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