RF Amplifier Design

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RF Amplifier Design [Copy link]

RF Amplifier Design
Stability analysis is an integral part of RF design. Learn the basics of stability analysis, including how to determine if a device is unconditionally stable or potentially unstable.
If you are designing an active RF circuit, you should always start with a thorough stability analysis.
We will also learn how the arrangement of these circles on the Smith chart can produce unconditionally stable or potentially unstable devices.
First, why is stability analysis so important?
At high frequencies, unavoidable parasitic effects can easily cause a circuit to oscillate. For example, a poor grounding scheme can cause coupling between different stages of a multistage amplifier and lead to instability.
In addition, some circuits in the RF signal chain may not have well-defined source or load impedances. For example, a low-noise amplifier (LNA) in a receiver needs to be connected to the outside world through an antenna. If a user brings their hand close to the antenna, the impedance of the antenna changes, so the LNA must remain stable for all possible source impedance values at all frequencies.
Sometimes, instability can produce strange signs, such as sudden changes in the amplifier's DC parameters, or high sensitivity of the circuit to the surrounding environment. This makes performing a properly thorough stability analysis a challenging and complex task.
Single-Stage RF Amplifier
Figure 1 shows the basic layout of an RF amplifier.

Figure 1. Schematic diagram of a basic single-stage RF amplifier.
In the figure above, matching networks are used on both sides of the transistor to transform the input impedance (Z1) and output impedance (Z2) to the desired values S and L. The subscripts S and L represent the source and load respectively.
To check the stability of the circuit, we model the active device as a two-port network (Figure 2). This two-port network will be characterized by its S parameters and connected to the impedances provided by the input and output matching networks, ZS and ZL.

Figure 2. Circuit for analyzing RF amplifier stability.
Using signal flow graph analysis, we can derive expressions for the reflection coefficients, ΓIN and ΓOUT, in terms of the transistor's S parameters:

Equation 1.

Equation 2.
The equation above allows us to check the stability of the two-port network. Remember that for passive circuits, the magnitude of Γ is between 0 and 1. Therefore, the reflected signal is smaller than the incident signal. However, for active devices, the reflected signal may experience gain rather than attenuation.
In other words, for certain S-parameter values, the magnitude of the input and output reflection coefficients of active devices can be greater than unity (|ΓIN| > 1 and/or |ΓOUT| > 1). This occurs even if the source and load terminations of the transistor are passive (|ΓS| < 1 and |ΓL|< 1). Also, note that reflection coefficients greater than unity correspond to impedances with negative real parts. Oscillations can occur when negative resistance is created at the input or output port.

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Unconditional Stability and Potential Instability
A two-port network is considered unconditionally stable if it can be connected to any source and load impedances without exhibiting oscillations. In this context, we usually mean any passive source and load impedances, in other words, assuming |ΓS| < 1 and |ΓL|< 1.Thus, for an unconditionally stable transistor, the loci of the input and output reflection coefficients are the entire area inside the unit circle of the Smith chart.
Transistors that are not unconditionally stable are often described as "potentially unstable" devices. Potentially unstable devices can become unstable for certain values of passive source and load impedances. Note that stability is frequency dependent: a transistor may be unconditionally stable over a certain frequency range, but unstable over a different frequency range. Therefore, stability should be evaluated at every frequency point for which transistor data is available.
In mathematical terms, unconditional stability requires |ΓIN| < 1 and |ΓOUT|< 1.Thus, we have:

Equation 3.

Equation 4.
When the above conditions are met, the real part of the transistor input and output port impedance is positive for all passive source terminations and load terminations. If these conditions are not met at a particular frequency, the transistor may be unstable at that frequency.

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Input and output stability circles
If we set |ΓIN| and |ΓOUT| equal to unity, we obtain the equations that specify the instability boundaries:

Equation 5.

Equation 6.
Equation 5 specifies that all load reflection coefficients (ΓL) are such that the magnitude of the input reflection coefficient is equal to unity (|ΓIN| = 1). Because this equation specifies the usable boundary ΓL values, it should be plotted on the ΓL plane. For the same reason, equation 6 specifies ΓS, so it should be plotted on the ΓS plane.
In their current form, these equations are easy to interpret but difficult to use. Fortunately, some mathematical manipulation allows both equations to be put into the standard form of the equation of a circle. Equation 5 becomes:

Equation 7.
Where cL represents the center of the circle and rL represents the radius of the circle.
To find cL, we use equation 8:

Equation 8.
Equation 9 gives us rL:

Equation 9.
In the above equation, Δ is the determinant of the S parameter matrix. It can be expressed as follows:

Equation 10.
Since equation 7 defines a circle that specifies the available boundary ΓL values, we call the above circle the output stability circle.
Similarly, equation 6 corresponds to a cS:

Equation 11.
and with radius rS:

Equation 12.
Because it specifies the available ΓS values, we call the circle produced by Equation 6 the input stability circle.