 11 a 1 + S 12 a 2 b2 = S 21 a 1 + S 22 a 2
- Definitions of S (^) ii:
S 11 =
b a 1 for a^2 = 0, i.e., input^ Γ^ for output terminated in Zo^.
S 21 =
b a 1 for a^2 = 0, i.e., forward transmission ratio with Zo^ load.
S 22 =
b a 2 for a^1 = 0, i.e., output^ Γ^ for input terminated in Zo^.
S 21 =
b a 2 for a^1 = 0, i.e., reverse transmission ratio with Zo^ source. lS 21 l^2 = Transducer power gain with Zo source and load.
- Definitions of ΓL, Γs, Γin and Γout:
ΓL =
ZL - Zo ZL + Zo , the reflection coefficient of the load
Γs =
Zs - Zo Zs + Zo , the reflection coefficient of the source
Γin =
Zin - Zo Zin + Zo = S^11 +
S 12 S 21 ΓL
1-S 22 ΓL , the input reflection coefficient
Γout =
Zout - Zo Zout + Zo = S^22 +
S 12 S 21 Γs 1-S 11 Γs , the output reflection coefficient
- Power Gain G, Available Gain GA, Transducer Gain GT :
1) G =
P L
P (^) in =^
1 - lΓinl^2 lS 21 l^2
1 - lΓLl^2 l1 - S 22 ΓLl^2
2) GA =
P (^) avout P (^) avs =^
1 - lΓsl^2 l1 - S 11 Γsl^2 lS 21 l^2
1 - lΓoutl^2
3) GT =
P L
P (^) avs =^
1 - lΓsl^2 l1 - ΓinΓsl^2 lS 21 l^2
1 - lΓLl^2 l1 - S 22 ΓLl^2
=
1 - lΓsl^2 l1 - S 11 Γsl^2 lS^21 l
2 1 - lΓLl^2 l1 - ΓoutΓLl^2
The expressions for Γin and Γout are
- Γin = S 11 +
S 12 S 21 ΓL
1-S 22 ΓL
- Γout = S 22 +
S 12 S 21 Γs 1-S 11 Γs
For a unilateral network, S 12 =0 and
- Γin = S 11 if S 12 =0 (unilateral network)
- Γout = S 22 if S 12 =0 (unilateral network)
The transducer gain GT can be expressed as the product of three gain contributions
GT =GsGo GL, where
Go = lS 21 l^2
Gs =
1 - lΓsl^2 l1 - ΓinΓsl^2 and
GL =
1 - lΓLl^2 l1 - S 22 ΓLl^2
Z (^) o
Z (^) o
ΓΓΓΓ s ΓΓΓΓ (^) in ΓΓΓΓ out L
Input Matching Circuit G (^) s
Output Matching Circuit G (^) L
Transistor [S] G (^) o
If the device is unilateral, or sufficiently so that S 12 is small enough to be ignored, the
unilateral transducer gain GTU is simplified because
GsU =
1 - lΓsl^2 l1 - S 11 Γsl^2 , where the subscript U indicates unilateral gain.
In practice, the difference between GT and GTU is often quite small, as it is desirable for devices to be unilateral if possible.
The components of GTU can also be expressed in decibel form, so that
GTU (dB) = Gs (dB) + Go (dB) + GL (dB).
We can maximize Gs and GL by setting Γs = S 11 * and ΓL = S 22 * so that
Gsmax =
1 - lS 11 l^2 and
GLmax =
1 - lS 22 l^2 , so that
GTUmax =
1 - lS 11 l^2 lS 21 l^2
1 - lS 22 l^2
Note that, if lS 11 l=1 or lS 22 l=1, GTUmax is infinite. This raises the question of stability,
which will be examined next.
Stability
In a two-port network, oscillations are possible if the magnitude of either the input or output reflection coefficient is greater than unity, which is equivalent to presenting a negative resistance at the port. This instability is characterized by
noise performance of an amplifier unless accomplished in connection with an analysis of the amplifier noise figure.
Constant Gain Circles
Because of the form of the unilateral transducer gain, values of the input reflection coefficient Γin that produce constant gain also lie on circles on the Smith chart. The derivation of the radius and center of these circles is found on pg. 621-626 of Pozar and pp. 102-105 of Gonzalez.
Noise in Amplifiers
The lower limit of amplifier signal capability is set by noise. Three sources of noise in transistor amplifiers are
Thermal noise due to random motion of charge carriers due to thermal agitation: Available noise power P (^) av=kTB.
Shot noise due to random flow of carriers across a junction, which produces a noise current of i (^) n^2 = 2qIdc B
Partition noise due to recombination in the junction, which
produces a noise current of i (^) p^2 =
2kT re ' αo^ (1 -^ αo^ ) B
In these expressions, kT=-174 dBm in 1 Hz bandwidth, B is the bandwidth, q is the charge of the electron, and the other parameters are elements of the transistor equivalent circuit.
The noise figure F of an amplifier is defined as the ratio of the total available noise power at the output of the amplifier to the available noise power at the output that would result only from the thermal noise in the source resistance. Thus F is a measure of the excess noise added by the amplifier. Amplifier noise can also be characterized by an equivalent noise temperature of the source resistance that would provide the same available noise power output. This equivalent noise temperature is given by
Te =(F - 1)T (^) o
Because of the interaction of the various noise sources and resistances of a microwave transistor, the noise figure of an amplifier generally varies as
F = Fmin +
R N
GS lYs^ - Yoptl
(^2) , where
Ys is the source admittance presented to the transistor Yopt is the optimum source admittance that results in minimum noise figure Fmin is the minimum noise figure
R (^) N is the equivalent noise resistance of the transistor, and Gs is the real part of Ys
If instead of admittance we use reflection coefficients, it will come as no surprise that we can find circles of constant noise figure on the Smith chart, as derived in Pozar on pg.
We can generally expect to be provided with Fmin, Γopt and R (^) N for a given device and frequency. These parameters can, of course, be derived also from direct measurement of noise figure under conditions of optimum source impedance. It is unusual for noise figure and gain circles to be concentric, as maximum gain conditions are not the same as minimum noise figure conditions.
The noise figure of cascaded amplifiers is given by the numerical (not dB) relationship
F = F 1 +
F 2 -
GA1 ,
where F 1 and GA1 are the noise figure and available gain of the first stage, and F 2 is the noise figure of the second stage. This applies to lossy stages and networks as well.
Power Amplifiers
The design of power amplifiers involves less emphasis on noise parameters, and more emphasis on linearity and intermodulation, as well as efficiency and thermal considerations. To design a power amplifier, one must use large-signal S-parameters and be aware of nonlinear effects.
Where careful design of the input matching network is required to realize the full capabilities of low-noise amplifiers, in power amplifiers more emphasis tends to be on optimizing the output matching network. There are, however, special problems associated with the very low input impedance that can be found in bipolar power devices, which require special treatment in the input matching network if wideband operation is to be achieved.
A key issue for multi-stage amplifiers is the ability to cascade individually designed stages without a requirement for retuning or redesign to account for the characteristics of the driving or following stages. In many cases, the use of balanced amplifiers permits the benefit of 3 dB coupler interstages, which direct reflected power to the isolated port rather than the driving stage. As we will see in later lectures, there are special problems of nonlinear oscillations arising from interaction between signal harmonics and modes of the output matching structure.
Bias Circuits and Bias Circuit Instabilities
Once the microwave amplifier is designed, it remains to provide the dc bias voltages and currents required for the active device. This is no simple problem, as the arrangements to introduce the biases can disturb the microwave circuit. Generally, high impedance microstrip traces can be used as decoupling inductors, but caution must be exercised not to create a low frequency oscillator circuit in the bias network.
A common cause of trouble is the use of an inductor with a large bypass capacitor, which can create a resonator in the MHz region that can support oscillation of the active element, which will have very high gain at lower frequencies.
Bias-circuit instabilities are a common source of problems in amplifiers and other active circuits. These generally result from the use of inductors and capacitors in the bias circuit without regard to resonances or situations where 180° phase shift can occur.
Two examples of circuit configurations that can promote bias oscillations at low frequencies are
- The use of an inductor with bypass capacitors on both sides to filter dc supply to earlier stages of an amplifier; this can have 180° phase shift at a video frequency where the active element has substantially more gain than at microwave frequencies.
- The use of an inductor with bypass capacitor to isolate dc supply input to the base or gate of the active element; this can form a resonator for video frequency oscillations.
Examples of bias circuits that are prone to parasitic oscillation are shown here:
+
In the circuit on the left, the potential for 180° phase shift across the inductor can be seen. The cure is to use inductors that are lossy, either because of resistive loading or by use of lossy ferrite cores to provide RF isolation. The circuit on the right exhibits a resonant circuit that can result in parasitic oscillation; in this case, the small current required for base or gate bias permits the use of resistance in series with the inductor. It is possible to use a wire-wound resistor to accomplish both functions at the same time.
It is generally helpful to sketch the low-frequency equivalent circuit to be sure that no instabilities can exist. Resistive or lossy elements are required to guarantee stability in the bias circuit.
In the past, emphasis was placed on achieving the minimum number of components for reasons of cost and reliability. With the introduction of integrated circuits, this concept has been superseded, and active feedback bias circuits are generally used to insure the stability of device operating conditions.
