The Rectangular Wave

Note that this is in terms of angles. Firstly, to avoid confusing the symbol x with distance (distance being also often called x ), it helps to first change it to a more familiar symbol for angle, like 0. We can then also use the summation sign to write the above expansion in a more compact form  

applying our conversion rule to go from angles to time (time being called the t-plane here), we get 

The cn in this equation can also be explicitly written out, using the same conversion rule (from angle to time). We must remember though, that 2a in the 0-plane corresponds 

We have provided the full expansion of the voltage on the drain of the mosfet (ignoring the leakage spike and possible ringing). 

Note that we now have multiplied all the unity height cn s by the actual wave amplitude (since we know that all the coefficients scale accordingly). Also, the first term is the average value of the waveform. Note that we could also have written this expansion with alternating signs as follows  

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However, the Mathieu parameters a and q, defined by Eqs. 2.9 and 2.10, can still be employed for the description of the DIT theoretical stability diagram, considering the U and V values as the average values of the dc and ac components of the rectangular wave voltage applied to the intermediate electrode. They are defined as... 

As shown in Fig. 2.23, the duty cycle d is described as the ratio between x and the total period T of the rectangular wave. Unlike the sinusoidal wave, the rectangular wave can be generated with different pulsing times for Fi and V2 produced by the digital circuitry. In a DIT, the az value is a function of both the dc offset and the duty cycle d. In fact, the dc component U can be generated either by an imbalance between Vi and V2 or by variation of the duty cycle d. 

With this approach, first the V values (usually 103V) are lower than that employed in QIT (usually in the 104V range) second the ions are ejected from the trap by scanning the rectangular wave frequency. 

Now we take the rectangular wave discussed above and make it a little more realistic by introducing nonzero rise and fall times. By a similar procedure as for a rectangular wave, we can get the following equation (for the case of equal rise and fall times) ... 

The very simple approaches shown in Figures 8.31 and 8.32 rely on some assumptions. First, the rectangular wave is nearly ideal with instantaneous rise and quite stable during a half period. Second, the electrodes are big enough to keep electrode polarization at a minimum. 

In seeking a relationship for the value of the MTF for a system, we consider the contribution from a single component and offer the output as the input for the next component in the system chain. Hence, the incoherent image I x) of the rectangular wave grating, formed by convolution of the object Lijc) with the line-spread function h x) of the first component, is given by... 

Fig. 14. Calculated scattering intensity in the rectangular-wave model, eqs. (16) and (17), with Nj =Nj = 3 ...

Fig. 17. Schematic diagram of the rectangular wave model, showing (from top to bottom) the modulation of c-axis lattice spacing, the scattering factor, the turn angle, and the resulting total phase advance.

Ding, L. Kumashiro, S. Ion Motion in the Rectangular Wave Quadrupole Field and Digital Operation Mode of a Quadrupole Ion Trap Mass Spectrometer. Rapid Commun. Mass Spectrom. 2006, 20,3-8. 

At a frequency of 5 Hz the burst-time depended on the potential of the rectangular wave. When the largest potential was set at +0.8 V, the negative potential was closely related to the time for cell-burst. For example, when the modulation was set at 0.8 V and -0.4 V, a very long time (144 s) was required for the cell-burst. However, when the two potentials were 0.8 V and -0.8 V, the cell-burst occurred within 4 s. On the other hand, a less remarkable effect of the potential application on HeLa cells was observed when the positive potential was changed. By keeping the negative potential at -0.8 V, the burst-time was 45 s at a positive potential of 0.4 V. The increase of the positive potential caused the decrease in the burst time. 

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