Patent Publication Number: US-2021174777-A1

Title: Methods and systems for estimating coil impedance of an electromagnetic transducer

Description:
RELATED APPLICATION 
     The present disclosure claims priority to U.S. Provisional Patent Application Ser. No. 62/944,090, filed Dec. 5, 2019, which is incorporated by reference herein in its entirety. 
     FIELD OF DISCLOSURE 
     The present disclosure relates in general to estimating a coil impedance of an electromagnetic transducer, in particular a haptic transducer. 
    
    
     BACKGROUND 
     Vibro-haptic transducers, for example linear resonant actuators (LRAs), are widely used in portable devices such as mobile phones to generate vibrational feedback to a user. Vibro-haptic feedback in various forms creates different feelings of touch to a user&#39;s skin, and may play increasing roles in human-machine interactions for modern devices. 
     An LRA may be modelled as a mass-spring electro-mechanical vibration system. When driven with appropriately designed or controlled driving signals, an LRA may generate certain desired forms of vibrations. For example, a sharp and clear-cut vibration pattern on a user&#39;s finger may be used to create a sensation that mimics a mechanical button click. This clear-cut vibration may then be used as a virtual switch to replace mechanical buttons. 
       FIG. 1  illustrates an example of a vibro-haptic system in a device  100 . Device  100  may comprise a controller  101  configured to control a signal applied to an amplifier  102 . Amplifier  102  may then drive a haptic transducer  103  based on the signal. Controller  101  may be triggered by a trigger to output to the signal. The trigger may for example comprise a pressure or force sensor on a screen or virtual button of device  100 . 
     Among the various forms of vibro-haptic feedback, tonal vibrations of sustained duration may play an important role to notify the user of the device of certain predefined events, such as incoming calls or messages, emergency alerts, and timer warnings, etc. In order to generate tonal vibration notifications efficiently, it may be desirable to operate the haptic actuator at its resonance frequency. 
     The resonance frequency f 0  of a haptic transducer may be approximately estimated as: 
     
       
         
           
             
               
                 
                   
                     f 
                     0 
                   
                   = 
                   
                     1 
                     
                       2 
                       * 
                       π 
                       * 
                       
                         
                           C 
                           * 
                           M 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where C is the compliance of the spring system, and M is the equivalent moving mass, which may be determined based on both the actual moving part in the haptic transducer and the mass of the portable device holding the haptic transducer. 
     Due to sample-to-sample variations in individual haptic transducers, mobile device assembly variations, temporal component changes caused by aging, and use conditions such as various different strengths of a user gripping of the device, the vibration resonance of the haptic transducer may vary from time to time. 
       FIG. 2A  illustrates an example of a linear resonant actuator (LRA) modelled as a linear system including a mass-spring system  201 . LRAs are non-linear components that may behave differently depending on, for example, the voltage levels applied, the operating temperature, and the frequency of operation. However, these components may be modelled as linear components within certain conditions. 
       FIG. 2B  illustrates an example of an LRA modelled as a linear system, including an electrically equivalent model of mass-spring system  201  of LRA. In this example, the LRA is modelled as a third order system having electrical and mechanical elements. In particular, Re and Le are the DC resistance and coil inductance of the coil-magnet system, respectively; and Bl is the magnetic force factor of the coil. The driving amplifier outputs the voltage waveform V(t) with the output impedance Ro. The terminal voltage V T  (t) may be sensed across the terminals of the haptic transducer. The mass-spring system  201  moves with velocity u(t). 
     An electromagnetic load such as an LRA may be characterized by its impedance Z LRA  as seen as the sum of a coil impedance  4   011  and a mechanical impedance Z mech : 
         Z   LRA   =Z   coil   +Z   mech   (2)
 
     Coil impedance Z coil  may in turn comprise a direct current (DC) resistance Re in series with an inductance Le: 
         Z   coil   =Re+s*Le   (3)
 
     Mechanical impedance Z mech  may be defined by three parameters including the resistance at resonance R RES  representing an electrical resistance representative of mechanical friction of the mass-spring system of the haptic transducer, a capacitance C MES  representing an electrical capacitance representative of an equivalent moving mass M of the mass-spring system of the haptic transducer, and inductance L CES  representative of a compliance C of the mass-spring system of the haptic transducer. The electrical equivalent of the total mechanical impedance is the parallel connection of R RES , C MES , L CES . The Laplace transform of this parallel connection is described by: 
     
       
         
           
             
               
                 
                   
                     
                       Z 
                       mech 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     1 
                     
                       ( 
                       
                         
                           1 
                           
                             R 
                             RES 
                           
                         
                         + 
                         
                           1 
                           
                             
                               L 
                               CES 
                             
                             * 
                             s 
                           
                         
                         + 
                         
                           
                             C 
                             MES 
                           
                           * 
                           s 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The resonant frequency f 0  of the haptic transducer can be represented as: 
     
       
         
           
             
               
                 
                   
                     f 
                     0 
                   
                   = 
                   
                     1 
                     
                       ( 
                       
                         2 
                         * 
                         π 
                         * 
                         
                           
                             
                               L 
                               CES 
                             
                             * 
                             
                               C 
                               MES 
                             
                             * 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The quality factor Q of the LRA can be represented as: 
     
       
         
           
             
               
                 
                   Q 
                   = 
                   
                     
                       
                         
                           R 
                           RES 
                         
                         * 
                         Re 
                       
                       
                         
                           R 
                           RES 
                         
                         + 
                         Re 
                       
                     
                     * 
                     
                       
                         
                           C 
                           MES 
                         
                         
                           L 
                           CES 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Referring to equation (6), it may appear non-intuitive that the expression involves a subexpression describing the parallel connection of resistances Re and R RES   
     
       
         
           
             ( 
             
               
                 i 
                 . 
                 e 
                 . 
               
               , 
               
                 
                   
                     R 
                     RES 
                   
                   * 
                   Re 
                 
                 
                   
                     R 
                     RES 
                   
                   + 
                   Re 
                 
               
             
             ) 
           
         
       
     
     while in  FIG. 2B  these resistances are shown in a series connection. However, such may be the case where a driving voltage Ve is oscillating but then abruptly turns off and goes to zero. The voltage amplifier shown in  FIG. 2B  may be considered to have a low source impedance, ideally zero source impedance. Under these conditions, when driving voltage Ve goes to zero, the voltage amplifier effectively disappears from the circuit. At that point, the top-most terminal of resistance Re in  FIG. 2B  is grounded as is the bottom-most terminal of resistance R RES  and so resistances Re and R REs  are indeed connected in parallel as reflected in equation (6). 
     Electromagnetic transducers, such as LRAs or microspeakers, may have slow response times.  FIG. 3  is a graph of an example response of an LRA, depicting an example driving signal to the LRA, a current through the LRA, and a back electromotive force (back EMF) of the LRA, wherein such back EMF may be proportional to the velocity of a moving element (e.g., coil or magnet) of the transducer. As shown in  FIG. 3 , the attack time of the back EMF may be slow as energy is transferred to the LRA, and some “ringing” of the back EMF may occur after the driving signal has ended as the mechanical energy stored in the LRA is discharged. In the context of a haptic LRA, such behavioral characteristic may result in a “mushy” feeling click or pulse, instead of a “crisp” tactile response. Thus, it may be desirable for an LRA to instead have a response similar to that shown in  FIG. 4 , in which there exists minimal ringing after the driving signal has ended, and which may provide a more “crisp” tactile response in a haptic context. Accordingly, it may be desirable to apply processing to a driving signal such that when the processed driving signal is applied to the transducer, the velocity or back EMF of the transducer more closely approaches that of  FIG. 4 . 
     SUMMARY 
     In accordance with the teachings of the present disclosure, the disadvantages and problems associated with estimating a coil resistance of an electromagnetic transducer may be reduced or eliminated. 
     In accordance with embodiments of the present disclosure, a method may include selecting, based on a condition of an electromagnetic load, a selected measurement technique from a plurality of impedance measurement techniques for measuring an impedance of the electromagnetic load and performing the selected measurement technique to generate an estimate of the impedance of the electromagnetic load. 
     In accordance with these and other embodiments of the present disclosure, a system for estimating impedance of an electromagnetic load may be configured to select, based on a condition of the electromagnetic load, a selected measurement technique from a plurality of impedance measurement techniques for measuring an impedance of the electromagnetic load and perform the selected measurement technique to generate an estimate of the impedance of the electromagnetic load. 
     In accordance with these and other embodiments of the present disclosure, a host device may include an electromagnetic load and a subsystem coupled to the electromagnetic load and configured to select, based on a condition of the electromagnetic load, a selected measurement technique from a plurality of impedance measurement techniques for measuring an impedance of the electromagnetic load and perform the selected measurement technique to generate an estimate of the impedance of the electromagnetic load. 
     Technical advantages of the present disclosure may be readily apparent to one having ordinary skill in the art from the figures, description and claims included herein. The objects and advantages of the embodiments will be realized and achieved at least by the elements, features, and combinations particularly pointed out in the claims. 
     It is to be understood that both the foregoing general description and the following detailed description are examples and explanatory and are not restrictive of the claims set forth in this disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete understanding of the present embodiments and advantages thereof may be acquired by referring to the following description taken in conjunction with the accompanying drawings, in which like reference numbers indicate like features, and wherein: 
         FIG. 1  illustrates an example of a vibro-haptic system in a device, as is known in the art; 
         FIGS. 2A and 2B  each illustrate an example of a Linear Resonant Actuator (LRA) modelled as a linear system, as is known in the art; 
         FIG. 3  illustrates a graph of example waveforms of an electromagnetic load, as is known in the art; 
         FIG. 4  illustrates a graph of desirable example waveforms of an electromagnetic load, in accordance with embodiments of the present disclosure; 
         FIG. 5  illustrates a block diagram of selected components of an example mobile device, in accordance with embodiments of the present disclosure; 
         FIG. 6  illustrates a block diagram of selected components of an example integrated haptic system, in accordance with embodiments of the present disclosure; 
         FIG. 7  illustrates an example system for improving transducer dynamics, in accordance with embodiments of the present disclosure; 
         FIG. 8  illustrates an example of a linear resonant actuator (LRA) modelled as a linear system and including a negative resistance, in accordance with embodiments of the present disclosure; 
         FIG. 9  illustrates a flow chart for example operation of a haptic state machine, in accordance with embodiments of the present disclosure; 
         FIG. 10  illustrates a flow chart for example operation of a haptic state machine when using a thermal model for coil impedance estimation, in accordance with embodiments of the present disclosure; 
         FIG. 11  illustrates a flow chart for example operation of a haptic state machine when using background calibration for coil impedance estimation, in accordance with embodiments of the present disclosure; and 
         FIGS. 12A and 12B  (which may be referred to herein as together as “ FIG. 12 ”) illustrate a table which summarizes the possible voice coil impedance estimation approaches that the haptic state machine may employ, in accordance with the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The description below sets forth example embodiments according to this disclosure. Further example embodiments and implementations will be apparent to those having ordinary skill in the art. Further, those having ordinary skill in the art will recognize that various equivalent techniques may be applied in lieu of, or in conjunction with, the embodiment discussed below, and all such equivalents should be deemed as being encompassed by the present disclosure. 
     Various electronic devices or smart devices may have transducers, speakers, and acoustic output transducers, for example any transducer for converting a suitable electrical driving signal into an acoustic output such as a sonic pressure wave or mechanical vibration. For example, many electronic devices may include one or more speakers or loudspeakers for sound generation, for example, for playback of audio content, voice communications and/or for providing audible notifications. 
     Such speakers or loudspeakers may comprise an electromagnetic actuator, for example a voice coil motor, which is mechanically coupled to a flexible diaphragm, for example a conventional loudspeaker cone, or which is mechanically coupled to a surface of a device, for example the glass screen of a mobile device. Some electronic devices may also include acoustic output transducers capable of generating ultrasonic waves, for example for use in proximity detection type applications and/or machine-to-machine communication. 
     Many electronic devices may additionally or alternatively include more specialized acoustic output transducers, for example, haptic transducers, tailored for generating vibrations for haptic control feedback or notifications to a user. Additionally or alternatively, an electronic device may have a connector, e.g., a socket, for making a removable mating connection with a corresponding connector of an accessory apparatus, and may be arranged to provide a driving signal to the connector so as to drive a transducer, of one or more of the types mentioned above, of the accessory apparatus when connected. Such an electronic device will thus comprise driving circuitry for driving the transducer of the host device or connected accessory with a suitable driving signal. For acoustic or haptic transducers, the driving signal will generally be an analog time varying voltage signal, for example, a time varying waveform. 
       FIG. 5  illustrates a block diagram of selected components of an example host device  502 , in accordance with embodiments of the present disclosure. As shown in  FIG. 5 , host device  502  may comprise an enclosure  501 , a controller  503 , a memory  504 , a force sensor  505 , a microphone  506 , a linear resonant actuator  507 , a radio transmitter/receiver  508 , a speaker  510 , and an integrated haptic system  512 . 
     Enclosure  501  may comprise any suitable housing, casing, or other enclosure for housing the various components of host device  502 . Enclosure  501  may be constructed from plastic, metal, and/or any other suitable materials. In addition, enclosure  501  may be adapted (e.g., sized and shaped) such that host device  502  is readily transported on a person of a user of host device  502 . Accordingly, host device  502  may include but is not limited to a smart phone, a tablet computing device, a handheld computing device, a personal digital assistant, a notebook computer, a video game controller, or any other device that may be readily transported on a person of a user of host device  502 . 
     Controller  503  may be housed within enclosure  501  and may include any system, device, or apparatus configured to interpret and/or execute program instructions and/or process data, and may include, without limitation a microprocessor, microcontroller, digital signal processor (DSP), application specific integrated circuit (ASIC), or any other digital or analog circuitry configured to interpret and/or execute program instructions and/or process data. In some embodiments, controller  503  interprets and/or executes program instructions and/or processes data stored in memory  504  and/or other computer-readable media accessible to controller  503 . 
     Memory  504  may be housed within enclosure  501 , may be communicatively coupled to controller  503 , and may include any system, device, or apparatus configured to retain program instructions and/or data for a period of time (e.g., computer-readable media). Memory  504  may include random access memory (RAM), electrically erasable programmable read-only memory (EEPROM), a Personal Computer Memory Card International Association (PCMCIA) card, flash memory, magnetic storage, opto-magnetic storage, or any suitable selection and/or array of volatile or non-volatile memory that retains data after power to host device  502  is turned off. 
     Microphone  506  may be housed at least partially within enclosure  501 , may be communicatively coupled to controller  503 , and may comprise any system, device, or apparatus configured to convert sound incident at microphone  506  to an electrical signal that may be processed by controller  503 , wherein such sound is converted to an electrical signal using a diaphragm or membrane having an electrical capacitance that varies as based on sonic vibrations received at the diaphragm or membrane. Microphone  506  may include an electrostatic microphone, a condenser microphone, an electret microphone, a microelectromechanical systems (MEMs) microphone, or any other suitable capacitive microphone. 
     Radio transmitter/receiver  508  may be housed within enclosure  501 , may be communicatively coupled to controller  503 , and may include any system, device, or apparatus configured to, with the aid of an antenna, generate and transmit radio-frequency signals as well as receive radio-frequency signals and convert the information carried by such received signals into a form usable by controller  503 . Radio transmitter/receiver  508  may be configured to transmit and/or receive various types of radio-frequency signals, including without limitation, cellular communications (e.g., 2G, 3G, 4G, LTE, etc.), short-range wireless communications (e.g., BLUETOOTH), commercial radio signals, television signals, satellite radio signals (e.g., GPS), Wireless Fidelity, etc. 
     A speaker  510  may be housed at least partially within enclosure  501  or may be external to enclosure  501 , may be communicatively coupled to controller  503 , and may comprise any system, device, or apparatus configured to produce sound in response to electrical audio signal input. In some embodiments, a speaker may comprise a dynamic loudspeaker, which employs a lightweight diaphragm mechanically coupled to a rigid frame via a flexible suspension that constrains a voice coil to move axially through a cylindrical magnetic gap. When an electrical signal is applied to the voice coil, a magnetic field is created by the electric current in the voice coil, making it a variable electromagnet. The coil and the driver&#39;s magnetic system interact, generating a mechanical force that causes the coil (and thus, the attached cone) to move back and forth, thereby reproducing sound under the control of the applied electrical signal coming from the amplifier. 
     Force sensor  505  may be housed within enclosure  501 , and may include any suitable system, device, or apparatus for sensing a force, a pressure, or a touch (e.g., an interaction with a human finger) and generating an electrical or electronic signal in response to such force, pressure, or touch. In some embodiments, such electrical or electronic signal may be a function of a magnitude of the force, pressure, or touch applied to the force sensor. In these and other embodiments, such electronic or electrical signal may comprise a general purpose input/output signal (GPIO) associated with an input signal to which haptic feedback is given. Force sensor  505  may include, without limitation, a capacitive displacement sensor, an inductive force sensor (e.g., a resistive-inductive-capacitive sensor), a strain gauge, a piezoelectric force sensor, a force sensing resistor, a piezoelectric force sensor, a thin film force sensor, or a quantum tunneling composite-based force sensor. For purposes of clarity and exposition in this disclosure, the term “force” as used herein may refer not only to force, but to physical quantities indicative of force or analogous to force, such as, but not limited to, pressure and touch. 
     Linear resonant actuator  507  may be housed within enclosure  501 , and may include any suitable system, device, or apparatus for producing an oscillating mechanical force across a single axis. For example, in some embodiments, linear resonant actuator  507  may rely on an alternating current voltage to drive a voice coil pressed against a moving mass connected to a spring. When the voice coil is driven at the resonant frequency of the spring, linear resonant actuator  507  may vibrate with a perceptible force. Thus, linear resonant actuator  507  may be useful in haptic applications within a specific frequency range. While, for the purposes of clarity and exposition, this disclosure is described in relation to the use of linear resonant actuator  507 , it is understood that any other type or types of vibrational actuators (e.g., eccentric rotating mass actuators) may be used in lieu of or in addition to linear resonant actuator  507 . In addition, it is also understood that actuators arranged to produce an oscillating mechanical force across multiple axes may be used in lieu of or in addition to linear resonant actuator  507 . As described elsewhere in this disclosure, a linear resonant actuator  507 , based on a signal received from integrated haptic system  512 , may render haptic feedback to a user of host device  502  for at least one of mechanical button replacement and capacitive sensor feedback. 
     Integrated haptic system  512  may be housed within enclosure  501 , may be communicatively coupled to force sensor  505  and linear resonant actuator  507 , and may include any system, device, or apparatus configured to receive a signal from force sensor  505  indicative of a force applied to host device  502  (e.g., a force applied by a human finger to a virtual button of host device  502 ) and generate an electronic signal for driving linear resonant actuator  507  in response to the force applied to host device  502 . Detail of an example integrated haptic system in accordance with embodiments of the present disclosure is depicted in  FIG. 6 . 
     Although specific example components are depicted above in  FIG. 5  as being integral to host device  502  (e.g., controller  503 , memory  504 , force sensor  506 , microphone  506 , radio transmitter/receiver  508 , speakers(s)  510 ), a host device  502  in accordance with this disclosure may comprise one or more components not specifically enumerated above. For example, although  FIG. 5  depicts certain user interface components, host device  502  may include one or more other user interface components in addition to those depicted in  FIG. 5  (including but not limited to a keypad, a touch screen, and a display), thus allowing a user to interact with and/or otherwise manipulate host device  502  and its associated components. 
       FIG. 6  illustrates a block diagram of selected components of an example integrated haptic system  512 A, in accordance with embodiments of the present disclosure. In some embodiments, integrated haptic system  512 A may be used to implement integrated haptic system  512  of  FIG. 5 . As shown in  FIG. 6 , integrated haptic system  512 A may include a digital signal processor (DSP)  602 , a memory  604 , and an amplifier  606 . 
     DSP  602  may include any system, device, or apparatus configured to interpret and/or execute program instructions and/or process data. In some embodiments, DSP  602  may interpret and/or execute program instructions and/or process data stored in memory  604  and/or other computer-readable media accessible to DSP  602 . 
     Memory  604  may be communicatively coupled to DSP  602 , and may include any system, device, or apparatus configured to retain program instructions and/or data for a period of time (e.g., computer-readable media). Memory  604  may include random access memory (RAM), electrically erasable programmable read-only memory (EEPROM), a Personal Computer Memory Card International Association (PCMCIA) card, flash memory, magnetic storage, opto-magnetic storage, or any suitable selection and/or array of volatile or non-volatile memory that retains data after power to host device  502  is turned off. 
     Amplifier  606  may be electrically coupled to DSP  602  and may comprise any suitable electronic system, device, or apparatus configured to increase the power of an input signal Y IN  (e.g., a time-varying voltage or current) to generate an output signal V OUT . For example, amplifier  606  may use electric power from a power supply (not explicitly shown) to increase the amplitude of a signal. Amplifier  606  may include any suitable amplifier class, including without limitation, a Class-D amplifier. 
     In operation, memory  604  may store one or more haptic playback waveforms. In some embodiments, each of the one or more haptic playback waveforms may define a haptic response a(t) as a desired acceleration of a linear resonant actuator (e.g., linear resonant actuator  507 ) as a function of time. DSP  602  may be configured to receive a force signal V SENSE  indicative of force applied to force sensor  505 . Either in response to receipt of force signal V SENSE  indicating a sensed force or independently of such receipt, DSP  602  may retrieve a haptic playback waveform from memory  604  and process such haptic playback waveform to determine a processed haptic playback signal V IN . In embodiments in which amplifier  606  is a Class D amplifier, processed haptic playback signal V IN  may comprise a pulse-width modulated signal. In response to receipt of force signal V SENSE  indicating a sensed force, DSP  602  may cause processed haptic playback signal V IN  to be output to amplifier  606 , and amplifier  606  may amplify processed haptic playback signal V IN  to generate a haptic output signal V OUT  for driving linear resonant actuator  507 . 
     In some embodiments, integrated haptic system  512 A may be formed on a single integrated circuit, thus enabling lower latency than existing approaches to haptic feedback control. By providing integrated haptic system  512 A as part of a single monolithic integrated circuit, latencies between various interfaces and system components of integrated haptic system  512 A may be reduced or eliminated. 
     The problem illustrated in  FIG. 3  may result from a linear resonant actuator  507  with a high quality factor q with a sharp peak in impedance at a resonant frequency f 0  of linear resonant actuator  507 . 
       FIG. 7  illustrates an example system  700  for improving dynamics of an electromagnetic load  701 , in accordance with embodiments of the present disclosure. In some embodiments, system  700  may be integral to a host device (e.g., host device  502 ) comprising system  700  and electromagnetic load  701 . 
     In operation, a haptic waveform generator  722  of a system  700  of a host device may generate (in the absence of a pilot tone generated by pilot tone generator  718 ) a raw transducer driving signal x′(t) comprising a haptic waveform signal or audio signal. In some embodiments, raw transducer driving signal x′ (t) may be generated based on stored haptic waveforms and/or dynamically-generated haptic waveforms stored by haptic waveform generator  722  or a memory (e.g., memory  604 ) accessible to haptic waveform generator  722 . 
     Raw transducer driving signal x′ (t) may be received by combiner  726  which may combine raw transducer driving signal x′(t) with a correction term from multiplier  725  to generate transducer driving signal x(t) in order to effectively cancel some or all of a coil impedance of electromagnetic load  701 , as described in greater detail below. Also as described below, by effectively reducing the coil resistance of electromagnetic load  701 , system  700  may also reduce an effective quality factor q of electromagnetic load  701 , which may in turn decrease attack time and minimize ringing occurring after the raw transducer driving signal has ended. Although  FIG. 7  depicts a virtual negative resistance being applied by way of a combiner  726 , in some embodiments a negative impedance filter may be applied to raw transducer driving signal x′ (t) to generate transducer driving signal x(t) to achieve the same or a similar effect of effectively reducing coil impedance of electromagnetic load  701 . An example of such a negative impedance filter is described in U.S. patent application Ser. No. 16/816,790, filed Mar. 12, 2020, and entitled “Methods and Systems for Improving Transducer Dynamics,” which is incorporated by reference herein in its entirety. 
     Transducer driving signal x(t) may in turn be amplified by amplifier  706  to generate a driving signal V(t) for driving electromagnetic load  701 . Responsive to driving signal V (t), a sensed terminal voltage V T  (t) of electromagnetic load  701  may be converted to a digital representation by a first analog-to-digital converter (ADC)  703 . Similarly, sensed current I(t) may be converted to a digital representation by a second ADC  704 . Current I(t) may be sensed across a shunt resistor  702  having resistance R s  coupled to a terminal of electromagnetic load  701 . The terminal voltage V T  (t) may be sensed by a terminal voltage sensing block  707 , for example a volt meter. 
     As shown in  FIG. 7 , system  700  may include an impedance estimator  714 . Impedance estimator  714  may include any suitable system, device, or apparatus configured to estimate, based on sensed terminal voltage V T (t), sensed current I(t), and/or any other measured parameters of electromagnetic load  701 , one or more components of the electrical and/or mechanical impedances of electromagnetic load  701 , and generate one or more control signals. For example, one control signal generated by impedance estimator  714  may include a negative impedance Re_neg generated based on an estimate of DC coil impedance Re of electromagnetic load  701 . As another example, impedance estimator  714  may also generate a voltage offset V OFFSET  and a current offset I OFFSET , which may be subtracted from sensed terminal voltage V T (t) and sensed current I(t), respectively, by combiners  710  and  712 , respectively, to cancel any measurement offsets that may be present and detected by impedance estimator  714 . As a further example, and as described in greater detail below, impedance estimator  714  may generate one or more control signals for communication to haptic state machine  716 . 
     Also shown in  FIG. 7  are two bandpass filters (BPF)  730  and  732 , respectively, filtering the offset-cancelled versions of the sensed terminal voltage V T (t) and sensed current I(t). Bandpass filters  730  and  732  may filter out haptic playback content of driving signal V (t) from entering impedance estimator  714 , and thus possibly biasing it away from an accurate estimate of DC coil impedance Re. It may be desirable, in some situations, to bypass use of these bandpass filters  730  and  732 , for example, in situations where there is no haptic playback. Such a moment may occur when a low latency initial Re estimate must be made just prior to playback of the haptic waveform, as described below with reference to  FIG. 9  in initial state  904 . Because bandpass filters  730  and  732  may add latency to input signals for impedance estimator  714 , it may be desirable to bypass bandpass filters  730  and  732 . 
     Examples of approaches for estimating one or more components of the electrical and/or mechanical impedances of electromagnetic load  701  and generating a negative coil impedance value Re_neg are described in, without limitation, U.S. patent application Ser. No. 16/816,790 filed Mar. 12, 2020 and entitled “Methods and Systems for Improving Transducer Dynamics;” U.S. patent application Ser. No. 16/816,833 filed Mar. 12, 2020 and entitled “Methods and Systems for Estimating Transducer Parameters;” U.S. patent application Ser. No. 16/842,482 filed Apr. 7, 2020 and entitled “Thermal Model of Transducer for Thermal Protection and Resistance Estimation;” and U.S. patent application Ser. No. 16/369,556 filed Mar. 29, 2019 and entitled “Driver Circuitry;” all of which are incorporated by reference herein in their entireties. 
     As mentioned above and described in greater detail below, a system  700  may effectively reduce a coil impedance of electromagnetic load  701  by applying a negative resistance correction term to raw transducer driving signal x′(t) to generate transducer driving signal x(t), which may reduce an effective quality factor q of the transducer, which may in turn decrease attack time and minimize ringing occurring after the raw transducer driving signal has ended. Quality factor q of a transducer may be expressed as: 
     
       
         
           
             
               
                 
                   q 
                   = 
                   
                     
                       
                         
                           R 
                           RES 
                         
                         * 
                         Re 
                       
                       
                         
                           R 
                           RES 
                         
                         + 
                         Re 
                       
                     
                     * 
                     
                       
                         
                           C 
                           MES 
                         
                         
                           L 
                           CES 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     In equation (7), as DC resistance Re increases, the numerator term R RES *Re increases more rapidly than the denominator term R RES +Re. Therefore, quality factor q generally increases with increasing DC resistance Re. Accordingly, one way system  700  may minimize quality factor q is to effectively decrease DC resistance Re. In some embodiments, system  700  may ideally decrease the effective DC resistance Re to a point in which critical damping occurs in electromagnetic load  701 . 
     Turning briefly to  FIG. 8 ,  FIG. 8  illustrates an example of electromagnetic load  701  modelled as a linear system including electrical components  802  and electrical model of mechanical components  804  and including a negative resistance resistor  806  with negative impedance Re_neg inserted in series with electromagnetic load  701 , in accordance with embodiments of the present disclosure. The addition of negative impedance Re_neg may lower quality factor q because effectively it subtracts from DC resistance Re thereby reducing the overall DC electrical impedance. 
     In practice, negative resistors do not exist. Instead, system  700  may be configured to behave substantially like the circuit shown in  FIG. 8 , including a mathematical model of negative impedance Re_neg in series with a mathematical model of electromagnetic load  701 . In operation, system  700  (e.g., at the output of combiner  726 ) may in effect compute a voltage V m  that would occur at the junction of negative impedance Re_neg and DC resistance Re as shown in  FIG. 8 , if, in fact, it were possible to place a physical resistor with negative impedance Re_neg in series with electromagnetic load  701 . Computed voltage V m  may then be used to drive electromagnetic load  701 . 
     In essence, system  700  may implement a sensorless velocity control feedback loop for electromagnetic load  701 . The feedback loop may use a dynamic estimate of parameters of electromagnetic load  701  and generate feedback (e.g., negative impedance Re_neg) to cancel most of the electrical and mechanical impedance of electromagnetic load  701 . In the case of DC coil resistance Re, its estimate must be quite accurate (e.g., &lt;1% error) in order for the feedback loop of system  700  to achieve stability and achieve the desired negative impedance effect. The electrical and mechanical impedance of electromagnetic load  701  may change in response to the stimulus applied to it (e.g., amplitude and frequency of driving signal V(t)), ambient temperature conditions, and/or other factors. 
     Turning back to  FIG. 7 , system  700  may also comprise a pilot tone generator  718 . Pilot tone generator  718  may comprise any system, device, or apparatus configured to, responsive to one or more control signals received from haptic state machine  716 , generate a pilot tone significantly below or above a resonance frequency of electromagnetic load  701 . Accordingly, pilot tone generator  718  may be capable of driving signals at frequencies and amplitudes that may affect electrical parameters of electromagnetic load  701 , while producing little or no perceptible haptic effects at electromagnetic load  701 . As shown in  FIG. 7 , the output of pilot tone generator  718  may be combined by combiner  724  with the output of haptic waveform generator  722 . 
     Haptic state machine  716  may comprise any system, device, or apparatus configured to, responsive to haptic trigger events, control signals, and/or other information received from other components of system  700 , generate control signals to other components of system  700  to control operation of such other components in order to sequence pilot tone generation of pilot tone generator  718 , haptic waveforms generated by haptic waveform generator  722 , and operation of impedance estimator in order to accurately and efficiently (from a time latency perspective) estimate coil resistance of electromagnetic load  701 . 
     To better understand the functionality of haptic state machine  716 , it may be illustrative to consider various possible conditions for system  700  and electromagnetic load  701 . For example, possible conditions for electromagnetic load  701  may include:
         a haptic trigger has been received by system  700  but electromagnetic load  701  is not yet stimulated to produce haptic vibration;   a haptic playback event is ongoing at electromagnetic load  701  and electromagnetic load  701  is stimulated;   a haptic playback event just ended and electromagnetic load  701  is no longer stimulated by the driving signal, but electromagnetic load  701  may be in motion; or   a haptic playback event has not occurred for a significant period of time and electromagnetic load  701  has experienced no motion since the end of the prior haptic event.       

     Possible conditions for a haptic playback waveform generated by haptic waveform generator  722  may include:
         the haptic playback waveform is pre-stored in a memory;   the haptic playback waveform is dynamically generated as it is played back;   the haptic playback waveform has spectral content concentrated in the frequency region near and including the resonant frequency of electromagnetic load  701 ;   the haptic playback waveform has significant spectral content in the frequency region below the resonant frequency of electromagnetic load  701 ;   the haptic playback waveform has significant spectral content in the frequency region above the resonant frequency of electromagnetic load  701 ; and   the haptic playback waveform has significant spectral content in the frequency regions at, below, and above the resonant frequency of electromagnetic load  701 .       

     Possible conditions for pilot tones generated by pilot tone generator  718  may include:
         the pilot tone is at a frequency significantly below the resonant frequency of electromagnetic load  701  such that impedance of electromagnetic load is dominated by DC coil resistance Re;   the low-frequency pilot tone may be played during a haptic playback event;   the low-frequency pilot tone may be played for some minimum amount of time during and after a haptic playback event such that sufficient cycles of the pilot tone have occurred to provide a first estimate of DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701 ;   the low-frequency pilot tone may be played when a pre-determined period of time (e.g., one minute) has occurred since the end of a previous haptic playback event;   the pilot tone is at a frequency significantly above the resonant frequency of electromagnetic load  701  such that impedance of electromagnetic load  701  is dominated by DC coil resistance Re and coil inductance Le;   the high-frequency pilot tone may be briefly played immediately after a haptic trigger but immediately before the playback of the haptic playback waveform (e.g., a short enough period of time such that a user perceives no delay between the haptic trigger and the haptic effect responsive to the haptic trigger);   the high-frequency pilot tone may be played for some minimum amount of time during and after a haptic playback event such that sufficient cycles of the pilot tone have occurred to provide a first estimate of DC coil resistance Re for frequencies above the resonant frequency of electromagnetic load  701 ; and   the pilot tone is a combination of the low-frequency and high-frequency tones described above played for some minimum amount of time during and after the haptic playback event such that sufficient cycles of the tones provide first estimates of DC coil resistance Re at a frequency significantly above the resonant frequency of electromagnetic load  701 , coil resistance Re at a frequency significantly below the resonant frequency of electromagnetic load  701 , and coil inductance Le.       

     In all cases, the pilot tones may be at amplitudes low enough that a user is unable to perceive the presence of the pilot tones, either through tactile perception or auditory perception. 
     To better understand the functionality of haptic state machine  716 , it may also be illustrative to consider various possible approaches for estimating various entities of system  700  and electromagnetic load  701 . For example, possible estimation approaches for determining DC coil resistance Re and coil inductance Le of electromagnetic load  701  may include:
         a least-squares fitting technique to determine a relationship between voltage and current as seen by electromagnetic load  701 ; and   a thermal model to predict change in DC coil resistance Re over time, which may:
           use a timer to track elapsed time between significant events, such as a start of a haptic playback event, end of a haptic playback event, and time between haptic events;   be used to predict how much electromagnetic load  701  heats up during a haptic playback event;   be used to predict how much electromagnetic load  701  cools down after a prior haptic playback event; and   be used to distinguish between changes to DC coil resistance Re due to heating and cooling and ambient temperature.   
               

     As another example, possible estimation approaches for determining voltage offset V OFFSET  and current offset I OFFSET  may include:
         playback of a zero-value haptic playback signal in order to estimate DC offsets of sensed terminal voltage V T  (t) and sensed current I(t); and   playback of some integer number of cycles of a pilot tone to estimate DC offsets of sensed terminal voltage V T  (t) and sensed current I(t) (e.g., the coil impedances Re and Le may be estimated together with the DC offsets by a least-squares fit procedure; such method may provide a low-latency approach to obtain fresh estimates of the sensor offset and coil impedance between the time of the haptic trigger and the start of haptic playback);   wherein, in either case, the sampled values resulting from such signals may be, respectively, accumulated and the offsets determined by dividing by the number of samples.       

     As a further example, a thermal model may be used to determine a relationship between voltage and current as seen by electromagnetic load  701 . 
       FIG. 9  illustrates a flow chart for example operation of haptic state machine  716  without use of a thermal model for coil impedance estimation, in accordance with embodiments of the present disclosure. Upon powering up of system  700 , haptic state machine  716  may begin in sleep state  902 . Haptic state machine  716  may remain in sleep state  902  until a request for a haptic playback event is received (e.g., a haptic trigger), at which point haptic state machine  716  may proceed to initial state  904 . 
     At initial state  904 , haptic state machine  716  may cause pilot tone generator  718  to play a pilot tone significantly above the resonant frequency of electromagnetic load  701  for an integer number of cycles while impedance estimator  714  collects samples of sensed terminal voltage V T  (t) and sensed current I(t). After the end of the integer number of cycles, impedance estimator  714  may estimate voltage offset V OFFSET  and current offset I OFFSET  to apply during the upcoming haptic playback event, and also estimate DC coil resistance Re for frequencies above the resonant frequency of electromagnetic load  701 . Once impedance estimator  714  has completed estimates for voltage offset V OFFSET , current offset I OFFSET , and DC coil resistance Re, haptic state machine  716  may proceed to playback startup state  906 . 
     During playback startup state  906 , haptic state machine  716  may cause haptic waveform generator  722  to play the haptic playback waveform responsive to the haptic trigger received during sleep state  902 , and may also cause pilot tone generator  718  to play a pilot tone significantly below the resonant frequency of electromagnetic load  701  contemporaneously with the haptic playback waveform. In some embodiments, haptic state machine  716  may cause pilot tone generator  718  to play a pilot tone significantly above the resonant frequency of electromagnetic load  701  contemporaneously with the haptic playback waveform. Also during playback startup state  906 , haptic state machine  716  may cause impedance estimator  714  to collect samples of sensed terminal voltage V T  (t) and sensed current I(t), now compensated by voltage offset V OFFSET  and current offset I OFFSET , respectively, and estimate DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701 . In embodiments in which haptic state machine  716  causes pilot tone generator  718  to play a pilot tone significantly above the resonant frequency of electromagnetic load  701  contemporaneously with the haptic playback waveform during playback startup state  906 , impedance estimator  714  may also estimate DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701  and/or estimate coil inductance Le. If and when a first estimate of DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701  is made, haptic state machine  716  may proceed to playback tracking state  908 . On the other hand, if and when a request has been made for the haptic playback event to end (e.g., at the end of the haptic playback waveform generated by haptic waveform generator  722  responsive to the haptic trigger), haptic state machine  716  may proceed to mute state  910 . 
     As specific examples, the resonant frequency of electromagnetic load  701  may be 150 Hz in haptic applications. A pilot tone significantly below such resonant frequency may be a tone of between 10 Hz and 40 Hz. A pilot tone significantly above such resonant frequency may be a tone of between 500 Hz and 2.5 KHz. Because a human ear may be most sensitive to tones in the 2-KHz range, a tone around such frequency may cause a perceptible acoustic response. To mitigate this possibility, the pilot tone significantly above resonant frequency may be set to very low signal magnitudes (e.g., 100 mV) and played back for a very short duration (e.g., 5 millseconds) between a haptic trigger and a playback of a haptic waveform such that any residual audio transient is masked by an onset of the haptic playback itself. 
     In the case that electromagnetic load  701  does not have an audible response in the region of the pilot tone above the resonant frequency, it may be advantageous to play both a pilot tone below the resonant frequency and a pilot above the resonant frequency at the same time along with transducer driving signal x(t), for a number of reasons. The higher-frequency pilot may enable an estimate for DC coil resistance Re earlier in time as compared to when the lower-frequency pilot may be able to provide an estimate for DC coil resistance Re. In that case, system  700  may begin tracking changes to DC coil resistance Re sooner. As the lower-frequency pilot reaches a time at which a more accurate estimate of DC coil resistance Re is achieved with the lower-frequency pilot rather than the higher-frequency pilot, impedance estimator  714  may switch its output negative impedance Re_neg accordingly. A tradeoff may exist between length of estimate and accuracy—the high-frequency pilot may enable a reasonably accurate estimate sooner, but the lower frequency pilot enables a more accurate estimate for longer term tracking of changes to DC coil resistance Re. Because accuracy of estimate DC coil resistance Re (e.g., less than 1% error may be required for feedback loop stability, as described above) may be critical for the feedback control loop to be stable and provide value, playing the two pilot tones together may be advantageous. Finally, the estimate of the lower-frequency pilot estimate may be able to provide a calibration reference value for the higher-frequency pilot, because the lower-frequency pilot may enable the more accurate estimate. It is known that DC coil resistance Re may increase with frequency due to eddy current losses in the magnet of electromagnetic load  701 , thus the coil impedance estimate with the high-frequency pilot tone may be higher than the actual DC coil resistance Re needed for the feedback control loop. Having the two estimates from the high- and low-frequency pilots simultaneously may allow the estimate from the high-frequency pilot tone to be calibrated to match the estimate from the low-frequency pilot tone in order to infer an unbiased DC coil resistance Re from the estimate based on the high-frequency pilot tone. Depending on construction of electromagnetic load  701 , the difference between estimates of the low-frequency pilot tone and high-frequency pilot tone may be negligible or may amount to several percent. 
     During playback tracking state  908 , haptic state machine  716  may cause pilot tone generator  718  to continue playing the pilot tone(s) generated during playback startup state  906 , may cause impedance estimator  714  to continue to collect samples of sensed terminal voltage V T  (t) and sensed current I(t) as compensated by voltage offset V OFFSET  and current offset I OFFSET , respectively, and continue to estimate DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701 . In embodiments in which haptic state machine  716  causes pilot tone generator  718  to play a pilot tone significantly above the resonant frequency of electromagnetic load  701  contemporaneously with the haptic playback waveform during playback startup state  906  (and playback tracking state  908 ), impedance estimator  714  may also continue to estimate DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701  and/or estimate coil inductance Le. If and when a request has been made for the haptic playback event to end (e.g., at the end of the haptic playback waveform generated by haptic waveform generator  722  responsive to the haptic trigger), haptic state machine  716  may proceed to mute state  910 . 
     In mute state  910 , haptic state machine  716  may cause haptic waveform generator  722  to cease playing the haptic playback waveform, cause pilot tone generator  718  to ramp down (or step down) the magnitude(s) of any pilot tone(s) it is generating, cause impedance estimator  714  to cease collection of samples of sensed terminal voltage V T  (t) and sensed current I(t) and generation of impedance estimates. This ramp down function may be important because simply stepping down the pilot tone may result in a broadband transient that triggers a human-perceptible acceleration response, and the ramp down function may reduce or eliminate such transient artifact. During mute state  910 , haptic state machine  716  may cause impedance estimator  714  to save (e.g., to a memory accessible to impedance estimator  714 ) all final estimated impedance values and any other estimated values during the haptic playback event. After pilot tone generator  718  has muted the pilot tone(s), haptic state machine  716  may proceed again to sleep state  902 . 
       FIG. 10  illustrates a flow chart for example operation of a haptic state machine when using a thermal model for impedance estimation, in accordance with embodiments of the present disclosure. 
     Upon powering up of system  700 , haptic state machine  716  may begin in sleep state  1002 . Haptic state machine  716  may remain in sleep state  1002  until a request for a haptic playback event is received (e.g., a haptic trigger), at which point haptic state machine  716  may proceed to initial state  1004 . At initial state  1004 , haptic state machine  716  may cause haptic waveform generator  722  to playback a zero waveform for a pre-determined period of time, during which impedance estimator  714  may collect samples of sensed terminal voltage V T  (t) and sensed current I(t). After the end of the predetermined period of time, impedance estimator  714  may estimate voltage offset V OFFSET  and current offset I OFFSET  to apply during the upcoming haptic playback event. Also during initial state  1004 , haptic state machine  716  may cause impedance estimator  714  to use a thermal model (e.g., as described in U.S. patent application Ser. No. 16/842,482 filed Apr. 7, 2020 and entitled “Thermal Model of Transducer for Thermal Protection and Resistance Estimation,” referenced above), to estimate an initial value of DC coil resistance Re based on a prior haptic playback event&#39;s estimated DC coil resistance Re, ambient temperature, and elapsed time since the prior haptic playback event. Once impedance estimator  714  has completed estimates for voltage offset V OFFSET , current offset I OFFSET , and DC coil resistance Re, haptic state machine  716  may proceed to playback startup state  1006 . 
     During playback startup state  1006 , haptic state machine  716  may cause haptic waveform generator  722  to play the haptic playback waveform responsive to the haptic trigger received during sleep state  1002 , and may also cause pilot tone generator  718  to play a pilot tone significantly below the resonant frequency of electromagnetic load  701  contemporaneously with the haptic playback waveform. Also during playback startup state  1006 , haptic state machine  716  may cause impedance estimator  714  to collect samples of sensed terminal voltage V T  (t) and sensed current I(t), now compensated by voltage offset V OFFSET  and current offset I OFFSET , respectively, and estimate DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701 . Further in playback startup state  1006 , impedance estimator  714  may update its thermal model. If and when a first estimate of DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701  is made, haptic state machine  716  may proceed to playback tracking state  1008 . On the other hand, if and when a request has been made for the haptic playback event to end (e.g., at the end of the haptic playback waveform generated by haptic waveform generator  722  responsive to the haptic trigger), haptic state machine  716  may proceed to mute state  1010 . 
     During playback tracking state  1008 , haptic state machine  716  may cause pilot tone generator  718  to continue playing the pilot tone generated during playback startup state  1006 , and may cause impedance estimator  714  to continue to collect samples of sensed terminal voltage V T  (t) and sensed current I(t) as compensated by voltage offset V OFFSET  and current offset I OFFSET , respectively, continue to estimate DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701 , and continue to update the thermal model of impedance estimator  714 . If and when a request has been made for the haptic playback event to end (e.g., at the end of the haptic playback waveform generated by haptic waveform generator  722  responsive to the haptic trigger), haptic state machine  716  may proceed to mute state  1010 . 
     In mute state  1010 , haptic state machine  716  may cause haptic waveform generator  722  to cease playing the haptic playback waveform, cause pilot tone generator  718  to ramp down (or step down) the magnitude of any pilot tone it is generating, and cause impedance estimator  714  to cease collection of samples of sensed terminal voltage V T  (t) and sensed current I(t), generation of impedance estimates, and updating of the thermal model. During mute state  1010 , haptic state machine  716  may cause impedance estimator  714  to save (e.g., to a memory accessible to impedance estimator  714 ) all final estimated impedance values and any other estimated values during the haptic playback event. Further, in mute state  1010 , haptic state machine  716  may cause impedance estimator  714  to reset a timer. After the timer has been reset, haptic state machine  716  may proceed again to sleep state  1002 . Notably, during sleep state  1002 , the timer may continue to run so that it may be used to estimate DC coil resistance (e.g., via a thermal model) when haptic state machine  716  again enters the initial state  1004 . 
       FIG. 11  illustrates a flow chart for example operation of haptic state machine  716  when used in a background calibration mode for coil impedance estimation, in accordance with embodiments of the present disclosure. If it has been a significant time since a prior haptic event, haptic state machine  716  may begin in sleep state  1102 . Upon entering sleep state  1102 , a counter may become active and may be reset to some positive predetermined value, for example, a value which corresponds to 60 seconds. A countdown may then begin (count down timer  1103 ), for example, once per second, and proceeds toward zero if no haptic events are requested during such period of time. If a haptic event happens to occur before expiration of the timer, then haptic state machine  716  may proceed according to either of  FIG. 9  or  FIG. 10 . If, however, a haptic event does not occur before expiration of the counter, haptic state machine  716  may proceed to initial state  1104 . 
     At initial state  1104 , haptic state machine  716  may cause pilot tone generator  718  to play a pilot tone significantly below the resonant frequency of electromagnetic load  701  for one complete cycle of the pilot tone while impedance estimator  714  collects samples of sensed terminal voltage V T  (t) and sensed current I(t). After the end of the pilot tone, impedance estimator  714  may estimate voltage offset V OFFSET  and current offset I OFFSET  to apply during the next haptic playback event, the next time it occurs, and also estimate DC coil resistance Re for frequencies below the resonant frequency of electromagnetic load  701 . Once impedance estimator  714  has completed estimates for voltage offset V OFFSET , current offset I OFFSET , and DC coil resistance Re, haptic state machine  716  may proceed to mute state  1110 . 
     In mute state  1110 , haptic state machine  716  may cause pilot tone generator  718  to ramp down (or step down) the magnitude of the pilot tone it is generating, and cause impedance estimator  714  to cease collection of samples of sensed terminal voltage V T  (t) and sensed current I(t), and generation of impedance estimates. During mute state  1110 , haptic state machine  716  may cause impedance estimator  714  to save (e.g., to a memory accessible to impedance estimator  714 ) all final estimated impedance values and any other estimated values during the haptic playback event. Haptic state machine  716  may proceed again to sleep state  1102 . Notably, during sleep state  1102 , the countdown timer may be reset. 
       FIG. 12  provides a summary of different example methods for estimating coil impedance, all of which may be managed by haptic state machine  716 , shown in its various modes in  FIGS. 9, 10, and 11 . For example, highlighted in  FIG. 12  are three different methods used to achieve an initial estimate of DC coil resistance Re prior to playback of a haptic waveform. These three methods may form the basis, respectively, for which mode haptic state machine  716  operates in, wherein  FIGS. 9, 10, and 11 , each represent a mode. It may be desirable to calculate an initial estimate of DC coil resistance Re as close in time to a haptic event request as possible, and without human-perceptible delay between the request for the haptic event and the haptic effect. A haptic event, for example, may be in response to a virtual-button push with an acknowledgement (e.g., a short click or vibration) by the device back to the user that the virtual button was actually pushed. Approaches to achieve an initial estimate of DC coil resistance Re without perceptible delay are set forth in the first three rows of the table in  FIG. 12 . In two of these three approaches (“Thermal Model” and “Background Calibration”), an initial estimate of DC coil resistance Re may be based on a recent past estimate. In one approach (“High Frequency Pilot”), the initial estimate of DC coil resistance Re may be formed after a request for a haptic event is made, but the high-frequency pilot approach may estimate Re very quickly (for example, in under 5 milliseconds) so that there is little or no human-perceptible delay between the event request and the perceived effect itself. Achieving a low-latency estimate of DC coil resistance Re may become increasingly important in devices which may employ force-sensing virtual buttons, which have their own latency between the user interaction with the virtual button and the device hosting the virtual button actually detecting such user interaction. The latency of this user interaction detection is typically much longer than the latency of the initial estimate of DC coil resistance Re, using one of the three approaches outlined in  FIG. 12 . For this reason, it may be important for a satisfying user experience that the initial estimate of DC coil resistance Re be as low latency as possible in order not to add human-perceptible delay between a user interacting with a virtual button and receiving a haptic effect in acknowledgement of such user interaction. 
     Also highlighted in  FIG. 12  are three different approaches to track DC coil resistance Re during playback of haptic waveforms. These three approaches are shown in the last three rows of the table of  FIG. 12 . In these approaches, some kind of stimulus (e.g., a high-frequency pilot or low-frequency pilot or the haptic waveform itself if sufficiently broadband) may be used to excite electromagnetic load  701  so that an estimation technique, such as least squares, may continuously estimate a relationship between sensed terminal voltage V T  (t) sensed current I(t) in order to estimate accurately the DC coil impedance Re and possibly coil inductance Le. 
     Which of the approaches set forth in  FIG. 12  may be used to achieve an initial estimate of DC coil impedance Re and a continuously-updated estimate of DC coil impedance Re may depend on a response of electromagnetic load  701 , especially its acoustic response. A decision regarding choice of approach may be made during haptic product development. 
     Although the foregoing discusses application to a linear electromagnetic load, it is understood that systems and methods similar or identical to those disclosed may be applied to other linear or non-linear systems. 
     Further, although the foregoing contemplates use of a negative resistance filter to implement a model of an LRA, in some embodiments a mathematical equivalent to an LRA may be used in lieu of a model. 
     As used herein, when two or more elements are referred to as “coupled” to one another, such term indicates that such two or more elements are in electronic communication or mechanical communication, as applicable, whether connected indirectly or directly, with or without intervening elements. 
     This disclosure encompasses all changes, substitutions, variations, alterations, and modifications to the example embodiments herein that a person having ordinary skill in the art would comprehend. Similarly, where appropriate, the appended claims encompass all changes, substitutions, variations, alterations, and modifications to the example embodiments herein that a person having ordinary skill in the art would comprehend. Moreover, reference in the appended claims to an apparatus or system or a component of an apparatus or system being adapted to, arranged to, capable of, configured to, enabled to, operable to, or operative to perform a particular function encompasses that apparatus, system, or component, whether or not it or that particular function is activated, turned on, or unlocked, as long as that apparatus, system, or component is so adapted, arranged, capable, configured, enabled, operable, or operative. Accordingly, modifications, additions, or omissions may be made to the systems, apparatuses, and methods described herein without departing from the scope of the disclosure. For example, the components of the systems and apparatuses may be integrated or separated. Moreover, the operations of the systems and apparatuses disclosed herein may be performed by more, fewer, or other components and the methods described may include more, fewer, or other steps. Additionally, steps may be performed in any suitable order. As used in this document, “each” refers to each member of a set or each member of a subset of a set. 
     Although exemplary embodiments are illustrated in the figures and described below, the principles of the present disclosure may be implemented using any number of techniques, whether currently known or not. The present disclosure should in no way be limited to the exemplary implementations and techniques illustrated in the drawings and described above. 
     Unless otherwise specifically noted, articles depicted in the drawings are not necessarily drawn to scale. 
     All examples and conditional language recited herein are intended for pedagogical objects to aid the reader in understanding the disclosure and the concepts contributed by the inventor to furthering the art, and are construed as being without limitation to such specifically recited examples and conditions. Although embodiments of the present disclosure have been described in detail, it should be understood that various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the disclosure. 
     Although specific advantages have been enumerated above, various embodiments may include some, none, or all of the enumerated advantages. Additionally, other technical advantages may become readily apparent to one of ordinary skill in the art after review of the foregoing figures and description. 
     To aid the Patent Office and any readers of any patent issued on this application in interpreting the claims appended hereto, applicants wish to note that they do not intend any of the appended claims or claim elements to invoke 35 U.S.C. § 112(f) unless the words “means for” or “step for” are explicitly used in the particular claim.