1. Introduction

Field Effect Transistor (FET) RF switches are used in nearly all wireless communication devices, from cell phones to Wi-Fi to GPS systems [1-9]. With a growing communications market, their numbers will also grow. However, the continued growth is contingent on the constant evolution and refinement of RF switches. Specifically, newer communications architectures and implementations for fourth generation communications (4G), such as 3GPP Long Term Evolution (LTE) and IEEE802.11n, require higher frequencies and corresponding bandwidths while minimizing the distortion [1, 5, 7-9]. Minimizing distortion translates to maximizing linearization. There have been many steps over the past several years to improve the linearity of switches to meet the demanding specifications. Correspondingly, defining and modeling FET as switches have also improved [12-18], although most work has been done viewing the FET as simply a microwave device or as an amplifier [19-34]. This author has found no analysis specifically tying modulation of the ON resistance (R_on) to linearity or what inherent characteristics in a FET, such as transconductance, might lead to ON resistance sensitivity to modulation.

2. Background

The wireless performance requirements in today’s RF devices call out very stringent IP3 specifications depending upon the architecture and options used (65 dBm for GSM and WCDMA) – and have been very difficult to meet [35 – 37]. Nonetheless, researchers and companies have striven the create switches to meet these requirements. In 2006, Mil’shtein and Liessner et. al. introduced methods to improve linearity of FET switches [38-40]. In 2010, Andrew Dearn and Liam Devlin of Plextek Ltd. presented high linearity SPDT switches with IP3 at 1.9 GHz up to 54 dBm, with the second harmonic recorded at 80 dBc for that device [41]. Also, in 2010, Michael Yore, et.al. of Triquint Semiconductor reported a SP7T pHEMT switch with IMD2 typically better than -100 dBm and IMD3 typically better than -105dBm [42]. As of 2013, commercially, Skyworks currently offers an ultra-high linearity SP2T switch, SKY13405-409LF, capable of a typical IIP3 of 68 dBm [37].

3. Analysis

Looking at the DC I-V characteristics of a FET switch, the channel resistance for a switch when it is in the ON state is effectively the inverse slope of the Ids versus V_ds for a given V_gs at low values of V_ds:

(1)

In this linear region of I_ds versus V_ds, this ratio is essentially a constant with a constant V_gs. However, this ratio – or equivalently the resistance, R_on, - changes as a function of V_gs. Since R_on is a function of V_gs, taking the derivative of (1) with respect to V_gs will look as follows:

(2)

(3)

Reviewing the definition of the transconductance, g_m (4), and substituting it into equation (3), along with equation for R_on (1), we establish a relationship between the sensitivity of R_on to V_gs, V_ds, and I_ds in equation (5).

(4)

(5)

Equation (5) establishes a direct correlation between the change in R_on to its dc value, the value of the dc current (I_ds), and the transconductance (g_m), to the variation in the gate voltage (∂V_gs). It is seen that the smaller R_on is, the smaller the sensitivity to change in V_gs. In addition, as the dc I_ds increases, the variation in R_on also diminishes. This is because a larger I_ds implies a smaller R_on; and the smaller R_on is, the smaller the impact that any variation on it has on the overall signal.

Now we need to relate this derivation to switch linearity for a series switch, and to the 3^rd intermodulation products in particular. In Figure 1 is shown the simple power transfer circuit of a switch in the ON state. P_in and P_out are described in equations (6) and (7). Manipulating equations (6), (7), and (8), and realizing that R_on is typically much less than the Z_L, we can derive the relationship of P_in to P_out relative to R_on. The associated insertion loss is described by equation (10).

Figure 1. Switch ON state with load

(6)

(7)

(8)

(9)

(10)

Using a Taylor Series expansion [43], a weakly nonlinear response can be modeled around a quiescent point, such as a switch either in the ON state or OFF state:

(11)

(12)

(13)

Deriving for a two-tone solution, as used for IP3 measurements, equation (13) becomes as follows:

(14)

Now we can proceed with deriving the appropriate IM3 calculation. From equation (9) and (14), we can separate out the IM3 products:

(15)

If P_in is a constant value A, and we used the simplified attenuation factor in (9) we can now define the G function from (15) as:

(16)

To find G₃, from equation (13) we will use equation (17), and identifying ∂R_on and ∂V_gs, and calling k the RF coupling, we derive equation (20).

(17)

(18)

(19)

(20)

Where δ is defined as:

(21)

Plugging (20) into (15) yields the final equation for finding the amplitude of 3^rd order intermodulation product with a modulated R_on. The logarithmic version of IM3 in dB is found in equation (23).

(22)

(23)

To calculate the IIP3, first P_in must be properly defined. Since ∂R_on is much less than Z_L,

(24)

Since measurements are made using dBm as the standard, the amplitude is multiplied by a factor of 1000 to properly scale.

(25)

And the IIP3 is shown below:

(26)

However, note that I_ds is not arbitrary – it is tied to the power input (or, equivalently, the RMS input voltage), the ON resistance, and the load impedance as follows:

(27)

And therefore:

(28)

With δ represented now as:

(29)

And as expected, the power cancels out in this equation since compression and dispersive effects are not in the model. This leaves the IIP3 a function of R_on, g_m, Z_L, and the coupling factor k. Note that this analysis only shows the affect and sensitivity of a switch due to transconductance in the region of interest and fluctuations to R_on; it does not provide a model for a specific pHEMT structure otherwise.

4. Results

A series of plots were made to show the predicted impact on switch linearity. R_on was swept over a range of 0.1Ω to 2Ω, and four different transconductance values were selected: 500mS, 250mS, 100mS, and 10mS. The RF coupling factor was selected to be at 50%, 100%, and 10% in three separate plots. The load impedance was set to 50Ω. Figures 2, 3, and 4 show the results. Figures 5 and 6 display the affect of a 2:1 VSWR on the load, with Figure 5 calculated for 100Ω and Figure 6 determined for a 25Ω. Note that the diminished Z_L also represents potential loading caused by an OFF switch element on the switch.

5. Discussion

Looking at the linearity for the ON state of RF FET switches, analysis and calculations suggest the hypothesis of a sharp transconductance profile for optimum linearity. This reduces the affect of RF coupling onto the gate of the FET. Using the equations established, the plots in figures 2 – 4 show the clear advantage to minimizing the g_m value in a switch FET when it is in the ON state. Higher g_m in the ON state produces more distortion at the output, even with minimal RF coupling. Referring to figure 4, which shows results based on a coupling factor of k = 0.1, a device with a R_on = 1 ohm and a g_m = 500mS can only achieve an IIP3 of 70dBm, whereas a device with the same R_on but with a g_m = 100mS can reach an IIP3 = 92dBm. This is a 22dB improvement. As the coupling factor increases to 0.5 and 1.0 as shown in figures 2 and 3 respectively, the improvement achieved by the lower g_m increases even further: at k = 0.5 there is a 31dB gain, and at k = 1.0 there is a >80dB improvement. Despite that, a high transconductance is preferred for the transition of switch states and higher I_ds. A sharp, peaked, impulse-shaped transconductance profile is suggested to be best, since it allows for a lower g_m during a switch ON state. The coupling factor, k, affects the intermodulation products as expected - more coupling results in less linearity and poorer IP3 values. While this occurs regardless of g_m, it is demonstrated in figures 2 -6 and discussed above, that a lower g_m compared to a higher g_m improves the IP3 performance. It is noticed in Figures 5 and 6, that a lesser load impedance actually improves the linearity. The reason for this is that the higher I_ds is, the greater the linearity; the compression and dispersion effects are not modeled here.

Figure 2. IIP3: k=0.5

Figure 3. IIP3: k=1.0

Figure 4. IIP3: k=0.1

Figure 5. IIP3: k=0.5, ZL = 100 ohms

Figure 6. IIP3: k=0.5, ZL = 25 ohms

6. Conclusions

After extensive analysis, a new approach to characterizing FET switches has been proposed, analyzed, and mathematically supported. A new analytical tool has been created to provide guidance to designing switches with high linearity, and it has been shown that a lower transconductance value when the FET is in the ON state yields superior linearity over a FET with higher transconductance, suggesting a different transconductance profile than what is often proposed, such as an impulse.