Summary

A direct up-conversion mixer with wide IF bandwidth is designed and fabricated for 60-GHz applications. Large-signal has been verified to estimate the accurate trend of impedance versus frequency at IF port. According to the experimental results, the up-conversion mixer has 3.5-GHz IF bandwidth which is feasible for high-speed data transmission.

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Chapter 4

Frequency Tripler Using Second-Order Harmonic Current Injection Technique

4.1 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 low-frequency oscillator to the required higher frequency band for the purpose of up/down conversion in transceivers.

In general, frequency triplers and higher order multipliers have not seen prominence and detailed investigation benefited by doublers, due to higher circuit complexity and lower achievable conversion gain and efficiency [7]-[10], [16]-[17].

Driving the transistors into strongly non-linear region to obtain the square wave then filtering out the desired harmonic as existing frequency triplers do has the disadvantage of low conversion efficiency. Since most power are wasted in the undesired frequency components. Due to its poor conversion efficiency, designers usually have to boost the third harmonic in the following stage which results in more power consumption.

In this work, a novel technique of generating the third-order harmonic is proposed and analyzed. Applying this technique, third-order harmonic can be generated under low power consumption and the circuit itself is uncomplicated compared with other published methods. Furthermore, the proposed technique is especially suitable for the communication systems that utilize I/Q signals for image rejection. Analytical

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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 third-order harmonic and suppress the undesired harmonics.

According to the simulation results, the proposed harmonic current injection frequency tripler (HCI-FT) has -5.6 dB conversion gain under only 2.6 mW dynamic power consumption with fundamental input power of +2 dBm. The proposed HCI-FT is fabricated using 0.18-μm standard CMOS technology for the verification of theoretical results.

In Section 4.2 the analyses relating to the second-order current injection is investigated. The design considerations are mentioned in Section 4.3. The chip implementation and simulation results are presented in Section 4.4. Finally, a summary is given in Section 4.5.

4.2 Second-Order Current Injection

The proposed harmonic current injection frequency tripler (HCI-FT) was developed from the concept shown in Fig. 4.1. We try to generate third-order harmonic using mixers. The easiest way to accomplish our purpose is using two Gilbert cell mixers in cascade, as Fig. 4.2 shows.

Fig. 4.1 Block diagram of generating third-order harmonic

Fig. 4.2 The straightforward implementation of our idea

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The first mixer is used to generate the second-order harmonic for feeding the second mixer. In RF communication circuit design, we are all familiar with Gilbert cell mixer, and it is not difficult to design, however, it seems inefficient to generate the second-order harmonic in this way. There is still some other more efficient and easier ways for generating this desired harmonic, Fig. 4.3 is an example. At the joint of a source-coupled pair, there is an inherent second-order harmonic if the

fundamental signal was applied at the gate terminal. Thus, a simplified and more efficient circuit of implementing the function of the block diagram above is as Fig. 4.4 demonstrates.

Fig. 4.3 A source-coupled pair

Fig. 4.4 Simplified circuit

The second-order harmonic is generated by the source-coupled pair and it propagates to the transconductance stage of the mixer. After current commutating at switching stage, both fundamental and the third-order are generated. Nonetheless, we

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still strive to simplify this circuit, since it takes three steps to generate the final second-order current for commutating as Fig. 4.4 marks.

Fig. 4.5(a) is the fully-simplified circuit we finally come out with. The second-order harmonic generation part is directly folded to the bottom of switching stage in a cascode configuration. Not only it simplifies the circuit complexity, the current-reuse structure also makes it advantageous in low-power circuit implementation.

Fig. 4.5 (a) Fully-simplified circuit and its (b) conceptual circuit

Fig. 4.5(b) shows the conceptual circuit of the fully-simplified circuit. The second-order current is generated by the bottom differential pair, and this current is injected into the upper differential pair. The fundamental input signal at the gate of the upper differential pair makes current commutation; it works as the mixing stage of a conventional mixer. Therefore, the mixing frequency, fo and 3fo are generated at output.

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4.2.1 Output Current versus Injection Phase

To evaluate the feasibility of the fully-simplified circuit in Fig. 4.5(a), we have to identify its mechanism first. For the convenience of analyzing this circuit, we substitute the second-order harmonic generation part into an ideal current sink which draws a DC current and injects an AC current (the second-order current) at the same time, as Fig. 4.6 indicates.

f0 f0

Fig. 4.6 Further simplified for analyzing

A third-order harmonic current is expected at the output (drain terminal) as this second-order current is injected. Let the second-order injection current expresses as:

Iinj=|Iinj|∠Iinj (4.1) where |Iinj| is the injection magnitude, and ∠Iinj is the injection phase.

It is intuitively that the larger injection magnitude would results in larger output third-order current. However, the larger injection magnitude implies the larger fundamental input power is needed. This is definitely not the way for low-power design. So, we are interested in the injection phase. We want to know what happened to the output current under various injection phase.

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(a)

(b)

(c)

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Fig. 4.7 Normalized output current – (a) the fundamental (b) the second-order (c) the third-order

Therefore, in the preliminary investigations, injection magnitude is fixed and output current under various injection phase is observed. Fig. 4.7 shows the results.

The fundamental, the second-order harmonic and the third-order harmonic current are observed at the output. They are normalized to the corresponding harmonic current magnitudes without injection and shown in dB scale, respectively.

From Fig. 4.7(c), the third-order output current at 0º injection phase is 2.7 dB more than that without injection, and is near 7dB more than that of 180º injection phase. Moreover, at 0º injection phase, the undesired fundamental current has its minimum. If we define the harmonic rejection ratio (HRR) as:

HRRn = Desired harmonic

20 log

Undesired n-th order harmonic

× ⎛⎜⎝ ⎠

⎞⎟ (4.2)

In our case, the desired harmonic is the third-order harmonic, and the undesired harmonics are fundamental and the second-order harmonic.

Not only the third-order harmonic current is maximized at 0º injection phase, fundamental current is also minimized at this injection phase, thus the best HRR1. If an improper injection phase was chosen, say ∠Iinj =180º, it would lead to the worst results, in which the third-order harmonic current is minimized and fundamental current is maximized. From the above investigation, the 0º injection phase is the optimal injection phase.

It will be more easily to know what happened to the third-order output current if we partition the circuit shown in Fig. 4.6 into a DC current part (without current injection) and a AC current part (with current injection) as Fig. 4.8 depicts. The part without current injection is simply a differential pair biased by a constant DC current.

As we all know, the transistors of an ideal differential pair draw the bias current alternatively as two switches. Therefore, there is a square-wave-shaped current at its output, which contains the third-order harmonic component. In a word, there exists a

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third-order harmonic current at the output of “Part A” shown in Fig. 4.8 even without current injection. Things will be much clearer if the output current is shown in a complex plane, which enable us to observe magnitude and phase simultaneously. Fig.

4.9(a) demonstrates the overall third-order output current and the third-order output current without current injection. It is evident that the phase of the overall third-order current at ∠Iinj =0º (which we choose to be the optimal injection phase above) is in-phase with the phase that without current injection.

f0 f0

Fig. 4.8 Conceptual circuit partition

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(a) (b)

Fig. 4.9 Normalized (a) overall output third-order current and output third-order current without current injection (b) injected contribution in complex plane

The overall third-order output current (Iout, overall) is the linear combination of that without injection (Iout, w/o injection) and that with injection (Iout, injected). Moreover, it is reasonable to assume that under some injection level, the phase of the third-order output current without current injection is unchanged. Then, the contribution of the injected current can be defined as:

, , , /

out injected out overall out w o injection

I =I −I (4.3)

Fig. 4.8(b) shows the contribution of the injected current. We can conclude that if

∠Iinj =0º, then ∠Iout, injected = ∠Iout, w/o injection, therefore the |Iout, overall| is maximized.

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4.2.2 Injection Node

So far, the output current what we most concern with is analyzed. However, we also wanna know what happened to the injection node X in Fig. 4.6.

A source-coupled pair is inherently to generate a second-order harmonic signal at the joint as Fig. 4.3 shows. Let the VmRef be the reference voltage without second-order injection current. It is reasonable that under some injection level, VmRef is supposedly unchanged or its variation is negligible.

f0 f0

Fig. 4.10 Source-coupled pair with second-order current injection

Re

Fig. 4.11 second-order harmonic voltage of Vm (a) without (b) with second-order

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current injection

Fig. 4.11 shows the second-order harmonic voltage of Vm in complex plane. As mentioned above, under some injection level VmRef is unchanged, thus the resulting voltage Vm’ with second-order current injection is the linear combination of vector VmRef and the vector injected contribution. This is like what we do in Section 4.2.1, which partitions the overall result into an intrinsic part and an extrinsic part.

Fig. 4.12 Normalized third-order output current under different injection level

Fig. 4.12 demonstrates the magnitude of the third-order output current under various injection level with normalized to the non-injection condition in dB scale.

Still, the optimal injection phase is 0º.

Coincidentally, the 0º injection phase results in the maxima of the second-order voltage (Vm’) at the injection node at the same time, and these maxima are in-phase with VmRef. Fig. 4.13 shows the detail in complex plane. The dot which locates on the unit circle is the VmRef, and the injection phases which result in the maxima third-order output current at different injection level are marked with crosses.

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Applying the assumption made above, if the injected contribution is in-phase

with VmRef, a maximum third-order current is obtained at the output.

The injected contribution here is in voltage-domain, however, our input (injection current) is in current-domain. The relationship between them should be explained. Since the node Vm is a low impedance node, the first pole of this node may be far from the frequency of the second-order harmonic if the fundamental frequency is not too high. Therefore, the voltage contributed by the injection current at this node is almost in-phase with the injection current. In fact, the resulting voltage of the injection current has a little negative phase shift since the parasitic capacitances make this node capacitive.

Fig. 4.13 Normalized third-order output current under different injection level in complex plane

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4.2.3 Optimal Injection Phase Calculation

To further verify the simulation results, some paper and hand calculation were done for cross reference. We are not trying to obtain the exact solutions; instead we tried to approximate the numerically-converged results using some analytical equations under some reasonable simplifications and assumptions.

Before calculating the optimal injection phase, some fundamentals should be well-constructed. First of all, in the small-signal case, the transconductance (Gm1) is used to linearly characterize the current variation at the vicinity of the bias point of a transistor. It is convenient to express the I/V relationship as:

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1 gs

d m

i =G ×v (4.4)

where id is the small-signal drain current and the vgs is the small-signal input voltage.

However, the expression in (4.4) is implicit, since Gm1 contributes not only a magnitude scaling from vgs to id, but also introduces a phase shift [19]-[20], as (4.5) indicates. That is, Gm1 is a complex in practice (see Fig. 4.8(a)).

0 1 information is especially important in our calculations, since we are handling with injection phase.

Secondly, phasor is often utilized in the calculation of linear system. Thus, for convenience, (4.5) can be expressed as (4.6) shows.

1 (1 )

Phasor would be helpful while manipulating sinusoidal signals in a linear system.

Finally, in the large-signal case, power series is often used for characterizing some nonlinear effects. The power-series approach is useful in some instances and it gives the designers a good intuitive sense of the behavior of many types of nonlinear circuits. The I/V relationship can be further expressed as:

2 3 4 5

1 2 gs 3 gs 4 gs 5 gs

d m gs m m m m

i =G ×v +G ×v +G ×v +G ×v +G ×v +L (4.7) Gm2, Gm3 and other high-order terms should be involved to characterize the nonlinear relationship between id and vgs. Also, they are complexes which would introduce phase shifts as Gm1 does.

The normalized Gm1, Gm2, and Gm3 under various gate biases at 1GHz are shown in Fig. 4.14. The X-axis shows the real part, and the Y-axis shows the imaginary part.

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(a)

(b)

(c)

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Fig. 4.14 Normalized (a) Gm1 (b) Gm2 (c) Gm3 under various gate biases

Furthermore, Gm1, Gm2, and Gm3 are functions of frequency as well. Both their magnitudes and phases are different at different frequency, Fig. 4.15 demonstrates this fact.

1GHz 10GHz 20GHz

(a)

1GHz 10GHz 20GHz

(b)

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frequencies. The X-axis shows the real part and the Y-axis shows the imaginary part.

It is obvious that at different bias or frequency both the magnitude and phase response are different. This should be kept in mind while doing calculation.

After constructing some bases above, we are going to derive the optimal injection phase for the simplified circuit shown in Fig. 4.6. The calculations here are in voltage-domain, so the final result would be optimal “voltage” injection phase.

However, as we explained in the last of Section 4.2.2, the phase of the injection voltage is of the same phase with the injection current. Therefore, if we got the optimal voltage injection phase, we got the optimal current injection phase.

In addition, we calculate only the contribution on the output third-order current from the injection current, as the “Part B” in Fig. 4.8 indicates. Since we know from the previous sections that, if this contribution from certain injection phase is in-phase

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with the intrinsic one, as the “Part A” in Fig. 4.8 indicates, then this injection phase is the optimal injection phase.

At the gate of the upper differential pair, there is a fundamental signal with magnitude A, and a second-order signal with magnitude B and a phase advance Φ is injected at the source, thus:

( ) cos

Applying (4.7) to calculate the output current, abundant harmonic frequencies would emerge. But what we concern about is the third-order harmonic, as (4.9) demonstrates.

Phasors are commonly used in the linear system. In a nonlinear system, phasors are failed to deal with the numerous harmonic frequencies. In our case, however, the third-order harmonic is especially picked out as if there are no other frequencies in our system. So, phasor could be used for manipulating this “single frequency” system.

Let

be the i-th order transconductance of the transistor.

Then,

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2

If, the order higher than five are discarded, and let X be the phasor of the third-order harmonic in Iout, injected:

3 3

Since the injection node is a low-impedance node, it is reasonable to assume that A>>B. Several terms could be neglected to simplify our calculation. Furthermore, it is intuitive that desired third-order current is mainly generated by the term, Gm2, since the upper differential pair serves as a switching stage in a mixer. Therefore, to generate the desired third-order current efficiently, Gm2 should be maximized. If Gm2

is chosen to be its local maximum, then Gm3 would be zero. This further indicates that, Gm4 is also at its local maximum and Gm5 is zero. This may not be exactly true in practice, but it is a reasonable assumption to simplify our calculation. (4.12) is simplified as (4.13) indicates:

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( 2 ) 3 ( By substituting the parameters (including, Gm2, Gm4, A, and B) extracted from the simplified circuit in Fig. 4.6 into (4.16), the phasor X versus injection phase Φ can be plotted.

Fig. 4.16(a) shows the calculated phasor X (or the calculated output third-order current excited by the injection current) versus injection phase Φ in complex plane with normalized to the maximum magnitude of simulated result. The magnitudes of phasor X under various injection phases are uniform. At 0º injection phase, the phasor X locates on the minus X-axis, which is almost the same result as we derived in Section 4.2.1.

Fig. 4.16(b) gives the simulated result for comparison. The calculated result shows great agreement on the resulting phase versus injection phase Φ to the simulated result. The calculated result has 27% error refer to the simulated result;

however, the exact value is not what we concern about. Both the discrepancies in value and shape (circle and ellipse) can be due to the simplifications we made during the calculation.

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(a)

(b)

Fig. 4.16 Normalized (a) calculated and (b) simulated output third-order current excited by the injection current

4.2.4 Injection-to-Bias Ratio

In the above analyses, the optimal injection phase at some specific Ibias and |Iinj| is derived. But will the optimal injection phase be the same at other Ibias or |Iinj|? Sweeps on Ibias and |Iinj| were done to answer this question.

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However, it will lead to an unfair condition, say if |Iinj| was fixed, and Ibias was swept. With the increment of the DC current Ibias, the AC component |Iinj| would become more and more negligible.

So, first of all, the parameter “injection-to-bias ratio” should be defined for fair comparison.

Injection-to-bias ratio : ^inj

bias

I

α ≡ I (4.14)

Fig. 4.17 shows the normalized magnitude of the third-order output current versus injection phase from α=0 to α=0.4. The maximum at different α is still located on 0º injection phase. Fig. 4.18 provides the normalized third-order output current in complex plane, each ellipse show the third-order output current from 0º to 360º injection phase. α=0 to α=0.4 form the concentric ellipses from inside out.

Fig. 4.17 Normalized magnitude of the third-order output current versus injection phase from α=0 to α=0.4

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Fig. 4.18 Normalized third-order output current versus injection phase in complex plane from α=0 to α=0.4

Fig. 4.19 shows the normalized magnitude of the third-order output current versus injection phase from Ibias=1 mA to Ibias=3 mA. The maximum under different Ibias is still located on 0º injection phase. Fig. 4.20 provides the normalized third-order output current in complex plane, each ellipse show the third-order output current from 0º to 360º injection phase. Ibias=1 mA to Ibias=3 mA form the interlaced ellipses from left to right. We can conclude that the optimal injection phase is invariant at different α and Ibias.

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Fig. 4.19 Normalized magnitude of third-order output current versus injection phase from Ibias=1mA to Ibias=3mA

Fig. 4.20 Normalized third-order output current versus injection phase in complex plane from Ibias=1mA to Ibias=3mA

Also, the output current HRRs are investigated as Fig. 4.21 and Fig. 4.22 demonstrate. We can tell from theses two contour plots that the larger α results in the better HRR1 and poorer HRR2, and the smaller α results in the poorer HRR1 and better

Also, the output current HRRs are investigated as Fig. 4.21 and Fig. 4.22 demonstrate. We can tell from theses two contour plots that the larger α results in the better HRR1 and poorer HRR2, and the smaller α results in the poorer HRR1 and better

在文檔中 使用三倍頻器產生本地震盪訊號的升頻混波器 (頁 51-0)