Table 2.1: Difference between Taguchi approach and classical DOE.
Taguchi approach Classical DOE
Standard approach Methods are not standardized Smaller number of experiments Larger number of experiments
Standard method of noise factor No standardized method of noise treatment Seeks to find stable condition Develops models
Used to solve engineering problems Used to solve scientific experiments
robust products. The objective of classical DOE is to gather scientific knowledge about fac-tor effects and their interactions. Difference between Taguchi approach and classical DOE is shown in Tab. 2.1 [46]. In this thesis, it is suitable for us to use the classical DOE to investigate the problem due to our data type.
2.8 Summary
In this chapter, we introduce the statistical methodology which is used in this work. Screen-ing design is the first step in this work to select the significant factors. After this step, we have three type central composite design and choose one type among them to construct the response surface model. The basic of the response surface model and adequacy check-ing are then introduced. Then we discuss desirability function which is used to solve the multiple responses and according to this index we can optimize successfully. Finally we compare the difference between Taguchi approach and classical DOE, and we note that the
34 Chapter 2 : Statistical Methodology
classical DOE is suitable for this work.
Chapter 3
Low Noise Amplifier
In this chapter, we discuss the low noise amplifier (LNA) circuit which will be one of our testing examples. Due to a large number of circuit simulations, it is necessary to produce the response data, where Hspice simulator is integrated in our method. This chapter is organized as follows: Section 3.1 describes characteristics of LNA circuit. Section 3.2 describes what problem we will discuss and Sec. 3.3 presents the usage of the circuit simulation. Finally, a summary of this chapter is given.
3.1 A LNA Circuit with Deep Submicron MOSFETs
LNA circuit is important to modern communication systems. The main object of LNA is to ensure the quality of signal in the process of receiving the signal. A LNA design presents
35
36 Chapter 3 : Low Noise Amplifier
a considerable challenge because of its simultaneous requirement for high gain, low noise figure, good input and output matching and unconditional stability at the lowest possible current draw from the amplifier. In this experiment, the working frequency of the tested LNA circuit is from 2.11 to 2.17 GHz, shown in Fig. 3.1. The cascade low noise amplifier is constructed two transistor placed cascaded. Lload and Rload are the compact models of on-chip spiral inductors needed in our LNA circuit. The choke inductor Lchoke working at high frequency is fixed at 1 uH. Cin is an external signal couple capacitor is also fixed at 20 pF.
3.1 : A LNA Circuit with Deep Submicron MOSFETs 37
Figure 3.1: The explored LNA circuit in our experiment.
3.1.1 Noise Figure
The parameters we used to diagnose the noise of LNA are noise factor (F ) and noise figure (N F ). The noise factor of a low-noise amplifier is defined as the signal-to-noise ratio at the input divided by the signal-to-noise ratio at the output. The equation for noise factor and noise figure is given by
F = SN Rin
38 Chapter 3 : Low Noise Amplifier
N F = 10log(F ), (3.2)
where SN Rin is the signal-to-noise ratio at the input and SN Rout is the signal-to-noise ratio at the output. N oiseinis the noise from the previous stage, N oiseoutis the noise at the output which is additional noise from amplifier(Noiseamp) added the noise from Noisein. Besides, in Eq. (3.1), N oiseamp is always not zero, therefore, F > 1 and NF > 0 dB is consequential. In other words, SN Rinmust be greater than SN Rout.
In a cascade amplifier the final stage has an input signal that consists of the original sig-nal and noise amplified by each successive stage. Each stage in the cascade chain amplifies signals and noise from previous stages and contributes some noise of its own. The overall noise factor for a cascade amplifier is:
F = 1 + (F1− 1) + (F2− 1)
where F is the overall noise factor in cascade, Fi is the noise factor of the ith stage, Apiis the gain of the ith stage. Fn is the overall noise factor of n stages in cascade. As show in Eq. (3.3), the noise factor of the entire cascade chain is determined by the the first stage noise factor because the noise factors of the second and subsequent stages are divided by
3.1 : A LNA Circuit with Deep Submicron MOSFETs 39
the previous stage gains when referred back to the input. High gain and low noise low-noise amplifiers typically use a low-low-noise amplifier circuit for only the first stage or two in the cascade chain to achieve an overall noise factor.
3.1.2 Stability Factor
Unconditional stability of the circuit is the target of the LNA designer. It is a critical concern in designing a low noise amplifier. The stability of circuit can be determined by S-parameter of transistors, and the matching network of every stage. S-S-parameters provided by the manufacturer of the transistor will aid in stability analysis of the LNA circuit. Two main methods exist in S-parameter stability analysis: numerical and graphical. Numerical analysis consists of calculating a term called Rollett Stability Factor K [11][12][13]. An intermitted quantity called delta(Δ) should be calculated first to simplify the final equation for the K-factor.
Δ = S11∗ S22− S21∗ S12, (3.4)
then
K = 1 + |Δ|2 − |S11|2− |S12|2
2 ∗ |S11| ∗ |S12| . (3.5)
When the K factor is greater than unity, the circuit will be unconditionally stable for any combination of source and load impedance. When K is less than unity, the circuit is poten-tially unstable and oscillation may occur with a certain combination of source and/or load
40 Chapter 3 : Low Noise Amplifier
impedance presented to the transistor. The K factor represents a quick check for stability at given frequency and given bias condition. A sweep of the K-factor over frequency for a given biasing point should be performed to ensure unconditional stability outside of the band of operation.