A Simple Model for DoD Inkjet Frequency Response

A simple linear model of piezo DoD inkjet print-head jetting output (drop speed, volume, momentum) provides an analytic prediction for the frequency response for steady state and initial printing streams from nozzles. The model has been applied to both existing commercial and development inkjet print-head devices. Introduction As a result of the industrial marketing push towards ever higher inkjet printing (jetting) frequencies and smaller drop sizes, and the rapid progress being made in applications of MEMS-based manufacturing, some drop-on-demand (DoD) inkjet print head designs rely on piezo-actuated (driven) resonant chambers to generate liquid droplets. The resonant oscillations set up residual pressure waves as a result of simple actuation drive pulses, e.g. see Bogy and Talke (1984) [1], Dijksman (1984) [2], Dijksman and Pierik (2012) [3] and these have to be tamed by a combination of print-head design and/or by waveform modifications, e.g. Wijshoff (2010) [4], Khalate et al (2011) [5] and (2012) [6], and as described within [7]. We will focus here on the piezo DoD printhead output, which for most inkjet users is the drop speed and the drop volume that is achieved for a specific range of frequencies. Residual response Piezo-based DoD print-heads are generally actuated by means of piezoelectric materials to which voltage is applied (or removed). For resonant devices, the print-heads include main chambers with inlets and outlets for ink supply and jetting nozzles. In the simplest case, in which refill modes are ignored, the main chamber and jet nozzle can be represented as a single Helmholtz resonator having a frequency fH and Q-factor Q. Print-heads have mechanically (and fluidic) coupled arrays of such chambers and both jetting and nonjetting nozzles, so finite cross-talk between neighboring channels can seriously disrupt the output of a specific channel (or pin#). Using the well-known exponentially decaying form of residual waves associated with a damped single mode resonant chamber after completion of a (piezo-driven) drop-on-demand print-head actuation pulse, the effect of any number of similar pulses can be computed analytically, in the absence of any dynamic changes, by simple summation of terms, as found for linear superposition of waves in many other physical systems. This simple method completely ignores the details of the actuation pulse duration, but changes to the shape of the frequency response introduced by more realistic representations of the “waveform” are found to be rather small. Therefore the simple model results presented here provide a standard basis to benchmark DoD inkjet print-head designs. Multi-pulse train modelling is deceptively simple in principle, because it assumes a linear response. There is plenty of evidence for such behavior in the academic literature [8, 9] reporting DoD drop speeds and drop volumes are linear in the piezo drive voltage. The multi-pulse trains can be single pulses repeated at a constant frequency, or structured pulses having ‘grayscale’ sub-drops with fixed time spacing that are also repeated at a constant frequency. Compensation pulses and modified pulse amplitudes or structured timing within a ‘single’ pulse, and influence of multi-pulse trains on the other nozzle residuals (‘cross-talk’) can be easily modelled by appropriate adaptations of the multi-pulse train approach. To understand experimental frequency sweep responses of inkjet print-heads first assume that the print-head output behavior (drop speed and volume) is linear in the applied drive voltage, as ink drops are known to behave like this at low print rates for DoD print heads from a wide range of manufacturers. The linear models predict print-head frequency dependence by making summations over suitable multi-pulse trains. The drop speed and volume are then proportional to drive voltage and each other at all frequencies. Whatever the precise pulse-train formation, it appears that the multi-pulse train approach allows very simple, calculable and exact analytic prediction results for steady state and ‘first drop’ behavior over all frequencies. Thus the predictive power of the model derives from the linear assumption and exact results for the (normalized) drop speed and volumes jetted by chambers with independently specified fH and Q. From this baseline, experimental results probe and measure the assumptions of linearity, fH and Q; they have already been used to help identify some early prototype print-head build quality issues. Furthermore the linear multi-pulse train model explains many features of the observed drop speed and drop volume over the whole range of print frequencies up to and in some instances beyond fH. One early success of the multi-pulse train model showed that the actual response to a drive voltage with a unimodal pulse duration measured by the pulse width at half height (PW) did correspond closely to the optimum value (OPW) for the resonant chamber: OPW=1⁄2/fH even when PW ≠ OPW. This experimental fact helped simplify the mathematics (presented below) but is not intrinsic to the linear assumption and the physics of the chamber response to being driven off-resonance: driving at (or close to) peak requires lower drive voltage than that needed for the same output off-resonance. Likewise the Q-factor influences the efficiency of the piezo-driven print-head, but given the value of Q (and fH) does not influence the form of the frequency response. Derivation The resonant response of the DoD print-head to single 1 dpd (sub-drop per drop) pulse excitation is assumed to be proportional to a single, exponentially-damped, cosine term after a time t: cos(2πfHt)exp(-πfHt/Q) (1) The cosine term in (1) shows that the response involves the chamber resonant frequency fH. The πfH/Q in (1) is controlled by the damping factor ζ=1/(2Q), where 0 ≤ ζ < 1 will permit residual oscillations. For damping ζ with a low Q > 1⁄2, the oscillation 8 © 2015 Society for Imaging Science and Technology frequency is lowered to fH√(1-ζ2), but in piezo DoD inkjet printheads the damping correction to this frequency can be neglected. The residual decays to e-π ≈ 4.32% after a time equivalent to the ‘flat background’ frequency B = fH/Q. Since inkjet print-head drop speed specifications are typically ±5% or better, B is a reasonable estimate for the expected low frequency range for a flat response, although a lower flat frequency band may be anticipated (B/2) for precision applications. Q is approximately the number of oscillation cycles occurring before the residual response dies away to e-π ≈ 4.32%. Figure1 displays the decay with time for case Q=9; Figure 2 shows the steady state frequency response for this case and Figure 3 compares these for cases Q=3, 6 and 9 according to the multi-pulse train results predicted by equation (2) below. Actuation responses are summed linearly over all the earlier decaying responses (see Appendix below for the underlying math); for multi-pulse trains of frequency f, response R(f) is given in the steady state limit, with a= 2πfH√(1-1/(2Q)2)/f and b=πfH/(Qf), by R(f)=(1-cos(a)exp(-b))/(1+exp(-2b)-2cos(a)exp(-b)) (2) Figure 1. Decay of residual response for Q=9 during time after a single pulse. Dimensionless time units (fH=1) are used so that one cycle takes 1 time unit. Horizontal limit lines at ±exp(-π) represent some typical speed specifications. Figure 2. Predicted frequency response for DoD print-head system of Figure 1. Flat bandwidth B = 1/9 for frequency unit fH=1. Response can be interpreted as drop speed or volume (= 1 unit at low f). Typical drop speed ranges are shown. Figure 3. Jetting frequency response modelled for a resonant system of natural frequency fH =1 & Q-factors of 3, 6 or 9 (ζ = 1/6, 1/12 or 1/18, respectively). The efficiency for driving the resonant chamber at sub-harmonic peaks, rather than at lower frequencies, evidently increases with Q-factor, but drop response variations are also increased. (Note suppressions for clarity: the response axis 0.0 and small shifts for the low Q peak frequencies due to the factor √(1-ζ2).) Figure 4. Predicted Nth drop frequency response for the print-head of Figure 1. The response for N=1 is flat, but as N increases the frequency spectrum will build systematically towards the steady state limit shown. For Q=9 as Figure 1. At frequencies below the m=2 sub-harmonic of fH, the main changes predicted will arise due to the difference between the first and second drop