RC Circuits and Bioelectrical Impedance Analysis

Elliot Mylott*, Sabrina Hoffman, Ralf Widenhorn**

*emylott@pdx.edu, **ralfw@pdx.edu

Introduction • Bioelectrical impedance analysis is a tool used in doctor’s

offices or health clubs to estimate a person’s fat free mass (FFM) by analyzing a small AC current. From the FFM and a person’s weight their body fat percentage can be calculated.

• The body can be modeled as a RC circuit. Intra and extra cellular fluids are represented as resistors and cell membranes by capacitors because of their ability to retain charge.

• Compared to muscle tissue, fat cells have a very high impedance and conduct electricity poorly. The effect the body has on the current will be principally due to the FFM, which can be calculated by an empirically derived formula.

• Body mass index (BMI) relates weight to a person’s height. The combined information from BMI and FFM can be useful when determining a person’s health due to factors like weight and body fat percentage, which can be used to determine risk of diabetes or heart conditions.

Figure 1: Circuit to determine a body’s resistance and reactance (dashed lines) using a known reference resistor.

Conclusion A lab with this device is relevant to a general physics course for pre-health students. Students can apply their knowledge of AC circuits and use phasor diagrams and algebra to explore the relationships between the different voltages and phase angles. In the lab students could actually calculate their body fat percentage. It needs to be emphasized that measurements should be done on student volunteers and no student should be placed on the spot to have their data taken.

Useful references • Yang, Yuxiang et al. 2006. Design and preliminary evaluation of a portable device for the measurement of

bioimpedance spectroscopy. Physiol. Meas. 27 1293

• Kyle, Ursula G. et al. 2004. Bioelectrical impedance analysis--part I: review of principles and methods. Clinical Nutrition. 23 1226-1243

Apparatus • This apparatus enables students to qualitatively explore the relationship between

impedance, resistance, reactance and phase angle.

• Students are also able to study the electrical characteristics of the body and how they can be modeled with known components, which requires frequencies (50 kHz is commonly used) that are too high for many student laboratory voltage probes .

Procedure Biological Measurement

1. Turn the multimeter to DCV and connect it to the Ground and Gain jacks of the BIA device.

Theory • The impedance (Z) of an RC circuit (Fig. 1) has two components: capacitive

reactance (XC) and resistance (RZ)

𝑍 = 𝑋𝐶2 + 𝑅2

• The phase shift (𝜑) is given by:

tan 𝜑 =𝑋𝐶

𝑅

• The phase can be found experimentally by comparison of the AC voltage across a known resistor (Fig. 2).

• These two equations can be rearranged to solve for the R (body resistance) and XC (body reactance) as functions of Z and 𝜑:

𝑅 =𝑍2

1+tan2 𝜑 𝑋𝐶 = 𝑅 tan 𝜑 (1)

• As show in Fig. 1, all the components are in series and so have the same current passing through them. Therefore the gain (K) of the voltages across Z and Rreference is given by:

𝐾 =𝑉1

𝑉2=

𝑍∗𝐼

𝑅𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒∗𝐼=

𝑍

𝑅𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (2)

Figure 3: Completed Apparatus

Figure 4: Correct method to hold BIA electrodes.

2. Connect the similar colored jacks of the BIA device and the hand electrodes using banana cables.

3. Turn on the BIA device and stand with your feet apart and hands extended perpendicular to your body. The wire of the electrode should only make contact with your palms (Fig. 4). Note the voltage.

4. Connect the voltmeter to the Ground and Phase jacks and repeat steps 3.

5. Note how the voltages vary from person to person. A goal of the lab could be to calculate the FFM from these values.

Figure 2: This apparatus can be modified and used with an oscilloscope to display the AC Voltages across the body and reference resistor.

• The main components of the apparatus are a function generator , two differential amplifiers and a gain /phase detector.

• The gain/phase detector outputs DC voltages that are proportional to the gain (Vgain) and phase (Vphase) between the two input signals V1 and V2 (Fig. 1) which can be measured with a regular voltage probe. The output voltages are in the range of one volt and vary by a few tenths of a volt for different values of the gain and phase. A larger value for Vphase

corresponds to a smaller phase angle and larger values for Vgain correspond to larger values for Z.

↑ 𝜑 ↔ ↓ 𝑉𝑝ℎ𝑎𝑠𝑒 ↑ Z ↔ ↑ 𝑉𝑔𝑎𝑖𝑛 (3)

• Four electrodes are needed to measure the impedance of deep body tissue and to avoid the effect of skin impedance (Fig. 1).

• The oscillator constructed for this apparatus supplies a small 50 kHz current to the handheld BIA electrodes.

This work was supported in part by NSF under Grant 1141078 as part of the TUES program.

Measurement of a AC circuit model of the human body

• Connect the color coded leads from the BIA apparatus to the body impedance models constructed from resistors and a capacitor on the circuit board.

• Note the change in both Vgain and Vphase as you move to the “body” with the lower reactance (two capacitors in parallel). Use the proportionalities in (3) to compare your results to Fig 5.

Biological Impedance • A person’s resistance comes from both intra and extra-cellular fluids and the reactance

from cell membranes.

• The phase angle is therefore related with the ability of the cells to retain charge and hydration levels. It is even used in diagnosing some serious illnesses.

• Similar to how the mass of a resistor can be calculated from its dimensions, resistance and resistivity, a person’s FFM can be estimated based on their impedance.

Figure 5: Phasor diagram for the example body model circuit