Chapter 1 Introduction
1.2 Thesis Organization
In Chapter 2, fundamentals about nonlinear circuit design are introduced.
Techniques relating to the analysis of nonlinear system are also mentioned.
In Chapter 3, the design and analysis of a wide IF bandwidth 60-GHz up-conversion mixer is described. Simulation and experimental results are both provided.
In Chapter 4, a harmonic current injection frequency tripler (HCI-FT) is proposed and analyzed. Detailed investigations were done to optimize the performance of HCI-FT. Chip was implemented with careful considerations.
Complete simulated results are given in the end.
In Chapter 5, the conclusion and future work of the thesis are given.
3
Chapter 2
Fundamentals in Nonlinear Circuit Design
2.1 Linearity and Nonlinearity
All electronic circuits are nonlinear: this is a fundamental truth of electronic engineering. The linear assumption that underlies most modern circuit theory is in practice only an approximation. Some circuits, such as small-signal amplifiers, are only very weakly nonlinear, however, and 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. In 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; 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, 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 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 Nonlinear Phenomena
2.2.1 Harmonic Generation
4
3
Assume the current of a nonlinear element is given by the expression:
I =aV+bV2+cV (2.1)
where a, b, and c are constants, real coefficients. We assume 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 (2.2)
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 (2.3)
After doing the same with the second term, the quadratic, and applying the well-known trigonometric identities for squares and products of cosines, we obtain:
1 2
and the third term, the cubic, gives
1 2
The total current in the nonlinear element is the sum of the current components in (2.3) through (2.5).
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 (2.3) through (2.5) 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 are rejected by filters. In others, such as transmitters, harmonics may interfere with other communication systems and must be reduced by filters or other means.
2.2.2 Intermodulation Distortion
5
All the mixing frequencies in (2.3) through (2.5) that arise as linear combination of two or more tones are often called Intermodulation (IM) products. IM products generated in an amplifier or communications receiver often present a serious problem, because they represent spurious signals that interfere with, and can be mistaken for, desired signals. IM products are generally much weaker than the signals that generate them; 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, desired signal. Even-order IM products
usually occur at frequencies well above or below the signals that generate them, and consequently are often of little concern. The IM products of greatest concern are usually the third-order ones that occur at 2ω1-ω2 and 2ω2-ω1, because they are the strongest of all odd-order products, are close to the signals that generate them, and often cannot be rejected by filters. Intermodulation is a major concern in microwave system.
2.2.3 Saturation and Desensitization
Recall that (2.5) included components ω1 and ω2 that varied as the cube of signal level. Such components are responsible for gain reduction and desensitization in the presence of strong signals.
In order to describe saturation, we refer to (2.1) to (2.5). From (2.3) and (2.5), and with V2=0, we find the current component at ω1, designated i1(t), to be:
3
1 1 1
( ) 3 cos( )
i t =⎛⎜⎝aV +4cV ⎞⎟⎠ ω1t (2.6)
If the coefficient c of the cubic term is negative, the response current saturates;
that is, it does not increase at a rate proportional to the increase in excitation voltage.
Saturation occurs in all circuits because the available output power is finite. If a circuit such as an amplifier is excited by a large and a small signal, and the large signal drives the circuit into saturation, gain is decreased for the weak signal as well.
Saturation therefore causes a decrease in system sensitivity, call desensitization.
2.2.4 AM-to-PM Conversion
AM-to-PM conversion is a phenomenon wherein changes in the amplitude of a signal applied to a nonlinear circuit cause a phase shift. This form of distortion can have serious consequences if it occurs in a system in which the signal’s phase is important; for example, phase- or frequency-modulated communication systems. Let
6
the response current at ω1 in the nonlinear circuit element is (2.6), where i1(t) is the sum of first- and third-order current components at ω1. Suppose, however, these components were not in phase. This possibility is not predicted by (2.1) through (2.5) because these equations describe a memoryless nonlinearity. In a circuit having reactive nonlinearities, however, it is possible for a phase difference to exist. The response is then the vector sum of two phasors:
3
1 1 1 1
( ) 3
4
I ω =aV + cV ejθ (2.7)
where θ is the phase difference. Even if θ remains constant with amplitude, the phase of I1 changes with variation in V1. It is clear from comparing (2.7) to (2.6) that AM-to-PM conversion is serious as the circuit is driven into saturation.
2.3 Harmonic Balance Analysis
Transient analysis methods predate harmonic balance (HB) methods. Thus, the existence of harmonic-balance analysis implies that transient methods are not adequate for many kinds of circuits. In fact, the methods are pleasantly complementary: HB works well where transient analysis does not, and transient analysis usually outperforms HB in the kinds of problems where it is applicable.
Three problems can make time-domain techniques impractical. First, matching circuits may contain such elements as dispersive transmission lines, transmission-line discontinuities, and multiport subnetworks described by S or Y parameters. These are difficult to analyze in the time domain. Second, the circuit’s time constants may be large compared to the period of the fundamental excitation frequency. When long time constants exist, it becomes necessary to continue the numerical integration of the equations through many—perhaps thousands—of excitation cycles, until the transient part of the response has decayed and only the steady-state part remains. This long
7
integration is an extravagant use of both computer time and the engineer’s patience;
furthermore, numerical truncation errors in the long integration may become large and reduce the accuracy of the solution. Although algorithms exist to ameliorate this difficulty, implementing them is an extra complication. Third, each linear or nonlinear reactive element in the circuit adds a differential equation to the set of equations that describes the circuit. A large circuit can have many reactive elements, so the set of equations that must be solved may be very large. For this reason, time-domain analysis is notoriously slow.
The greatest advantage of time-domain analysis is its ability to handle very strong nonlinearities in large circuits. Its robustness results in part from the fact that small time steps can be used in the time-domain integration. As long as the nonlinearities are continuous, the time steps can always be made short enough so that the circuit voltages and currents change very little between steps.
Followings are two examples which demonstrate how HB method solves problems. Fig. 1.1 shows a simple dc diode circuit, which we wish to analyze.
Knowing that the diode’s I/V characteristic is given by (2.8), we can easily write an equation for the circuit as (2.9) shows.
Fig. 2.1 A simple dc-biased diode
( )
9
This equation cannot be solved algebraically. It must be solved numerically or, if only moderate accuracy is adequate, graphically.The usual method is to estimate I, substitute it into (2.9), and see if it satisfies the equation. If it does not, I is modified and the process repeated until the equation is solved. A variety of numerical methods can be used for this purpose. We do not need a method that solves the problem completely; all we need is to improve an estimated solution. Then, we need only repeat the process a number of times, using the result of each iteration as the starting estimate for the next one. Eventually, the error is reduced to the point where it is deemed negligible.
Thus, we need four things:
1. An initial estimate of the solution;
2. A numerical method for improving an estimated solution;
3. A criterion for determining whether the process has indeed improved the solution at any particular iteration step;
4. A way to decide when the solution is adequate.
These needs are easily satisfied for the circuit in Fig. 2.1, but they might not be so clear in more complex circuits. Fig. 2.2 is a slightly more complicated problem, which consists of an RF impedance, Z(ω). We excite our diode, with the RF source, at the frequency ωp. We know from above that the diode generates harmonics of both current and voltage, and Z(ω) can be expected to vary with harmonic frequency; thus, we could write it Z(kωp), where k is the harmonic number.
Fig. 2.2 A diode excited by an RF circuit (a) can be divided into a pair of equivalent circuits, one describing the linear part (b), and another, the nonlinear part (c)
First, we assume that we know the diode voltage (consisting of its complex components at all harmonic frequencies, kωp). We then create the equivalent circuit in Fig. 2.2 (b), which can be analyzed easily in the frequency domain, giving:
( ) ( )
Of course, if Vs consists of a dc and a sinusoidal component, only two components of Vs, Vs(0) and Vs(kωp), are nonzero. Vs need not be sinusoidal, but for our present purposes, it must be periodic.
Using Fourier theory, we convert V(kωp) into a time waveform, V(t). We then create the circuit in Fig. 2.2 (c) and find the current in the diode junction algebraically from (2.8):
(
V t( ) 1I =Isat eδ −
)
(2.11)If necessary, we can find INL(kωp) by Fourier transformation. The only remaining problem is that we really don’t know V(kωp). However, we do know how to tell
10
11 0
whether a particular V(kωp) is a solution; substitute it into (2.10) and (2.11), and see if Kirchhoff’s current law is satisfied at all the harmonics:
( ) ( )
LIN p NL p
I kω +I kω = (2.12)
If (2.12) satisfied, we have a solution.
We now can summarize the solution process as follows:
1. Create an initial estimate of V(kωp), k=0, 1, …,K, where K is the maximum harmonic with which we need be concerned. This estimate may be extremely crude; for example, V(kωp)=0 for all k.
2. Use (2.10) to obtain ILIN(kωp).
3. Inverse-Fourier transform V(kωp) to obtain V(t).
4. Use (2.11) to determine INL(t).
5. Fourier transform INL(t)to obtain INL(kωp).
6. Substitute ILIN(kωp) and INL(kωp) into (2.12). Of course, (2.12) probably will not be satisfied. Define an error function at each harmonic, fk, where:
( ) ( ) 0, 1, ...,
k LIN p NL p
f =I kω +I kω k= K (2.13)
7. Modify V(kωp) and repeat the process from step 2. Use some appropriate numerical method that can be trusted to decrease |fk |.
8. Continue until all K+1 errors fk are negligibly small.
Above demonstrates how HB method solves a simple nonlinear circuit. For the more complicated circuits, HB methods follows the same procedure and extends the dimension of the I/V equations.
12
Chapter 3
60-GHz Up-Conversion Mixer with Wide IF Bandwidth
3.1 Introduction
Many applications require or benefit from high data rate communication, such as the high quality video transmission which requires the data rate exceeding 1 Gb/s. The wireless LAN at 2.4 GHz or 5 GHz can obviously not meet this kind of transmission requirement. 7 GHz of unlicensed bandwidth around 60 GHz is potentially to provide the possibility of over-gigahertz data transmission with extraordinary capacity.
Up-conversion mixer is an important building block in the transmitter circuit, which provides the frequency translation from IF (intermediate frequency) to RF (radio frequency). IF bandwidth is an important parameter to characterize the up-conversion mixer since the 60-GHz band is aimed to provide over-gigahertz data transmission. At millimeter-wave frequencies, however, CMOS technology provides lower conversion gain when compared with other processes due to its lossy silicon substrate. The researches about 60-GHz up-conversion mixer are relatively rare.
Nonetheless, CMOS has the advantages of low-cost and high-level integration with VLSI section; it is worth further research definitely.
In this work, a direct up-conversion mixer is designed and analyzed to provide wide IF bandwidth under low power consumption. The up-converted differential signal is converted to single-ended signal using a Marchand-type balun for the future integration with power amplifier at its next stage. Besides, a frequency tripler is integrated in this work to provide LO (local oscillator) signal for measurement
13
consideration. The proposed 60-GHz direct up-conversion mixer is fabricated using TSMC 0.13-um CMOS technology. According to the measured results, this up-conversion mixer provides 3.5-GHz IF bandwidth under 2.7 mW power consumption and a conversion gain of -6 dB. This chapter shows the design considerations to achieve the desired wide IF bandwidth.
The analysis using large-signal method is described in Section 3.2, and the circuit realization is mentioned in Section 3.3. In Section 3.4, chip implementation and experimental results are presented. Finally, a summary is given in Section 3.5.
14
3.2 Analysis on Impedance Using Large-signal Method
All electronic circuits are nonlinear: this is a fundamental truth of electronic engineering. However, the nonlinear circuits are characterized as either strongly nonlinear or weakly nonlinear. Sometimes weakly nonlinear circuits are managed as linear circuits, since the techniques relating to the analysis of linear circuits are uncomplicated in contrast to that of nonlinear circuits.
In the low-noise amplifier (LNA) design, small-signal S-parameter is used to analyze the input impedance and the power gain, since an LNA is of small-signal operation. LNAs are categorized as weakly nonlinear circuits, therefore small-signal methods is suitable for analysis.
Mixers, however, a relative large local oscillator (LO) signal is used for current commutating in most cases. Furthermore, new frequency components are generated by the mixers. They cannot be categorized to weakly nonlinear group anyhow.
Therefore, the small-signal S-parameter is not suitable for the analysis of mixers.
Some other methods take over this job.
Because the 60-GHz band is aimed to provide over-gigahertz data transmission, wide intermediate frequency (IF) bandwidth becomes an important consideration in up-conversion mixer design. Therefore, it is important to guarantee that the return loss at IF port is kept almost constant within a certain bandwidth. In this work, a Gilbert-cell based mixer is chosen as the topology. However, IF signal is directly ac-coupled into the switching stage of the mixer, instead of a transconductor as Fig.
3.X demonstrates. Since this node is potentially to provide a slowly-varying impedance versus frequency, therefore a steady return loss.
The first problem goes to the observation of the impedance versus frequency at the IF port. The small-signal method widely used in linear system would lead to
mistakes since the up-conversion mixer is essentially nonlinear. Harmonic balance (HB) method and time-domain method are commonly used in the analysis of nonlinear circuits. As mentioned in Chapter 2, HB method has several merits superior to time-domain method. Nonetheless, time-domain method is also employed due to its ability of manipulating AM-to-PM distortion. Fig. 3.1 shows the operation concept of HB and time-domain method.
Fig. 3.1 The operational concept of HB method and time-domain method
A nonlinear, dynamic system can potentially convert the signal amplitude variation into phase disturbance such as AM-to-PM distortion. The dispersive spectra shown in the lower part of Fig. 3.1 is the AM-to-PM effect. Although HB method is powerful in handling most nonlinear systems, it is failed to take the AM-to-PM effect into calculation. The time-domain method is innate to take all nonlinear effects into consideration.
Fast Fourier transform (FFT) is used to do the transformation between time-domain and frequency-domain information. There are two basic problems when using FFT to study the frequency spectrum of signals: the fact that we can only
15
measure the signals for a limited time; and the fact that the FFT only calculates results for certain discrete frequency values.
The first problem arises because the signal can only be measured for a limited time. Nothing can be known about the signal’s behavior outside the measured interval, and the Fourier transform makes an implicit assumption that the signal is repetitive:
that is, the signal within the measured time repeats itself for all time.
Most real signals have discontinuities at the ends of the measured time, and when the FFT assumes the signal repeats it will assume discontinuities that are not really there, as Fig. 3.2 shows.
Fig. 3.2 Actual signal and signal assumed by FFT
Since sharp discontinuities have broadband frequency spectra, these will cause the signal’s frequency spectrum to be spread out. The spreading means that signal energy which should be concentrated only at one frequency instead of leaks into all the other frequencies. This spreading of energy is called “spectral leakage”.
The effects of spectral leakage can be reduced by reducing the discontinuities at the ends of the signal measurement time. This leads to the idea of multiplying the
16
signal within the measurement time by some function that smoothly reduces the signal to zero at the end points hence avoiding discontinuities altogether. The process of multiplying the signal data by a function that smoothly approaches zero at both ends, is call “windowing,” and the multiplying function is called a “window function”.
In a word, a window function puts less weight on the ends of the data, since they are potentially to produce discontinuities; and puts more weight on the center of the data, since they are more reliable. It is like a “matched filter” in the communication system in some respects. In our analyses, a “Hamming” window function is applied to the time-domain data for the spectral leakage consideration.
Get back to the analysis of our circuit, since the topology of the mixer is
Get back to the analysis of our circuit, since the topology of the mixer is