Chapter 2 Basic Concepts in RF Circuit
2.2 Noise Basic
2.2.2 Noise model of MOSFET
2 2
2 m
n T
ox
g
K K
i f A f
f WLC f ω
= ⋅ ⋅ Δ ≈ ⋅ ⋅ ⋅ Δ (2-3)
where K is the process-dependent constant, and A is the area of the gate.
2.2.2 Noise Model of MOSFET
The dominant noise source in CMOS devices is channel noise, which basically is thermal noise originating from the voltage-controlled resistor mechanism of a MOSFET. This source of noise can be modeled as a shunt current source in the output circuit of the device. The channel noise of MOSFET is given by
f g kT
ind2 =4 γ d0Δ (2-4)
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where γ is bias-dependent factor, and g is the zero-bias drain conductance of the d0 device. Another source of drain noise is flicker noise and is given by Eqn. 2-3. Hence, the total drain noise source is given by
WLC f
At RF frequencies, the thermal agitation of channel charge leads to a noisy gate current because the fluctuations in the channel charge induce a physical current in the gate terminal due to capacitive coupling. This source of noise can be modeled as a shunt current source between gate and source terminal with a shunt conductance gg, and may be expressed as
f
and δ is the gate noise coefficient. This gate noise is partially correlated with the channel thermal noise, because both noise currents stem from thermal fluctuations in the channel and the magnitude of the correlation can be expressed as
j
where the value of 0.395j is exact for long channel devices. Hence, the gate noise can be re-expressed as
where the first term is correlated and the second term is uncorrelated to channel noise.
From previous introduction of MOSFET noise source, a standard MOSFET noise model can be presented in Fig. 2.4, where ind2 is the drain noise source, i is the ng2
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gate noise source, and v is thermal noise source of gate parasitic resistor rg2 rg.
2.2.3 Noise Figure of Cascaded Stages
For a cascade of m stages, the overall noise figure can be characterized by Friis formula
This equation indicates that the noise results from the decrease in each stage as the gain preceding the increase in stage. Hence, the first few stages in a cascade are the most critical for noise figure. But if a stage exhibits attenuation, then the noise figure of the following circuit is amplified when referred to the input of that stage.
2.2.4 Noise Factor of a Two Port Network
Noise factor F is a useful measure of the noise performance of a system. It is defined as the ratio of the available noise power Pno at output divided by the product of the available noise power at input Pni which times the networks’s numeric gain G, or equivalently defined as the ratio of the signal to noise power at the input to the signal to noise power at the output .
o
The noise factor is a measure of the degradation in signal to noise ratio due to the noise from the system itself. Since the noise factor relates to the input noise power, a
C
gsv
gsg
mv
gs11
standardized definition of noise source has been setup: a resistor at 290K. A more general expression of noise factor NF is called noise figure which is just noise factor expressed in decibels:
F
NF =10log (2-12) When several networks are cascaded, each has its own gain Gi and noise factor Fi. The total output of the noise is composed of all the noise from each stage but with different amount of contribution to the noise performance. The noise factor of a cascade networks is given as
1 ... From (2-13), the noise factor of the first stage is most critical and must be keep as low as possible and its gain should be as large as possible to suppress the noise in the following stage. The result is intuitive since there is less interference of noise effect when the signal level is high.
2.2.5 Optimum Source Impedance for Noise Design
The noise factor of a two port network can be given as [5]2
where Rn is the correlation resistance which showed the relative sensitivity of the noise figure to departures from the optimum conditions, and Zo is the characteristic impedance of the system. This equation expresses that there exists an optimum source reflection coefficient (Γopt) or equivalently an optimum source impedance (Zopt) at the input of the network in order to deliver lowest noise factor (Fmin). The value of Γ provides a constant noise factor value forming non-overlapping circles on the s
Smith chart. It is usually the case that the optimum noise performance trades with the maximum power gain.
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2.3 Linearity and Nonlinearity
All electronic circuits are nonlinear: a fundamental truth of electronic engineering. The linear assumption underlying most modern circuit theory is in practice only an approximation. Some circuits, such as small-signal amplifiers, are only very weak nonlinear; however, they are used in systems as if they were linear. In these circuits, nonlinearities are responsible for phenomena that degrade system performance and must be minimized. Other circuits, such as frequency multipliers, exploit the nonlinearities in their circuit elements; these circuits would not be possible if nonlinearities did not exist. Among these, it is often desirable to maximize the effect of the nonlinearities and even to maximize the effects of annoying linear phenomena.
The problem of analyzing and designing such circuits is usually more complicated than for linear circuits, and it is the subject of much special concern.
Linear circuits are defined as those for which the superposition principle holds.
Specifically, if excitations x1 and x2 are applied separately to a circuit having responses y1 and y2 respectively, the response to the excitation ax1+bx2 is ay1+by2, where a and b are arbitrary constants. This criterion can be applied to either circuits or systems.
This definition implies that the response of a linear and time-invariant circuit of system includes only those frequencies present in the excitation waveforms. Thus, linear and time-invariant circuits do not generate new frequencies. As nonlinear circuits usually generate a remarkably large number of new frequency components, this criterion provides an important dividing line between linear and nonlinear circuits.
Nonlinear circuits are often characterized as either strongly nonlinear or weakly nonlinear. Although these terms have no precise definitions, a good working
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distinction is that a weakly nonlinear circuit can be described with adequate accuracy by a Taylor series expansion of its nonlinear current/voltage (I/V), charge/voltage (C/V), or flux/current (Φ/I) characteristic around some bias current or voltage. This definition implies that the characteristic is continuous and has continuous derivatives.
And for most practical purposes, they do not require more than a few terms in its Taylor series. Virtually all transistors and passive components satisfy this definition if the excitation voltages are well within the component’s normal operating ranges; that is, below saturation.
2.4 Nonlinear Phenomena
2.4.1 Harmonic Generation
Assumption of the current nonlinear element is given by the expression:
2 3
I =aV bV+ +cV (2-15)
where a, b, and c are constants, real coefficients. Assuming that Vs is a two-tone excitation of the term:
1 1 2 2
( ) cos( ) cos( )
s s
V =v t =V ωt +V ω t (2-16)
Substituting (2.1) into (2.2) gives, for the first term,
1 1 2 2
( ) ( ) cos( ) cos( )
a s
i t =av t =aV ωt +aV ω t (2-17)
After doing the same with the second term, the quadratic, and applying the well-known trigonometric identities for squares and products of cosines, we obtained:
1 2
2 2 2 2 2
1 1 1 2
1 2 1 2 1 2
( ) ( ) { cos(2 ) cos(2 )
2
2 [cos(( ) ) cos(( ) )]
b s
i t bv t b V V V t V t
V V t t
ω ω
ω ω ω ω
= = + + + +
+ + + −
(2-18)
and the third term, the cubic, gives
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The total current in the nonlinear element is the sum of the current components in (2-17) through (2-19).
One obvious property of a nonlinear system is its generation of harmonics of the excitation frequency or frequencies. These are evident as the terms in (2-17) through (2-19) at mω1 and mω2. The mth harmonic of an excitation frequency is an mth-order mixing frequency. In narrow-band systems, harmonics are not a serious problem because they are far removed in frequency from the signals of interest and inevitably rejected by filters. In other systems, such as transmitters, harmonics may interfere with other communication systems and must be reduced by filters or other means.
2.4.2 Intermodulation Distortion
All the mixing frequencies in (2-17) through (2-19) that arise as linear combination of two or more tones, often called Intermodulation (IM) products. IM products were generated in an amplifier or communications receiver, often come with a serious problem. While the interfered spurious signals can be mistaken for desired signals. IM products are generally much weaker than the generating signals; however, a situation often arises wherein two or more very strong signals, which may be outside the receiver’s passband, generate an IM product that is within the receiver’s passband and obscures a weak and desired signal. Even-order IM products usually occur at frequencies well above or below the generating signals, and consequently are often of little concern because…... The IM products of greatest concern are usually the third-order ones that occur at 2ω1-ω2 and 2ω2-ω1, and because they are the
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strongest of all odd-order products and close to the generating signals, they often cannot be rejected by filters. Thus, intermodulation is a major concern in microwave system.
2.5 Fundamental of the Volterra Series
The nonlinearity of the system often leads to interesting and important phenomena, such as harmonics, gain compression, desensitization, blocking, cross modulation, intermodulation, etc. These distortions will degrade the performance of the system, while Volterra series will be used for distortion computations. It can provide designers some information to derive which circuit parameters or circuit elements they have to modify to obtain the required specifications. Therefore, Volterra series will be introduced in the following section.
In fact, the Volterra series describe a nonlinear system in a way which is equivalent to the Taylor series approximating an analytic function. A nonlinear system excited by a signal with small amplitude can be described by the Volterra series, which can be broken down after the first few terms. The higher the input amplitude, the more terms of that series need to be taken into account to describe the system behavior properly.
For very high amplitudes, the series diverges just as Taylor series. Hence, Volterra series are only suitable for the analysis of weak nonlinear circuits.
The Volterra series approach has been proven to be useful for hand calculations of small transistor networks. Since Volterra kernels retain phase information, they are especially useful for high-frequency analysis.
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The theory of Volterra series can be viewed as an extension of the theory of linear, first-order systems to weakly nonlinear systems. And a system is considered as the combination of different operators of different order in the Volterra series description, as shown in Fig. 2.5. Every block H1, H2, and Hn represents an operator of order 1, 2, …, respectively. The amount of operators must be used depend on the input amplitude. In general, the weakly nonlinear effects can be described accurately by taking into account third-order effects only.
In the time domain, the transformation on an input signal (x(t)) was performed by a nth-order Volterra operator that is given by:
∫
The n-dimension integral can be seen as an nth-order convolution integral. The function )hn(τ1,τ2,",τn is an nth-order Volterra kernel. The output of a nonlinear system can represent the sum of the output of a first-order Volterra operator with the output of a second-order one, a third-order one and so on, as shown in Fig. 2.5. The Volterra series of the nonlinear system can be expressed as)]
In the frequency domain, the nth-order Volterra kernel can be given by
∫
and is called the nth-order nonlinear transfer function or the nth-order kernel transform.Fig. 2.5 Schematic representation of a system characterized by a Volterra series
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2.6 Nonlinear Performance Parameters in Terms of Volterra Kernels
When a system that is described by a Volterra series up to order three, it is excited by the sum of two sinusoidal excitations A1cosω and 1t A2cosω . Then 2t the output is given by the sum of the responses listed in Table 2.1. From Table 2.1, the expression for the second and third harmonic distortion in terms of general Volterra are given by
Furthermore, among the intermodulation products, the third-order intermodulation products at 2ω1−ω2 and 2ω2 −ω1 is important. Since if the difference between ω and 1 ω is small, the distortions at 2 2ω1−ω2 and 2ω2 −ω1 would appear in the vicinity of ω and 1 ω . Table 2.1 shows the third-order 2 intermodulation distortion in terms of Volterra kernel transforms
)
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Table. 2.1 Different responses at the output of a nonlinear system described by Volterra kernels.
Order Frequency of response Amplitude of response Type of response
2nd harmonics
2
19 This effect causes some distortion at our desired frequency and damages the desired signals. Therefore third intercept point (IP3) is used to characterize this behavior. This parameter is measured by supplying a two-tone signal to the system.
This input signal must be chosen to be sufficiently small in order to remove higher-order nonlinear terms. In a typical test, A1=A2=A, hence the magnitude of third-order intermodulation products grows at three times the rate at which the fundamental signal on a logarithmic scale when input signal increases. The third-order intercept point is defined to be the point at which third-order intermodulation product equals to the fundamental signal, and the corresponding input signal is called input IP3 (IIP3) and the corresponding output signal is called output IP3 (OIP3). The AIP3, therefore, can be obtained by setting IM3 =1 and expressing as
Besides, a quick method of measuring IIP3 is as follows. As shown in Fig. 2.6, If the power of the two-tone signal, Pin, is small enough to ignore higher order nonlinear terms, then IIP3 can be expressed as
dBm
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(a) (b)
Output power (dBm)
Input power (dBm) OIP3
Pin IIP3
IMD3
Fundamental Signal
ω2
ω1 2ω2-ω1
2ω1-ω2
ΔP/2
ΔP ΔP
3ΔP/2
Fig. 2.6 (a) Growth of output components in an intermodulation test (b) Intermodulation distortion
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Chapter 3
General Consideration in RF Circuit Design
3.1 Low Noise Amplifier Basic
Low noise amplifier is the first gain stage in the receive path so its noise figure directly adds to that of the system. There, therefore, are several common goals in the design of LNA. These include minimizing noise figure of the amplifier, providing enough gain with sufficient linearity and providing a stable 50 Ω input impedance to terminate an unknown length of transmission line which delivers signal from antenna to the amplifier. Among LNA architectures, inductive source degeneration is the most popular method since it can achieve noise and power matching simultaneously, as shown in Fig. 3.1. The following analysis is based on this architecture.
3.1.1 Low Noise Amplifier Architecture Analysis
In Fig. 3.1, the input impedance can be expressed asgs Fig. 3.1 Common source input stage with inductive source degeneration
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as shown in (3.1), the input impedance is equal to the multiplication of cutoff frequency of the device and source inductance at resonant frequency. Therefore it can be set to 50 Ω for input matching while resonant frequency is designed to be equal to the operating frequency.
According to prior introduction, the equivalent noise model of common-source LNA with inductive source degeneration can be expressed as Fig. 3.2, where R is l
the parasitic resistance of the inductor, R is the gate resistance of the device. Note g that the overlap capacitance Cgd has also been neglected in the interest of simplicity.
Then the noise figure can be obtained by computing the total output noise power and output noise power due to input source. To find the output noise, we first evaluate the trans-conductance of the input stage. With the output current proportional to the voltage no Cgs and noting that the input circuit takes the form of series-resonant network, the transconductance at the resonant frequency can be expressed as
s
where Qin is the effective Q of the amplifier input circuit. So the output noise power density due to the source can be expressed as
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Furthermore, channel current noise of the device is the dominant noise contributor, and its noise power density associated with the correlated portion of the gate noise can be expressed as
, 2
The last noise term is the contribution of the uncorrelated portion of the gate noise, and its output noise power density can be expressed as
, 2
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According to (3.3), (3.4), (3.5) and (3.8), the noise figure at the resonant frequency can be expressed as
⎟⎟⎠
From (3.11), we observe that χ includes the terms which are constant, proportional to QL, and proportional to Q . It follows that (3.11) will contain terms L2 which are proportional to QL as well as inversely proportional to QL. A minimum noise figure, therefore, exits for a particular QL.
3.1.2 Optimizations of Low Noise Amplifier Design Flow
The analysis of the previous section can now be drawn upon in designing the LNA. In order to pick the appropriate device size and bias point to optimize noise performance given specific objectives for gain and power dissipation, a simple second-order model of the MOSFET transconductance can be employed which accounts for high-field effects in short channel devices. Assume that the drain current, Id, has the form
The power consumption of the LNA, therefore, can be expressed as
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The noise figure can be expressed in terms of PD and Vgs. Two parameters linked to power dissipation need to be accounted for.
)
The noise figure of the LNA, therefore, can be expressed as
)
In general, there are two approaches to optimize noise figure. The first approach assumes a fixed transconductance, Gm. The second approach assumes fixed power consumption.
(1) Fixed Gm optimization: To fix the value of the transconductance, Gm, we need only assign a constant value to ρ. Once ρ is determined, the optimization of the noise figure can be obtained by (3.17):
)
From (3.18), we can obtain the optimal width to get the minimal noise figure for a given Gm under the assumption of matched input impedance. In this approach, the designer can achieve high gain and low noise performance by selecting the desired transconductance, but its disadvantage is that we must sacrifice the power consumption to achieve minimum noise figure.
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(2) Fixed PD optimization: An alternative method of optimization fixes the power dissipation and adjusts device size and bias point to minimize the noise figure.
Once PD is determined, the optimization of the noise figure can be obtained by (3.19):
Then the optimum device size can be obtained to get the best noise performance for fixed power dissipation. In this approach, the designer can specify the power dissipation and find the optimal noise performance, but its disadvantage is that the transconductance is held up by the optimal noise condition.
3.1.3 Amplifier stability
The stability of an amplifier, or its resistance to oscillate, is a very important consideration in a design and can be determined from the S parameters, the matching networks, and the terminations. The non-zero S12 parameter of a two port networks as shown in Fig. 3.3 provides a feedback path by which the power transferred to the output can be feedback to the input and combined together. Oscillation may occur when the magnitude of reflection coefficient Γ or IN ΓOUT, defined as the ratio of the reflected to the incident wave, exceeds unity. It is expected that a properly designed amplifier will not oscillate no matter what passive source and load impedances are connected to it, which is said to be unconditionally stable and the reflection coefficient is given as
1
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conditionally stable. In such a case, input and load stability circles, the contour of Γ =1 and IN ΓOUT=1 for certain frequencies on the Smith chart, are useful to fine the boundary line for load and source impedances that cause stable and unstable condition.
The stability circles can be calculated directly from the S parameters of the two port network, so another convenient parameter, stability factor K, is defined and given as
|
| 2
|
|
|
|
|
| 1
21 12
2 2
22 2 11
S S
S
K = − S − + Δ (3.21)
where
2 21 12 22 11
2 | |
|
|Δ = S S −S S
The amplifier is unconditionally stable provided that
>1
K and |Δ|2<1 (3.22) or equivalently
>1
K and B1 <1 (3.23) where B1=1+|S11|2 −|S22|2 −|Δ|2.
ΓS Γin Γout ΓL
ΓS Γin Γout ΓL