For the PDF version of this article, click here.

Mobile wireless communications links demand sensitive and highly selective transceiver circuits. The dynamic range and sensitivity of the RF wireless link depends on the HF noise and linearity of the transistors used in the low-noise amplifiers. Correlation improves the simulation accuracy of noise analysis of the transistors.

Noisy transistors degrade the performance of mobile wireless receivers and prevent amplifiers and oscillators from meeting the stringent requirements imposed on them while working at frequencies in the GHz range. Small noise improvements at the device level can have a large impact on overall system performance. For instance, a degradation of the low-noise amplifier (LNA) noise figure (Nƒ) by even 0.2 dB can be detrimental to the RF link performance.

Various analytical equations were used to determine the minimum noise figure Nƒ_min of a bipolar transistor as a function of bias conditions and transistor parameters. Recently, a different approach for the Nƒ_min calculation of bipolar transistors was used to give Nƒ_min in terms of the small-signal parameters of the transistors. All of these approaches neglected an important effect that should be considered to get good high-frequency (HF) noise modeling: the correlation between collector and base shot noise sources.

At RF and high frequencies, the main noise sources are the base resistance thermal noise and the terminal current shot noises. Figure 1 shows a representation of these noise sources for the common emitter configuration. There are two ways to connect the base current shot noise source: either directly between the emitter and the base as shown in the figure, or between the emitter and an internal base node connected to the base through the base resistance.

The base and collector shot noises are generally correlated to each other. Correlation can be understood as it relates to Van der Ziel's theory^[1] by the fact that collector current has the same component as the emitter current, which is due to the injection of electrons from the emitter to the base. Consequently, the shot noise due to this current component is fully correlated with collector current shot noise. Base current due to hole injection from base to emitter has shot noise as well, but it has no relation to the collector current so it is not correlated. Since the base current shot noise is due to both components of hole and electron injections, the shot noise due to base current is partially correlated to the collector current shot noise.

Connecting the base shot noise directly between the external base and the emitter simplifies analytical noise analysis and allows direct modeling of noise from measured y parameters. The assumption to simplify the noise figure equation^[2] is equivalent to connecting the noise source directly between the external base and the emitter.

Some recent publications^[3-6] addressed the noise modeling of heterojunction bipolar transistors (HBTs), applying the correlation in the model as a time-delay approach^[4-6]. A thermodynamic approach was also developed to account for the correlation^[5], but even though it agreed with the measured data the delay could not be incorporated in compact models because it was empirical. A SPICE-like noise model was given^[2]; however, correlation was not taken into account.

Any noisy two-port network, such as the equivalent small signal circuit of the transistor shown in Figure 1, can be replaced by a noiseless network in addition to two external equivalent noise sources with a correlation term between the voltage noise power and the current noise power, as shown in Figure 2.

The four basic noise parameters, equivalent noise resistance Rn_eq, minimum noise figure Nƒ_min, and the real and imaginary part of the optimum source admittance for minimum noise figure G_opt and B_opt, respectively, can be obtained as functions of correlation matrix parameters^[2][7]. Applying this with suitable substitutions, the following expressions as functions of the voltage noise power S_vn, the current noise power S_in and the correlation term S_invn* between them are obtained (Equations 1-4):

The operators J(.) and K(.) represent the imaginary and real parts.

To determine the values of S_vn, S_in and S_invn*, earlier work^[8] acquired the expressions for the previously mentioned quantities, but without taking into consideration the correlation between S_ib and S_ic.

The correlation term can be expressed as (Equation 5):

where C is the correlation coefficient and I_c and I_b are the dc collector and base currents, respectively. Taking this correlation into consideration, the following was obtained:

A. Using the method of directly connecting the base shot noise source between the external base and emitter: (Equations 6-8)

B. Connecting the base shot noise source between the internal base and emitter: (Equations 9-11)

where r_B is the base resistance and Y₁₁ and Y₂₁ are the small signal Y parameters of the transistor, representing the input admittance and the current gain, respectively.

Nƒ_min is overestimated in existing noise models^[9], so the same device was used^[9] with peak ƒt = 150 GHz at Vce = 1 V to verify the previous equations at room temperature. The device used is a single finger device that has a CBEBC contact configuration with an emitter area of 0.2 × 10.16 µm². Simulations were done using HICUM model^[10] in Eldo^[11] in the frequency range from 1 GHz to 27 GHz. First, the correlation coefficient value was taken from^[1], c = -j/√3. However, after a comparison to measured data, it was found that the value c = -j/√2 gives better fitting for this production process.

For the first method, Equation 6 shows that S_vn has no dependence on the correlation term while Equation 7 and Equation 8 indicate that both S_in and S_invn* have dependencies on the correlation term. As for the second method, Equation 9, Equation 10 and Equation 11 show that all quantities have dependencies on the correlation term.

Figure 3 shows that Rn_eq will not change by the added correlation term for the first representation Equation 6, while it is changed for the second representation Equation 9.

Both representations will make G_opt decrease, but it will cause B_opt to increase as shown in Figure 4, which shows G_opt with and without correlation for both representations. This B_opt increase is also shown in Figure 5, which shows B_opt for the same cases.

Figure 6 shows Nƒ_min for the correlated and non-correlated terms for both methods of representation. From the figure it is obvious that method A (approximate representation) is valid only at relatively low frequencies. At higher frequencies it approaches the case of no correlation, while the accurate method of representation (method B) shows more consistent behavior. Figure 7 shows that the added correlation term simulated with the accurate method B helps to decrease the value of the simulated Nƒ_min, which matches the measured data obtained from^[9] better than the model with no correlation.

It is obvious that the approximation taken from the first representation is sufficient for Nƒ_min calculation at lower frequencies. However, for higher frequencies the additional accuracy obtained from the second representation is a must.

This approach can be easily integrated in compact models because it is simple and requires only the addition of one extra parameter for the correlation coefficient magnitude that can be extracted from the fitting of the measured Nƒ_min data and one equation (Eq. 5) to the model, which is needed to calculate the correlation noise power, and then the simulator can take this quantity into account when it performs noise simulation. This should provide better noise characterization.

To design an LNA, some key criteria must be met to have an acceptable operation. The criteria are mainly noise figure, input and output matching, reverse isolation, gain, linearity, and stability of the LNA. Table 1 shows the most important LNA design criteria for acceptable operation^[12].

The LNA design methodology that was stated in^[13] is applied to the cascode topology in Figure 8, where the noise figure is dominated by the noise characteristics of transistor Q1 along with some elements to achieve the design criteria^[12][13].

The aim of this design example is to demonstrate the potential use of a new noise model, so optimizing the design was not the scope of this work. Consequently, an analog synthesis tool was used[14] to obtain suitable values for the design elements and to keep the required design specifications within the accepted limits.

The same transistor model of the single transistor two-port results was used for simulation of LNA transistors Q1 and Q2. The LNA was simulated in the frequency range of 1.8 GHz ± 10 MHz.

The HICUM noise model was replaced with a simple one that groups all of the collector and base current components that contribute to shot noise into two single noise sources connected similar to the accurate connectivity of method B, then the new correlation model was added to this modified noise model.

The values obtained for different LNA components are listed in Table 2, with the geometry scaling of bias transistor Q2 chosen to be minimum area = 1. Table 1 shows the LNA design criteria that were achieved for the design's desired frequency range. It is worth mentioning here that this design is the basic primary step in the normal design cycle. No parasitics were extracted from a layout and the design was not refined; it is just a demonstration on the potential effect of using the modified noise model.

Spot Noise Figure (SNF), simulated with and without correlation, is shown in Figure 9. The correlation model had an effect on the SNF value and the decrease was around 0.1 dB (5%). This might seem to be a small value, but it should be taken into account that the inductively degenerated cascode LNA topology targets noise characteristics optimization. Any decrease in this already optimized SNF is a sign of how this modified noise model can help achieve better noise characteristics simulation accuracy.

Existing noise models do not accurately describe the high-frequency noise behavior of bipolar transistor models due to the absence of the correlation between collector current shot noise and base current shot noise in these models. This affects the accuracy of RF noise analysis, especially when targeting applications like LNAs, by overestimating some of the noise quantities.

The proposed solution results in more accurate modeling of the noise behavior depending on the suitable choice of the correlation coefficient. The value used is arbitrary, taken from literature just to demonstrate the validity of this idea based on the demand for modeling the correlation between collector and base shot noise sources in many recently published papers.

This solution requires only minor changes to the already existing noise models, so it is suitable for compact modeling for better simulation of HBT noise behavior. This was verified when the new model was added to the HICUM compact model. This modified model was used to design a 1.8GHz LNA, which was used to test the new noise model implementation in the compact model by observing the SNF behavior with and without the new model. A decrease of the SNF was seen, although this decrease might seem to be an insignificant value. The chosen topology targets noise characteristics optimization, so any decrease in this optimized SNF is evidence that the proposed model is giving more accurate noise simulation results.

1. A. Van Der Ziel, “Noise in solid-state devices and circuits,” John Wiley & Sons, 1986.

2. Voinigescu, et al., “A scalable high-frequency noise model for bipolar transistors with application to optimal transistor sizing for low-noise amplifier design,” IEEE J. Solid-State Circuits, vol. 32, no. 9, pp. 1430-1438, September 1997.

3. Niu, et al., “Transistor noise in SiGe HBT RF technology,” in IEEE Journal of Solid-State Circuits, vol. 36, no. 9, pp. 1424-1427, September 2001.

4. Escotte, et al., “Noise modeling of microwave heterojunction bipolar transistors,” IEEE Trans. Electron Devices, vol. 42, pp. 883-888, May 1995.

5. Moller, et al., “An improved model for high-frequency noise in BJTs and HBTs interpolating between the quasi-thermal approach and the correlated shot-noise model,” IEEE BCTM, Digest of Papers, Monterey, pp. 228-231, October 2002.

6. Rudolph, et al., “An HBT noise model valid up to transit frequency,” in IEEE Electron Device Letters, vol. 20, no. 1, pp. 24-26, January 1999.

7. H. Hillbrand and P. Russer, “An efficient method for computer- aided noise analysis of linear amplifier networks,” IEEE Transactions on Circuits and Systems, vol. CAS-23, no. 4, pp. 235-238, April 1976.

8. Cui, et al., “An examination of bipolar transistor noise modeling and noise physics using microscopic noise simulation,” IEEE BCTM, Proceedings 2003, pp. 225-228.

9. Sakalas, et al., “Analysis of microwave noise sources in 150 GHz SiGe HBTs,” RFIC Symposium, 2004. Digest of Papers. IEEE, pp. 291-294, June 6-8, 2004.

10. M. Schroter, “Staying current with HICUM,” IEEE Circuits and Devices magazine, vol. 18, no. 3, pp. 16-25, 2002.

11. Mentor Graphics, “Eldo User's Manual,” 2005.3 release.

12. B. Razavi, “RF Microelectronics,” Prentice Hall, 1998.

13. Shana'a, et al., “Frequency-scalable SiGe bipolar RF front-end design,” IEEE Journal of Solid-State Circuits, vol. 36, no. 6, pp. 888-895, June 2001.

14. Farouk, et al. “The design of a wideband matching network for a short wire monopole using analog synthesis,” Proceedings, International Conference on Electrical, Electronic and Computer Engineering, ICEEC 2004, pp. 567-570.

Mohamed Selim is a software development engineer with the analog/mixed-signal verification group at Mentor Graphics Corporation, providing internal and foundry support for Eldo device models. He holds a B.Sc. in electronics and communications engineering and a M.Sc. degree in electronics from Cairo University in Egypt, Faculty of Engineering. He is interested in compact models, noise in RF circuits, analog and RF modeling.

RSS    Save to Del.icio.us  Digg This