BME 294: Circuits, Instrumentation, and Measurements
Estimated study time: 10 minutes
Table of contents
Sources and References
Primary texts — Nilsson and Riedel, Electric Circuits, 11th ed. (Pearson). Webster and Clark, Medical Instrumentation: Application and Design, 4th ed. (Wiley).
Supplementary texts — Horowitz and Hill, The Art of Electronics, 3rd ed. (Cambridge). Sedra and Smith, Microelectronic Circuits, 8th ed. (Oxford). Northrop, Analysis and Application of Analog Electronic Circuits to Biomedical Instrumentation, 2nd ed. (CRC).
Online resources — MIT OCW 6.002 Circuits and Electronics and 6.301 Solid-State Circuits. Analog Devices Op Amp Applications Handbook (open). Texas Instruments The Designer’s Guide to Instrumentation Amplifiers. IEC 60601-1 general safety of medical electrical equipment (family of standards).
Chapter 1: Circuit Fundamentals
1.1 Elements and Laws
Circuits are networks of ideal elements — resistors, capacitors, inductors, independent and dependent sources. Kirchhoff’s current law states that the algebraic sum of currents into any node is zero; Kirchhoff’s voltage law, that the algebraic sum of voltages around any loop is zero:
\[ \sum_k i_k = 0, \qquad \sum_k v_k = 0 . \]Ohm’s law \( v = iR \) is one of many constitutive relations; capacitors satisfy \( i = C\,dv/dt \), inductors \( v = L\,di/dt \).
1.2 Analysis Methods
Nodal analysis writes KCL at each node, treating node voltages as unknowns. Mesh analysis writes KVL around each loop, treating loop currents as unknowns. Superposition, valid in linear circuits, allows independent analysis of each source. Thevenin and Norton equivalents reduce any linear two-terminal network to a voltage source and series resistance or a current source and parallel resistance, a computational convenience and a conceptual anchor.
Chapter 2: Transient Response
2.1 First-Order Networks
An RC network charging through a step has response
\[ v(t) = V_\infty + (V_0 - V_\infty) e^{-t/\tau}, \qquad \tau = RC . \]An RL network similarly has \( \tau = L/R \). Time constants set the settling time and bandwidth; 5τ reaches within 1% of steady state.
2.2 Second-Order Networks
Series RLC under a step follows
\[ L\frac{d^2 i}{dt^2} + R\frac{di}{dt} + \frac{i}{C} = 0 , \]with characteristic roots \( s = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2} \), damping ratio \( \zeta = \alpha/\omega_0 \). Overdamped, critically damped, and underdamped regimes produce different transient behaviours; underdamped circuits ring at \( \omega_d = \omega_0\sqrt{1 - \zeta^2} \). Biosignal acquisition systems are designed to be critically damped or slightly underdamped so that transients settle without excessive oscillation.
Chapter 3: Sinusoidal Steady State and Frequency Response
3.1 Phasors and Impedance
Under sinusoidal excitation at angular frequency \( \omega \), each element is represented by a complex impedance:
\[ Z_R = R, \qquad Z_C = \frac{1}{j\omega C}, \qquad Z_L = j\omega L . \]Kirchhoff’s laws apply to phasors unchanged; nodal and mesh analysis proceed as in DC with complex arithmetic. Power decomposes into real, reactive, and apparent components, with power factor \( \cos\phi \).
3.2 Transfer Functions and Bode Plots
A linear time-invariant circuit has transfer function \( H(s) = V_{out}(s)/V_{in}(s) \). Bode plots display \( 20\log_{10}|H(j\omega)| \) and \( \angle H(j\omega) \) against \( \log \omega \), revealing poles (slope changes by −20 dB/dec) and zeros (+20 dB/dec). Low-pass, high-pass, band-pass, and notch topologies filter biosignals to within their frequency-band of interest (e.g., 0.5–40 Hz for ECG, 0.5–100 Hz for EEG, 20–500 Hz for EMG).
3.3 Input–Output Relationships
For LTI circuits, output is the convolution of input with the impulse response:
\[ v_{out}(t) = \int_0^t h(t-\tau)\, v_{in}(\tau)\, d\tau . \]Equivalently, in the frequency domain, \( V_{out}(j\omega) = H(j\omega) V_{in}(j\omega) \). The choice of domain is pragmatic — transient shape in time, filter shape in frequency.
Chapter 4: Operational Amplifiers
4.1 The Ideal Op-Amp
Five assumptions: infinite open-loop gain, infinite input impedance, zero output impedance, infinite bandwidth, and equal input terminals (virtual short) when negative feedback is applied. With these, inverting, non-inverting, summing, difference, integrating, and differentiating configurations follow by inspection. For the inverting amplifier, \( v_o = -(R_f/R_i) v_i \); for the non-inverting, \( v_o = (1 + R_f/R_i) v_i \).
4.2 Real Op-Amp Limitations
Finite gain-bandwidth product (GBP) limits closed-loop bandwidth: \( f_{3dB} = \mathrm{GBP}/G \). Slew rate caps large-signal \( dv/dt \). Input offset voltage, bias currents, and input noise degrade the low-signal limit. For a typical JFET-input op-amp, input voltage noise density ≈ 10 nV/√Hz sets the minimum detectable signal over a given bandwidth.
4.3 Instrumentation Amplifiers
Three-op-amp instrumentation amplifiers combine high differential gain (set by a single gain resistor) with very high common-mode rejection (CMRR > 100 dB). They are the front-end of choice for biopotential recording, where the common-mode signal — 50/60 Hz mains pickup — can be 10⁶ times larger than the differential signal of interest. Active shielding (driven-ground circuits) further reduces pickup by actively driving cable shields to the common-mode voltage.
Chapter 5: Signal Conditioning and Conversion
5.1 Filters
Active filters (Sallen–Key, multiple-feedback) implement low-pass, high-pass, band-pass, and notch responses with controlled \( Q \) and gain. Butterworth filters maximize pass-band flatness; Chebyshev maximize roll-off at the cost of ripple; Bessel maximize phase linearity, preferred when biosignal waveform shape must be preserved.
5.2 Analog-to-Digital Conversion
Sampling at rate \( f_s \) aliases components above \( f_s/2 \) (Nyquist). Anti-alias filtering before the ADC is therefore mandatory. Quantization to \( N \) bits introduces noise of RMS \( q/\sqrt{12} \) where \( q = V_{FS}/2^N \), yielding theoretical SNR
\[ \mathrm{SNR} = 6.02 N + 1.76 \,\mathrm{dB} . \]Oversampling and noise shaping (delta-sigma ADCs) move quantization noise out of the band of interest and resolve 16–24 bits at biosignal bandwidths.
5.3 Digital-to-Analog Conversion
DACs reconstruct analog signals with a zero-order hold that introduces a \( \mathrm{sinc} \) frequency response; a reconstruction filter restores smoothness. Resolution, settling time, and glitch energy characterize DAC performance for actuator drive and stimulator output.
Chapter 6: Biomedical Instrumentation Systems
6.1 Transducers
Electrodes convert ionic currents in tissue to electronic currents in wires; Ag/AgCl electrodes with hydrogel interface minimize half-cell potential and drift. Thermistors and RTDs measure temperature; strain gauges measure force through Wheatstone-bridge configurations; piezoelectric transducers measure pressure and acceleration. Each transducer introduces its own noise, bandwidth, and linearity profile.
6.2 Case Studies
ECG front-end. Ag/AgCl electrodes → instrumentation amplifier (gain ≈ 10, CMRR > 100 dB) → right-leg drive → high-pass (0.05 Hz) → low-pass (40–150 Hz) → notch (50/60 Hz) → ADC. Electrode motion artifact dominates noise at rest; bandwidth and filter choices are clinical, not only electrical, decisions.
EEG. Microvolt-scale signals require very low-noise front-ends with careful shielding and active electrode designs. Multichannel EEG systems use multiplexed ADCs or per-channel converters with common reference or bipolar topologies.
EMG. Signal bandwidth is broader (20–500 Hz) and amplitudes larger (up to mV). Surface EMG emphasizes electrode placement over the motor point; intramuscular EMG uses fine-wire or needle electrodes.
6.3 Safety and Grounding
Single-fault safety principles demand that no single failure produce a hazard. Isolation, redundant ground paths, and battery operation are standard; GFCI outlets and isolated power systems protect from shock in clinical environments. Grounding philosophy — star grounds, chassis-to-signal separation, shield termination — determines whether the theoretical performance of an instrument survives the real environment of a hospital room.
