# A Phase Error Correction System for Bioimpedance Measurement Circuits
## 1 Introduction
Analog Devices (ADs) (Wilmington, MA, USA) offer MAX30001-MAX3002, both SD-based bioimpedance instrumentation ASICs, which deliver performance comparable to standard laboratory impedance analysers. However, they can only achieve a maximum bandwidth of 131 kHz [12; 14]. MAX30009, another AD solution, is their latest and most accurate SD-integrated solution. However, to obtain the level of accuracy claimed in the datasheet, phase errors must be calculated for each stimulus frequency, making it unsuitable for impedance spectroscopy applications [15]. A CMOS ASIC research realization presented in [10], co-authored in part by members of our team, featured the approach of passing I and Q demodulation signals through the same preamplification stages as the first stage of the monitoring sub-system to minimise relevant phase errors. However, whilst this approach worked, it only achieved \(<\)4% error for RC loads at frequencies up to 100 kHz. Phase errors were identified as a serious issue affecting bioimpedance measurement accuracy in [16], which focused solely on a CMOS phase shifter simulation. The theoretical paper did not demonstrate how they would integrate with a bioimpedance measurement system nor did it identify how phase error would be identified prior to compensating for it. The present work introduces a Phase Error Compensated Synchronous Demodulation system (PECSD), an SD-based bioimpedance measurement system with automatic phase error correction that can achieve less than 5% phase errors for RC loads for frequencies up to 300 kHz. The topology presented here achieves minimal error at reasonably higher frequencies and does not require recalibration for each phase unlike other implementations, including ASIC-based ones. Figure 1 illustrates an equivalent circuit for human skin impedance. It provides a structured display of tissue layers and their interaction with bioimpedance measurement systems and instrumentation (Figure 2) using a tetrapolar electrode configuration. Figure 1 also illustrates an unwanted element: the introduction of erroneous phase shifts by instrumentation, compromising results accuracy. Most bioimpedance applications that depend on phase shifts  
between injected and measured signals for accurate diagnostics are predominantly performed at relatively high frequencies, i.e., skin cancer and impedance myography [2; 17; 18]. Most injected current paths traverse through the membrane/capacitive layer. Consequently, any additional phase shifts introduced by the measurement systems contribute to overall phase shifts in the membrane capacitance. This dual-phase shift scenario presents challenges: users cannot discern the extent of erroneous phase shifts versus those originating from the tissue itself [20]. Moreover, manually correcting these errors becomes arduous due to their frequency dependency. Each swept frequency results in a different phase error. Addressing these challenges is pivotal to enhancing the accuracy and reliability of bioimpedance measurements. Figure 1: Equivalent circuit of human skin impedance showing (**a**) the system-introduced phase error on skin tissue and (**b**) the equivalent circuit model of one channel of (**a**), identifying the tissue’s layered structural circuit components. Ce and Re are associated with the electrode/electrolyte interface. Rint represents interface effects due to the electrolyte. Cp and Rp are parallel RC circuits of the epidermis, while Rdh represents the dermis and inner layers [16; 19]. Figure 2: Bioimpedance measurement topologies based on Synchronous Detection/Demodulation (SD), Magnitude and phase (MP), and Synchronous Sampling (SS). Ideally, V\({}_{\text{in}}\) (and V\({}_{\text{\_0}}\)) are in-phase with the injected current \(I_{\text{out}}\)[21; 22; 23; 24; 25].