This lecture ties the whole series together in one real design problem: an ECG front-end. It starts from first principles — how a 230 V, 50 Hz mains wire pushes microamp-level displacement currents into a person — follows them through electrode impedances into the amplifier, and arrives at the circuit techniques (instrumentation amplifier, driven-right-leg) that keep the 1 mV heartbeat readable.
A displacement current is the current id that flows “across the gap” of a capacitance. Although physical charge does not cross the gap between a capacitor’s plates, a time-varying voltage across it produces a time-varying electric field, and a current appears on either side:
Take a mains wire (live), a bulb, and the return (neutral), with air gaps modelled as capacitances Cp and Cg between the power wire and the bulb and between the bulb and the return. A DC source could only light the bulb momentarily, during transients, but an AC source can continuously drive a current through the gaps. In practice, though, the coupling capacitances are small and the impedances huge:
A 230 V 40 W incandescent lamp needs about 40/230 ≈ 0.17 A to light. Assume Cp = Cg = 100 pF (on the higher end). The series capacitance is CpCg/(Cp+Cg) = 50 pF, so at 50 Hz
Ztot = 1/(2π×50×50×10−12) ≈ 63.6 MΩ ⇒ Imax = 230/63.6×106 ≈ 3.6 µARMS,
nowhere near the 170 mA needed.
The general rule: the maximum current transferred to any system through capacitive coupling is limited by the highest impedance in the path (the coupling impedance). Whatever displacement current flows out of a live wire must come back into the neutral wire through some path, and the highest impedance in that path limits how much current can flow — other series impedances limit it even further.
Replace the bulb with a battery-powered circuit sitting in the fluctuating electric field of an AC source. The displacement current enters the circuit from the live side and leaves towards the neutral via the stray capacitances; what voltages it creates depends on the impedances inside the circuit:
Take-homes: a displacement current from live to neutral is not necessary for a wire to pick up interference (a common-mode voltage appears anyway), and a displacement current does not have to flow from the noise source through the conductors — there only needs to be a closed loop. Both common-mode and differential-mode interference can be observed; differential-mode interference is significant only if the impedances, and their mismatch between nodes, are large enough that id1Z1 − id2Z2 is comparable to the signal. Nodes tied together by small impedances see only common-mode interference.
What does Earth have to do with it? The earth is a huge conductor (not as good as copper, but much better than an air gap’s impedance). Powerlines spread over large distances, so the capacitance between earth and the power lines becomes significant — and on top of that, most commercial power grids have the neutral lines connected to earth at the distribution transformers (except isolated IT supplies, e.g. in operating theatres). So:
This displacement current generally creates common-mode voltages in circuits with nodes tied together via low impedances, but can also generate differential-mode voltages if different high-impedance nodes exist.
The body is a conductor sitting in the electric field of the mains wiring. A human touching bare earth is already directly grounded; a human wearing shoes, or sitting on an insulator (a wooden chair), acts as the midpoint of two capacitors in series: Cp between the body and the power line (air gap), and Cg between the body and ground. The body is like a plate in a parallel-plate capacitor with one end connected to ground, and the displacement current from mains has a return path through the body.
The voltage-divider rule for impedances gives the voltage at the central plate (the human body):
Generic values. vn(t) = 230√2 sin(2π×50t), Cp = 3 pF and Cg = 200 pF.
Vb,RMS = 230 × 3/(3+200) ≈ 3.4 V
id(t) = (3×200/203)×10−12 × 230√2 × 2π × 50 × cos(2π×50t) ≈ 2.14×10−7√2 cos(2π×50t), so Id,RMS ≈ 214 nA.
If the human were directly in contact with ground, id = Cp dvn/dt gives Id,RMS = 3×10−12×230×2π×50 ≈ 217 nA.
All of these are far below the threshold of perception (about 500 µA) — harmless to the person, but enormous compared to a 1 mV ECG.
Consider a single-lead ECG with electrodes on the left arm (LA), right arm (RA) and right leg (RL, the reference). LA and RA go to the amplifier; the reference electrode defines the baseline against which all other voltages are measured.
Why do we need a reference electrode? A practical amplifier amplifies not only differential signals but also common-mode signals, so the absolute voltages at the input terminals matter: if they leave the amplifier’s input common-mode range it saturates. We have just calculated that the body sits at ≈ 3.4 V RMS common-mode from the mains; a human can also be electrostatically charged to kilovolts. Merely connecting the LA and RA electrodes to a high-gain amplifier would easily saturate it unless the common-mode voltage is offset. A simple way is to measure all signals with respect to a fixed location on the body — connect the amplifier ground to a position where the ECG is least felt (the right leg). (As we will see, this does not completely eliminate the common-mode voltage at the inputs.)
Each wire, electrode and the body couples capacitively to the mains. The independent displacement currents must form a closed loop with the mains, and there are three paths:
Most of the current takes the least-impedance path. The amplifier input impedance is very high, so negligible current flows there; and the capacitance between the human body and earth is larger than the direct stray capacitances of the wires, so the impedance is lower. Therefore the current mostly flows through the body to earth, and the total body current is approximately ib ≈ idb + iLA + iRA.
If the amplifier ground and earth ground are connected (e.g. an AC-mains-powered ECG), all of this current exits through the RL electrode and its impedance ZRL:
This acts as a common-mode fluctuation on both amplifier terminals. For a battery-powered (floating) amplifier, only the current returning through the amplifier’s own stray capacitance (Cbg) flows through ZRL and the rest shunts through Cbg, so vcm is much smaller.
Electrodes have high impedance at skin contact, and the impedances ZLA, ZRA and ZRL are never equal (imperfect contact, gel, skin condition). Displacement currents flowing through mismatched impedances create differential voltages that act like signal:
And even with a perfect match between ZLA and ZRA, the common-mode voltage vcm can induce a differential voltage through the amplifier’s finite input impedance, which forms a potential divider with each electrode impedance:
with Zin the amplifier input impedance (assumed equal at both inputs). Any difference between ZLA and ZRA therefore converts common-mode voltage into differential voltage that the amplifier then amplifies — the reason that high input impedance matters.
Another phenomenon adds a DC differential voltage at the amplifier input. At every metal–electrolyte interface a half-cell potential forms (the electrolyte is the gel of a wet electrode, or sweat for a dry one); for Ag/AgCl electrodes the half-cell potential is Ehc ≈ 220 mV relative to the standard hydrogen electrode. A differential voltage forms if RA and LA electrodes have different half-cell potentials:
If they are equal the differential voltage is zero, but in practice they rarely are (different skin conditions, gel or sweat under each electrode, temperature, contact pressure, motion…). It adds on top of the other interference voltages, and although DC it can drift over time with electrode movement, loosening and sweating.
Why it limits the front-end gain. The offset can be as small as 1 mV for matched electrodes on well-prepared skin, but with aged electrodes and drying gel it can rise to tens or even hundreds of mV; ECG device standards require the front end to handle up to ±300 mV of offset. With a 3.3 V supply rail the maximum gain before saturating against a 300 mV offset is 3.3/0.3 = 11 — at which the 1 mV ECG would only be amplified to 11 mV. A 10-bit ADC at VFS = 3.3 V has a resolution of 3.3/210 = 3.2 mV and would no longer capture the ECG features with enough fidelity.
Two ways out: use a higher-resolution ADC (e.g. 24-bit) so the front-end gain can stay low while still capturing the ECG; or amplify with a small gain, pass the result through a high-pass filter to remove the DC offset, and let a second stage amplify the faster-changing signal.
| Issue | Type | Mitigation |
|---|---|---|
| Displacement currents through mismatched electrode impedances | Differential | Shield the electrode leads and keep them short |
| DC electrode offset voltage | Differential | Lower front-end gain plus a higher-resolution ADC (or high-pass then second gain stage); better electrodes and skin preparation reduce the mismatch |
| Common-mode current through finite input impedance and mismatched electrode impedances | Differential | Raise the input impedance by orders of magnitude with an instrumentation amplifier (also improves CMRR) |
| Displacement current through the body into the grounded reference electrode (RL) | Common-mode | Reduce the common-mode magnitude with the driven-right-leg circuit |
With perfectly matched resistors (R4/R3 = R2/R1) a difference amplifier theoretically has infinite CMRR, but this is practically difficult because of resistor tolerances — and the source impedances (which can vary at each electrode) add to the problem. Also, its input impedance is different at the two inputs:
An INA is a difference amplifier with signals fed through two non-inverting gain stages at each input. This gives:
In most modern ECG systems the patient is not grounded. Instead, active common-mode compensation is used, often referred to as the driven-right-leg (DRL) circuit (the DRL electrode need not be at the right leg). Its principle: sense the body’s common-mode signal, invert and amplify it, and feed it back through the RL electrode so that negative feedback drives vCM towards zero — leaving the amplifier less common-mode noise to reject.
Analysis (Webster, example 6.4). Assume a displacement current ib flows from the power line into the patient’s body and out through the RL electrode impedance ZRL; this creates vCM on the body. Write the first-stage outputs as vA = vCM + (Rf/Rg + ½)vDM and vB = vCM − (Rf/Rg + ½)vDM.
1. Sensing. Two equal averaging resistors Ra on vA and vB give the voltage at their centre vx = (vA+vB)/2 = vCM — the differential parts cancel. (This holds only as long as nothing else draws current from the centre point.)
2. Inversion. The auxiliary op-amp is an inverting amplifier with feedback resistor Rd. Its inputs are virtual shorts to the grounded non-inverting input, so the feedback current is if = −(vA+vB)/Ra = −2vCM/Ra, and the output is
3. Feedback to the body. One side of ZRL is at the body voltage vCM, the other at vy. Ohm’s law across ZRL:
Compared with vCM = ibZRL when the RL electrode was simply connected to ground, connecting the DRL has effectively shrunk the electrode impedance, ZRL,eff = ZRL/(1 + 2Rd/Ra), and so reduced the common-mode voltage on the body by the factor 1 + 2Rd/Ra.
Through negative feedback, the DRL therefore reduces the common-mode voltage on the body, mitigating its downstream effects — amplifier saturation and differential-voltage generation. We want the effective impedance as low as possible during normal operation, which keeps vCM = ibZRL,eff low; hence large values for Rd and small values for Ra so that Rd/Ra remains large.
Typical values. Ra = 20 kΩ, Rd = 5 MΩ, ib = 200 nA, ZRL = 50 kΩ. Without a DRL, vCM = 10 mV. With the DRL, 1 + 2Rd/Ra = 501, so vCM ≈ 10 mV/501 ≈ 20 µV — a reduction by a factor of about 500.
The DRL introduces a safety risk to the patient. If the auxiliary amplifier saturates (e.g. on a large transient vCM), a low-impedance path to ground could form through the op-amp’s output terminal. To limit the current that could flow through the body below regulatory values, a large resistance Ro ≈ 5 MΩ is added in series with the output; thus large (>1 MΩ) values for Rd and Ro are used to keep the current below regulatory limits (electrical safety).