Chapter 1 Introduction
1.2 Thesis Organization
In Chapter 2, fundamentals about nonlinear circuit are introduced. Techniques and principles relating to the analysis of nonlinear system are also mentioned.
In Chapter 3, the detailed design and analysis of a frquency quadrupler with spurs rejection is described. We also introduce the Volterra series to analyze the nonlinear behavior.
In Chapter 4, chip implement and measurement result is demonstrated including harmonic response, phase noise, and the verification of efficacy by sub-harmonic mixing.
In Chapter 5, the conclusion and future work of the thesis are given.
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Chapter 2
Harmonic Generation by Nonlinear Behavior
2.1 Linearity and Nonlinearity
A fundamental truth of electronic engineering is all electronic circuits are nonlinear. Generally, the linear assumption that underlies most modern circuit theory is practically nearly an approximation. Some circuits, such as small-signal amplifiers, are very weakly nonlinear, and are usually be regarded as linear in systems. However, in these circuits, nonlinear behavior are often crucial factors that degrade system performance and must be minimized. As to some circuits, such as frequency multipliers, exploit the nonlinearities in their circuit elements; these circuits would be hardly implement if nonlinearities did not exist. So the courses of these circuits are often desirable to maximize the effect of the nonlinearities, and even to maximize the effects of annoying linear phenomena. The difficulty of analyzing and designing such circuits is usually more severe than for linear circuits; it is the main subject.
Linear circuits are defined as those which can put the superposition principle into analyzing. 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, time-invariant circuit of system includes only those frequencies present in the excitation waveforms.
Thus, linear, 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
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circuits.
Nonlinear circuits are often characterized as either strongly nonlinear or weakly nonlinear. Although these terms have no precise definitions, a good working 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, has continuous derivatives, and for most practical purposes, does 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.
2.2 Harmonic Generation
2.2.1 Single-frequency Excitation
We will first describe the frequency spectrum at the output of the test circuit when it is excited with one sinusoidal source at a frequency 1. When the amplitude A1 of the input signal is small enough, then the output spectrum of the circuit only contain one frequency component above the noise floor, this is to say, the response corresponding to the circuit's linear behavior. This is a signal at the same frequency of the input signal, called the fundamental frequency. The amplitude of this signal change proportionally with the input amplitude.
When the input amplitude increased, the output spectrum contains signals at the frequencies 21 and 31, and these signals called the second and third harmonics, originate from second- and third-order nonlinear circuit behavior. Respectively, as we shall see below. Harmonics higher than the third, caused by higher-order behavior,
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come above the noise floor at even higher input amplitudes. It is seen that the amplitude of the nth harmonic increases with the nth power of the input amplitude: an increase of the input amplitude with 6dB yields an increase of the second harmonic at the output with 12dB, the third harmonic increases with 18dB and so on. At high input amplitudes, this is not true anymore. Then it is observed that the third-order nonlinear behavior also give rise to a component at the fundamental frequency which increases with the third power of the input amplitude. As a result, the fundamental response can increase faster than linear, which for an amplifier means that the gain slightly increase.
In this case one speak about gain expansion. On the other hand, if the increase is less than linear because the sign of the third-order contribution is opposite to the sign of the linear response, then a gain compression is observed in the output. Similarly, forth-order behavior gives a contribution to the second harmonic and so on. This situation is depicted in Figure 2.1. Signals caused by nonlinear behavior of order higher than five are not shown in this figure. Also it must be noted that a component at 0Hz is found at the output. This DC shift is caused by second-order, forth-order, or in general, even-order nonlinear behavior.
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1
1
1
1(2) (4)
(1) (3)
(5) (2)
(4) (3)
(5) (4) (5)
frequency
Figure 2.1 The differential harmonics at the output of an analog circuit excited by a sinusoidal signal at frequency 1. The numbers between brackets indicate the order of nonlinear behavior by which the signal is determined.
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The nonlinear behavior as discussed above in a qualitative way can be clarified mathematically with a simple example. Assume that the relationship between the input signal x(t) (which is either a current or a voltage) of a circuit and the output y(t) is given by the following relationship
y t( )K x t1
K2
x t
2K3
x t
3... (2.1) When the input-output relationship is given explicitly by an analytic relationship y t
f
x t
(2.2) Then the coefficients K1,K2,K3,... can be identified with the coefficients of a Taylor series of f : second-order and third-order nonlinearity coefficients, respectively, in general, high-order nonlinearity coefficients.Returning to our example, we assure that the input signal x(t) has the form x t
Acos
1t1
(2.6) Substituting this expression into equation (2.4) yields the output y(t) :
1
1 1
2 2
1 1
0Hz. Both signals are proportional to K2 and to the square of the input amplitude A.7
Therefore, these signals are denoted as second-order signals. And the third-order signals at the frequency 31 and at the fundamental frequency 1 are with the same reason. Assure K3 has the same sign as K1, in this case the third-order signal at the fundamental frequency as the same sign at the first-order signal. In other words, the amplitude of the fundamental signal has increased due to third-order behavior. This situation corresponds to expansion. If K1 and K3 have an opposite sign then we have compression.
2.2.1 Two-frequency Excitation
In the same way as in the previous section, the test circuit under consideration is now excited with two sinusoids A1cos(1t) and A2cos(2t), both applied at the same input port. When A1 and A2 are sufficiently low, the output spectrum contains two signals above the noise floor at the fundamental frequency 1 and 2 due to the circuit's linear behavior. Because in a linear circuit the superposition principle is valid, the two excitations don't produce any interfering signal. However, When A1 and A2
become larger, then, apart from the harmonics of 1 and 2, interfering signals grow above the noise floor at the frequency 1+2, ∣1-2∣, 21+2, ∣21-2∣,
1+22 and ∣-1+22∣. The signals at ∣1±2∣are caused by second-order nonlinear behavior and are called second-order intermodulation products. They increase with the first power of both A1 and A2. The other signals come from third-order behavior and are denoted as third-order intermodulation products. The signals at ∣1±2∣increase with the square of A1 and with the first power of A2, and so on.
For analog circuits such as amplifiers, the intermodulation products are usually unwanted. Therefore, they are denoted as intermodulation distortion. In communication circuits, these unwanted products are often denoted as spurious responses. Then consider again the test circuit with the input-output relationship given
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by equation (2.1). When the input signal x(t) consists of two signals of equal amplitude and with a different frequency 1 and 2:
x t
Acos
1t Acos
2t (2.8) Then the different responses are shown in Figure 2.2. It is assumed that the amplitude A is sufficiently low such that the circuit behaves in a weakly nonlinear way. The frequency w1 and w2 are have been given the value *10 MHz and *11 MHz. It is seen that harmonics of 1 and 2 are present at the output as well as intermodulation products. with the input-output relationship given by equation (2.1) to a combination of two sinusoidal signals with the same amplitude A. The frequency of the two input signals are 10 MHz and 11 MHz.9
2.3 Description of Nonlinearities in Analog Integrated Circuits
prior to the analysis of nonlinear behavior of analog circuits, it is necessary to describe the nonlinear devices that are present in analog circuits. The devices most commonly used in silicon analog integrated circuits are transistors, resistors, capacitors and diodes. In circuit analysis the devices mentioned above are described using an equivalent circuit. This equivalent circuit can be as simple as one circuit element (e.g. one resistor), or it consist of several circuit elements (e.g. transistor).
The elements of such equivalent circuit are nonlinear in general. The following circuit elements are used in analog integrated circuits as a part of the equivalent circuit of a device.
▬ A nonlinear conductance: the current through this element is an algebraic function of the voltage over the element.
▬ A nonlinear transconductance: the current through this element is an algebraic function of a voltage other than the voltage over the element.
▬ A nonlinear resistance: the voltage over this element is an algebraic function of the current through this element.
▬ A nonlinear transresistance: the voltage over this element is an algebraic function of a current different from the current through this element.
▬ A nonlinear capacitance: the charge on this element is an algebraic function of the voltage across the element.
▬ A nonlinear transcapacitance: the charge on this element is an algebraic function of the voltage across the element.
These circuit elements are referred to as basic nonlinearities, since they are the building elements for nonlinear equivalent circuits of devices such as transistor,
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diodes, integrated resistors,... Nonlinear voltage-controlled voltage sources and current-controlled current sources are seldom used in equivalent circuits of devices in analog integrated circuits.
2.3.1 Power Series Description of Basic Nonlinearities
A power series description of a basic nonlinearity contains the derivatives of the output quantity (current or voltage) with respect to the controlling quantities. These derivatives are evaluated in the quiescent point. An accurate description of a nonlinear device in terms of power series requires that the different derivatives that are considered be accurate. These derivatives are a function of the controlling quantities and of the model parameters that describe the nonlinear device. The main nonlinearities in transistor are transconductances, then we look into the derivation of the nonlinear coefficients as following sections.
2.3.2 Nonlinear Transconductance
For a nonlinear transconductance, the current through the element iOUT(t), is a nonlinear function f of the controlling voltage vC(t) elsewhere in the circuit. This function can be expanded into a power series around the quiescent point IOUT = f(VC):
iO U T
t f
v
C
t
f VC
cv
t and the AC current. The voltage vc(t) is the AC voltage that controls the conductance.The second term in equation (2.9) is a power series representing the AC part of the current. When the analysis of a circuit that contains a nonlinear conductance is limited to first- second- and third-order nonlinear behavior, then the power series in equation
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(2.9) can be broken down after the third term.
Defining the following coefficients In which we omitted the time dependence for simplicity, and in general
Leads to the expression of the AC current through the conductance
iOUT
t g v t1 c
K2,g1vc2
t K3,g1v tc3
... (2.14) In this expression, g1 is the small-signal transconductance of the linearized circuit.The coefficients in the second and third term, K2,g1 and K3,g1 are respectively the second- and third-order nonlinearity coefficients that describe the nonlinear element.
Similarly, the small-signal conductance is often referred to as the first-order coefficient. The subscript for K2 and K3 is the symbol that represents the linearized element, in this case g1.
2.3.3 Two-dimensional Transconductance
A two-dimensional transconductance is an element the current of which is controlled by two different voltage. In other words, the current iOUT(t) is a function f of two voltages uC and vC, which can be expressed in terms of AC values using a
two-dimensional power series expression around the quiescent point IOUT=f(UC,VC)
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The AC part of the current corresponds to the second term of equation(2.15), which is a two-dimensional power series. This series can be split into three series i1, i2 and i3, each corresponding to a part of the total AC current.
The time dependence of uc and vc has been omitted for simplicity. The third series, i3, contains nothing but cross-terms, which are terms that contain a nonzero power of both uc and vc. The meaning of the subscripts in the nonlinearity coefficients defined above is as
follows. Suppose that the first-order derivative of the total current with respect to u and v are respectively g1 and g2. Then a coefficient like Km,jg1&(m-j)g2 with m and j positive integers and m > j means
A three-dimensional transconductance is a current source that is controlled by three voltage. In other words, the current is a function f of three voltage u(t), v(t), and w(t).
Using a power series expansion around the quiescent value of the current can be split into a quiescent part IOUT = f(UC,VC,WC) and an AC part. This AC part is given by
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The AC current can be split into distinct parts, and this series implies the introduction of the following nonlinearity coefficients.
The meaning of the newly defined coefficients is illustrated with a simple modelfor the drain current of an nMOS transistor in saturation. Taking into account bulk effect and Early effect, the drain current is given by
2 1
coefficient and the surface inversion potential, respectively. The first derivatives of the current with respect to the controlling voltage vGS, vBS, and vDS are the small-signal parameters gm, gmb, go. Then from above equations the AC current is given by:
14 lines represent the variation of the current when two voltages change at the same time.
The last line describes the variation of the current with the three controlling voltages at the same of the three-dimensional drain current in general requires sixteen second- and third- order coefficients. Table 2.1 lists the expressions for the coefficients that
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Chapter 3
A Frequency Quadrupler with Spur Harmonics Suppression
3.1 Introduction
Active frequency multipliers are utilized in numerous applications to efficiently provide a source of high frequency microwave energy. They are commonly used in communication systems to enable frequency translation of a signal from a low-noise and low-frequency oscillator to the required higher frequency band for the purpose of up/down conversion in transceivers.
In general, frequency quadrupler and higher order multipliers have not seen more prominence and detailed investigation than doublers and triplers, due to higher circuit complexity and lower achievable conversion gain and efficiency. Therefore, at first we inspect some published works about quadrupler.
Begin with the work pronounced by professor Huei Wang, proceedings of 2010 EuMC conference[1]. The circuit schematic is shown in figure 3.1. A single-stage
V-band frequency quadrupler is proposed and manufactured in 0.25-μm SiGe BiCMOS technology. The maxima output power is -10 dBm with 11.7 mW dc power consumption. The concept is through driving the strongly non-linear devices to get nonlinear harmonics then filtering out the undesired harmonics. The conversion efficiency is poor by this approach. Another weak point is its poor spur-harmonic rejection of output signal because of low quality of output filter operating at high frequency, especially the spurs around the desired harmonic. In the design of frequency quadrupler, the third-order spur harmonic at frequency 3f0 is in particular
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hard to suppress. The measured dates shown in figure 3.2 clearly shows the poor harmonic rejection, especially the spur at 3f0. Therefore, to achieve the further suppression of spur harmonics inevitably need additional filters. It is an inconvenient concern. According to the paper, the merit of the circuit is the 36% bandwidth from 52 to 75 GHz. In my opinion, it takes advantages of the strongly non-linear devices of SiGe BiCMOS technology, and by large input signal to make the output power achieve the saturation point in the frequency band.
Next, the work pronounced by Infineon Technologies AG, proceedings of 2009 EuMIC conference[3]. This work is manufactured in a Silicon-Germanium production
Figure 3.1 The quadrupler circuit schematic of reference[1].
Figure 3.2 Measurement date of reference[1].
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technology, and the schematic is shown in figure 3.3. The DC consumption of this quadrupler is 43mW from 3.5 V supply voltage and RF output power is -5dBm at frequency 77GHz. From my point of view, it is a good work. The structure is using two doublers stacked one by one. Therefore, the cost is higher supply voltage for stacking, and the requirement of quadrature input signal for each Gibert mixer operation. So the circuit contains several TRLs for 90 degree phase shifting which increases the complexity of design and chip area, especially when operating in lower frequency application.
Next, the work pronounced by professor Euisik Yoon, proceedings of 2005 TMTT[2]. The schematic shown in figure 3.4 is manufactured in a 0.18- m CMOS process. The DC consumption of the quadrupler is 106mW with 1.8V VDD and it has -18 dBm RF output power at frequency 40 GHz. Apparently, the focal point of this paper is not with an emphasis on the quadrupler. It merely contains four transistors biased at the maxima forth-order derivative of transconductance. The disadvantages
Figure 3.3 Stacked quadrupler of reference[3].
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are high power consumption and low conversion efficiency. Although the design do not have any particularity, except for the linear superposition. It let us think of some possibilities that can upgrade the circuit.
Finally, in this work, a new technique of generating the forth-order harmonic at 4f0 is proposed and analyzed. It improves the generating efficiency by not only direct generation, but also mixing generation. Applying this technique, frequency 4f0 can be generated under low power consumption and the circuit itself is uncomplicated compared with other published methods. Analytical equations were developed to approximate the numerically-converged results of CAD tools for optimization with paper and hand calculation. Detailed analyses were done to maximize the frequency 4f0 and suppress the undesired harmonics. According to the experimental results, the output power of proposed quadrupler is -20dBm (contained 9.5 dB loss of the output buffer) under only 4.5 mW dynamic power consumption with fundamental input power of +8 dBm. The proposed quadrupler is fabricated using TSMC 90-nm
low-leakage CMOS technology for the verification of theoretical results.
Figure 3.4 Quadrupler schematic of reference[2].
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In Section 3.2, the architecture of the proposed quadrupler is introduced preliminarily. The nonlinear analysis by Voterra series are presented in Section 3.3.
The design procedure of the proposed quadrupler is illustrated in Section 3.4, including the design of quadrature all pass filter.
3.2 A Proposed Architecture of Frequency Quadrupler
In the last section we saw several types of methods to devise a quadrupler.
Principally, you can find some issues which dominate the performances such as power consumption, harmonic rejection ratio(HRR), and conversion gain...etc. But however, a technique which can advance the performance in all aspects is nearly impossible.
When a technique can improve some quality, a drawback maybe emerges at the same time. Consequently, ameliorating the drawback is the target for all designers. Return to the topic, in the beginning we should determine what features are our desires. Apart
When a technique can improve some quality, a drawback maybe emerges at the same time. Consequently, ameliorating the drawback is the target for all designers. Return to the topic, in the beginning we should determine what features are our desires. Apart