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Nepal Engineering Council · Registration ExaminationABmE · Ch 7

Chapter 7

Electronic Devices, Circuits and Machines for BME

ABME07·6 Sub-topics·78 MCQs

7.1

Integrated Circuit Technology and Device Models

ABmE0701

This section outlines integrated circuit technology and then develops the DC and AC (small-signal) models of the diode, the JFET, the bipolar junction transistor and the MOS transistor.

Integrated Circuit Technology • An integrated circuit contains many interconnected components fabricated together on a single monolithic silicon chip.

Advantages: very small size and weight, low cost in volume, high reliability (no soldered joints), low power consumption, matched and thermally coupled devices, and high speed from short interconnections.

Limitations: inductors and large capacitors cannot be fabricated economically, resistor values have poor absolute tolerance (though excellent matching), power dissipation is limited, and the device cannot be repaired. • Classification by scale:

SSI, MSI, LSI, VLSI and ULSI.

By technology: monolithic, thin/thick film and hybrid.

By function: analogue (linear), digital and mixed-signal. • Principal fabrication steps: crystal growth (Czochralski) and wafer preparation → oxidation → photolithography (photoresist, mask, exposure, development) → etching → diffusion or ion implantation for doping → epitaxial growth → metallisation → passivation → testing, scribing, packaging and final test.

Isolation between devices is by reverse-biased p-n junctions, dielectric isolation or, in CMOS, by wells. • Technologies: bipolar (fast, good analogue performance, higher power);

NMOS and PMOS; and CMOS, which dominates because it has almost zero static power dissipation, high noise immunity and high packing density.

BiCMOS combines both. • Relevance to biomedical instrumentation: instrumentation amplifiers, isolation amplifiers, ADCs and DACs, microcontrollers, ASICs in pacemakers and hearing aids, and MEMS sensors are all IC products; the constraints of low power, low noise, small size and biocompatible packaging are what make implantable electronics possible.

Diode Models • Ideal (Shockley) equation:

I = IS(eV/ηVT − 1), where IS is the reverse saturation current, η the ideality factor (1 for germanium, about 2 for silicon at low current) and VT = kT/q ≈ 26 mV at 300 K — a number worth memorising. • DC (large-signal) models, in increasing order of accuracy: the ideal switch (0 V drop when forward biased, open circuit when reverse biased); the constant voltage drop model — a battery of 0.7 V for silicon (0.3 V for germanium) in series with an ideal switch, which is the model used for almost all hand analysis; the piecewise-linear model — the cut-in voltage in series with the forward (bulk) resistance rf, giving a sloping characteristic; and the full exponential model, solved graphically with a load line or iteratively. • AC (small-signal) model: about a quiescent point Q the diode behaves as a dynamic resistance rd = ηVT/IQ — so at 1 mA a silicon diode looks like roughly 26-52 Ω — in parallel with its capacitance.

Two capacitances matter: the depletion (transition/junction) capacitance CT ∝ 1/(VR)n, dominant in reverse bias (the basis of the varactor), and the diffusion capacitance CD = τ·I/ηVT, dominant in forward bias and responsible for the reverse recovery time that limits switching speed. • Special diodes:

Zener (reverse breakdown used for regulation — 7.4), Schottky (metal-semiconductor, low 0.3 V drop and negligible stored charge, hence very fast), LED, photodiode, varactor and tunnel diode.

Bipolar Junction Transistor Models • A BJT is two back-to-back p-n junctions (npn or pnp) with a thin lightly doped base.

Its regions of operation are set by the junction biases: active (EB forward, CB reverse — used for amplification), saturation (both forward — switch ON), cut-off (both reverse — switch OFF) and reverse active. • DC relationships:

IE = IB + IC; α = IC/IE (0.95-0.99); β = hFE = IC/IB (50-300); and the conversions β = α/(1 − α) and α = β/(1 + β).

In the active region IC = ISeVBE/VT, with VBE ≈ 0.7 V assumed for silicon and VCE(sat) ≈ 0.2 V. • Biasing fixes the Q point and must stabilise it against variation of β and temperature; the standard circuit is the voltage-divider (self) bias with an emitter resistor, whose negative feedback gives the best stability (a low stability factor S), the alternatives being fixed bias (poor) and collector-feedback bias. • AC small-signal models: the hybrid-π model — a voltage-controlled current source gmvbe with gm = IC/VT, an input resistance rπ = β/gm = βVT/IC, and an output resistance ro = VA/IC where VA is the Early voltage; the T model, with re = VT/IE ≈ 26 mV/IE; and the h-parameter model (hie, hre, hfe, hoe), with the two-port relations V₁ = hiI₁ + hrV₂ and I₂ = hfI₁ + hoV₂.

The key point is that gm depends only on the bias current, so gain is set by the operating point. • The three configurations: common emitter — high voltage and current gain, medium input and output impedance, 180° phase inversion, the general-purpose amplifier; common base — voltage gain with current gain below 1, low input and high output impedance, no phase shift, used at high frequency; common collector (emitter follower) — voltage gain just under 1, high current gain, very high input and very low output impedance, used as a buffer and impedance matcher.

High-frequency response is limited by the internal capacitances and the Miller effect, characterised by fT, the gain-bandwidth product.

JFET and MOSFET Models • Field-effect transistors are unipolar and voltage-controlled, in contrast to the current-controlled bipolar transistor, and have very high input impedance, low noise, good thermal stability and no thermal runaway — which is why they are used at the front end of biopotential amplifiers and with microelectrodes (5.1). • JFET: a channel of n- or p-type silicon whose width is controlled by the reverse-biased gate junction.

It is depletion mode only — it conducts at VGS = 0 and is turned off by reverse gate bias.

The DC model in saturation (pinch-off) is Shockley's equation, ID = IDSS(1 − VGS/VP)², where IDSS is the drain current at zero gate voltage and VP the pinch-off voltage.

Its small-signal model is a transconductance source gmvgs with gm = 2IDSS(1 − VGS/VP)/|VP| = gm0(1 − VGS/VP), in parallel with rd, and an essentially infinite input resistance. • MOSFET: the gate is insulated from the channel by a thin oxide, giving an input resistance of 10¹²-10¹⁵ Ω.

It exists in enhancement mode (no channel at VGS = 0; a channel is induced when VGS > Vth — the type used in all digital circuits) and depletion mode (a channel exists at zero bias).

Its regions are cut-off (VGS < Vth), triode/linear (VDS < VGS − Vth), where ID = k[(VGS − Vth)VDS − VDS²/2] and the device acts as a voltage-controlled resistor, and saturation, where ID = (k/2)(VGS − Vth)²(1 + λVDS), with λ the channel-length modulation parameter.

Its small-signal parameters are gm = k(VGS − Vth) = 2ID/(VGS − Vth) = √(2kID) and ro = 1/(λID). • Comparison worth memorising: for the same current, a BJT has a much higher gm (and hence gain) than a MOSFET, but the MOSFET has far higher input impedance, lower noise at low frequency is debatable, better scaling, and no base current;

MOSFETs are susceptible to electrostatic damage of the gate oxide and must be handled accordingly.

7.2

Amplifiers, Oscillators and Filters

ABmE0702

This section covers the classification of amplifiers, untuned and tuned power amplifiers, operational and differential amplifier circuits, op-amp relaxation oscillators, sinusoidal oscillator and filter circuits, and the CMOS inverter relaxation oscillator.

Classification of Amplifiers • Amplifiers may be classified by frequency range (DC, audio, video, radio-frequency, microwave), by configuration (common emitter, base or collector), by coupling (RC, transformer, direct — direct coupling being essential for the DC-level biopotentials of 5.1), by signal magnitude (small-signal or large-signal/power), by number of stages, and — most importantly for power amplifiers — by class, that is by the conduction angle of the output device.

Class Conduction angle Maximum efficiency Distortion and use A 360° — the device conducts for the whole cycle 25 % (series-fed), 50 % (transformer-coupl ed) Lowest distortion, but large quiescent current and heat even with no signal; small-signal and high-fidelity stages B 180° — each device conducts half the cycle, in a push-pull pair 78.5 % Crossover distortion at the zero crossing; efficient AB Slightly more than 180° Between 50 and 78.5 % A small forward bias removes crossover distortion — the standard audio power amplifier C Less than 180° Over 90 % Severe distortion — usable only with a tuned load that restores the sine wave;

RF transmitters D Switching (on or off only) Over 90 %, approaching 100 % Pulse-width modulation with an output low-pass filter; used where efficiency and heat matter — portable and implantable equipment • Untuned (wideband) power amplifiers use resistive or transformer coupling and amplify a broad band — audio amplifiers are the example.

Tuned (narrowband) power amplifiers use an LC tank circuit as the load, so they amplify only a narrow band centred on the resonant frequency f₀ = 1/(2π√(LC)) with a bandwidth of f₀/Q.

Because the tank stores energy and reconstructs the missing part of the waveform, a tuned amplifier can be operated in class C at very high efficiency; they are used in radio transmitters, diathermy generators (6.2), RF stages of MRI (4.3) and telemetry (5.3). • Key amplifier specifications: gain (and its expression in dB — 20 log for voltage, 10 log for power), bandwidth and the gain-bandwidth product, input and output impedance, slew rate, noise figure, distortion and CMRR.

Negative feedback trades gain for stability, reduced distortion and noise, controlled input and output impedance, and increased bandwidth: with feedback factor β, the closed-loop gain is Af = A/(1 + Aβ), and the gain-bandwidth product remains constant.

Operational and Differential Amplifiers • The differential amplifier is the input stage of every op-amp and of every biopotential amplifier.

It amplifies the difference between its two inputs and rejects what is common to both:

Vout = Ad(V₁ − V₂) + Acm(V₁ + V₂)/2, and its figure of merit is the common-mode rejection ratio CMRR = Ad/Acm, usually quoted in dB.

A high CMRR requires matched transistors and a high-impedance tail — hence the use of a constant-current source in the emitter/source circuit, since the tail resistance directly sets Acm. • The ideal operational amplifier: infinite open-loop gain, infinite input impedance, zero output impedance, infinite bandwidth and CMRR, zero offset voltage and zero drift.

Real devices have gain of 10⁵-10⁶, input impedance of megohms (or 10¹² Ω with a FET input), a gain-bandwidth product of a few megahertz, a finite slew rate and small but significant input offset voltage, bias and offset currents. • The two golden rules for analysis with negative feedback:

(1) no current flows into either input;

(2) the op-amp drives its output so that V₊ = V₋ — the 'virtual short', and if the non-inverting input is grounded the inverting input becomes a 'virtual earth'.

Circuit Output Inverting amplifier Vo = −(Rf/R₁)Vin; input impedance = R₁ Non-inverting amplifier Vo = (1 + Rf/R₁)Vin; very high input impedance Voltage follower (buffer) Vo = Vin, gain 1 — used purely for impedance transformation Summing amplifier Vo = −Rf(V₁/R₁ + V₂/R₂ + …) — the basis of the R-2R DAC Difference (subtractor) amplifier Vo = (Rf/R₁)(V₂ − V₁) with matched resistors; its CMRR depends critically on resistor matching Instrumentation amplifier Two buffer op-amps feeding a difference amplifier: gain = (1 + 2R/RG)·(R₃/R₂), giving very high input impedance, high CMRR and gain set by one resistor — the standard biopotential front end (5.1) Integrator Vo = −(1/RC)∫Vin dt — a low-pass function; needs a parallel resistor to prevent DC saturation Differentiator Vo = −RC·dVin/dt — a high-pass function; noisy, and needs a series resistor for stability Comparator Open-loop; output saturates according to the sign of (V₊ − V₋) Schmitt trigger Comparator with positive feedback giving hysteresis, so noise cannot cause multiple transitions — used for QRS detection and for squaring up signals Log/antilog, precision rectifier, current-to-voltage converter Nonlinear and transducer-interface circuits; the I-V converter (transimpedance amplifier) is the standard photodiode front end Oscillators • An oscillator converts DC into a periodic AC waveform without an input signal, using positive feedback.

The Barkhausen criterion states that sustained oscillation requires loop gain |Aβ| = 1 and total phase shift around the loop of 0° (or 360°); in practice the gain is made slightly greater than 1 at start-up and limited by amplitude stabilisation. • Sinusoidal (harmonic) oscillators:

RC types for low frequencies — the Wien bridge oscillator (f = 1/(2πRC), needing a gain of exactly 3 and some form of automatic amplitude control, and giving a very clean sine wave) and the RC phase-shift oscillator (three RC sections each contributing 60°, f = 1/(2πRC√6), requiring a gain of at least 29); and LC types for high frequencies — the Hartley (tapped inductor), Colpitts (tapped capacitor) and Clapp oscillators, all with f = 1/(2π√(LC)); and the crystal oscillator, which uses the piezoelectric resonance of quartz to give outstanding frequency stability — used for every microprocessor clock and for precise timing in medical equipment. • Relaxation oscillators generate non-sinusoidal waveforms — square, triangular or sawtooth — by repeatedly charging and discharging a capacitor between two thresholds.

The op-amp astable multivibrator is the standard example: a Schmitt trigger (comparator with positive feedback through R₁ and R₂) whose output charges a capacitor through R towards the supply rail; when the capacitor voltage reaches the trigger threshold the output flips, and the cycle repeats.

With β = R₁/(R₁ + R₂), the period is T = 2RC·ln[(1 + β)/(1 − β)], which for the common choice R₂ = R₁ (β = 0.5) reduces to T ≈ 2.2RC.

Adding a diode network makes the charge and discharge times unequal, giving an adjustable duty cycle; feeding an integrator from the square wave gives a triangular wave — the basis of a function generator.

The 555 timer is the standard IC implementation, in astable or monostable mode. • The CMOS inverter relaxation oscillator is the simplest practical oscillator and is worth knowing in detail.

In its classic form, two CMOS inverters are cascaded, the output of the second fed back through a resistor R to the input of the first, which is also connected through a capacitor C to the output of the first (or to ground).

Because a CMOS inverter has a well-defined switching threshold at about half the supply voltage, very high input impedance and rail-to-rail output, the capacitor charges and discharges through R between the thresholds, and the circuit oscillates at approximately f ≈ 1/(2.2RC) (the exact constant depending on the threshold and the configuration).

Its merits are extreme simplicity, very low component count and power, operation from a single supply and a logic-level square-wave output ready to clock digital circuits; its limitation is frequency that varies with supply voltage, temperature and device threshold, so where accuracy matters the same inverter is used with a crystal (the Pierce oscillator) instead.

A third inverter is often added as a buffer, and a series resistor protects the input protection diodes.

This circuit appears in timing, tone generation, switched-capacitor clocks and low-power medical devices.

Filter Circuits • A filter passes some frequencies and attenuates others.

Types by response: low-pass, high-pass, band-pass, band-stop (notch) and all-pass.

The cut-off frequency is where the response falls to −3 dB (half power, 0.707 of the voltage). • Passive filters use only R, L and C: simple and needing no supply, but they cannot give gain, load the source, and need bulky inductors at low frequency.

A single RC section gives fc = 1/(2πRC) and a roll-off of 20 dB/decade (6 dB/octave) per pole. • Active filters use an op-amp with R and C only: they provide gain, present a high input and low output impedance so sections can be cascaded without interaction, avoid inductors entirely, and allow precise, tunable responses.

Their limitations are the need for a power supply and the op-amp's bandwidth and noise.

The standard topologies are the Sallen-Key (VCVS) and the multiple-feedback second-order sections, cascaded to obtain higher orders — an n-th order filter rolls off at 20n dB/decade. • Approximations:

Butterworth — maximally flat passband, moderate roll-off, the usual default;

Chebyshev — steeper roll-off at the cost of passband ripple;

Bessel — maximally flat group delay, therefore the best preservation of waveform shape, which matters for the ECG and other pulse-like biosignals; and elliptic (Cauer) — the steepest transition, with ripple in both bands. • Filters in biomedical instrumentation: a high-pass (0.05 Hz) filter removes electrode drift and baseline wander; a low-pass (100 Hz or 40 Hz) filter removes muscle artefact; a 50 Hz notch filter removes mains interference — though it also removes signal and can distort the QRS, so high CMRR is always the better first defence (5.1); and an anti-aliasing low-pass filter must precede every analogue-to-digital converter.

7.3

AC Circuits

ABmE0703

This section covers Faraday's law, series and parallel R-L, R-C and R-L-C circuits, resonance, star-delta transformation, and three-phase systems with their voltage, current and power relationships.

Faraday's Law and AC Generation • Faraday's first law: an emf is induced in a conductor whenever the magnetic flux linking it changes.

Faraday's second law: the magnitude of the induced emf is proportional to the rate of change of flux linkage — e = −N·dΦ/dt, the minus sign being Lenz's law, which states that the induced current opposes the change producing it (a consequence of conservation of energy).

Dynamically induced emf arises when a conductor moves in a field, e = Blv·sin θ; statically induced emf arises when the flux itself changes, and is self-induced (e = −L·di/dt) or mutually induced (e = −M·di/dt). • Generation of a sinusoidal emf: a coil of N turns and area A rotating at angular velocity ω in a uniform field B links a flux Φ = BA·cos ωt, so the induced emf is e = NBAω·sin ωt = Em sin ωt — a sine wave, maximum when the coil lies in the plane of the field (cutting flux fastest) and zero when it is perpendicular to it. • AC quantities:

RMS (effective) value = 0.707 Vm for a sine wave, defined as the DC value producing the same heating; average value over a half cycle = 0.637 Vm; form factor = RMS/average = 1.11; peak factor = Vm/RMS = 1.414.

Ammeters and voltmeters read RMS, so 230 V mains has a peak of about 325 V.

The j-Operator and Phasors • Sinusoids of the same frequency are represented as phasors, and the j-operator (j = √−1) denotes a 90° anticlockwise rotation, so that j² = −1 and 1/j = −j.

A phasor may be written in rectangular form Z = R + jX or polar form Z = |Z|∠φ, with |Z| = √(R² + X²) and φ = tan⁻¹(X/R).

Addition and subtraction are easiest in rectangular form, multiplication and division in polar form. • The three elements: in a resistor current and voltage are in phase and Z = R; in a pure inductor the current lags the voltage by 90° and Z = jXL = j2πfL; in a pure capacitor the current leads the voltage by 90° and Z = −jXC = −j/(2πfC).

The mnemonic is CIVIL — in a Capacitor I leads V, and V leads I in an L. • Power in AC circuits: true (active) power P = VI cos φ watts, reactive power Q = VI sin φ volt-amperes reactive and apparent power S = VI volt-amperes, with S² = P² + Q² (the power triangle).

Power factor = cos φ = P/S = R/Z, and is lagging for an inductive and leading for a capacitive load.

A pure reactance consumes no average power, since it returns the energy it stores.

Low power factor wastes capacity and is corrected by connecting capacitors across an inductive load.

Series and Parallel Circuits Circuit Impedance and phase Notes R-L series Z = √(R² + XL²), φ = tan⁻¹(XL/R), current lags Power factor lagging; the model of a motor winding or a relay coil R-C series Z = √(R² + XC²), φ = tan⁻¹(XC/R), current leads Power factor leading; the basis of the high-pass coupling network R-L-C series Z = √[R² + (XL − XC)²], φ = tan⁻¹[(XL − XC)/R] Inductive if XL > XC, capacitive if XC > XL, purely resistive at resonance Parallel circuits Add admittances:

Y = 1/Z = G + jB siemens Easier than adding impedances; total current is the phasor sum of the branch currents Resonance • Series resonance occurs when XL = XC, at the resonant frequency f₀ = 1/(2π√(LC)).

At resonance: the impedance is a minimum and purely resistive (Z = R), the current is a maximum (I = V/R), the power factor is unity, and the voltages across L and C are equal and opposite and may each be Q times the supply voltage — 'voltage magnification'.

Because it draws maximum current at f₀, the series circuit is called an acceptor circuit and is used to select a wanted signal. • Parallel resonance (the 'tank' circuit) occurs at approximately the same frequency, but the behaviour is the opposite: the impedance is a maximum (the dynamic impedance Zd = L/CR), the line current is a minimum, and a large circulating current flows between L and C.

It is therefore a rejector circuit, used as the tuned load of the amplifiers and oscillators in 7.2. • Quality factor:

Q = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C), and Q = f₀/BW, so the bandwidth BW = f₀/Q = R/(2πL).

A high Q means a sharp, highly selective response; a low Q a broad one.

Q also equals the voltage magnification factor in a series circuit.

Star-Delta Transformation • Networks that are neither series nor parallel can often be simplified by converting between the star (Y or T) and delta (Δ or π) forms. • Delta to star: each star resistance is the product of the two adjacent delta resistances divided by the sum of all three — R₁ = RaRb/(Ra + Rb + Rc). • Star to delta: each delta resistance is the sum of the products of the star resistances taken two at a time, divided by the opposite star resistance — Ra = (R₁R₂ + R₂R₃ + R₃R₁)/R₃. • For equal resistances the relations reduce to RΔ = 3RY, which is worth remembering for quick checks.

The same transformations apply to impedances in AC circuits, and are used in bridge networks (including the Wheatstone bridge of 5.1) and in three-phase calculations.

Three-Phase Systems • A three-phase supply consists of three emfs of equal magnitude and frequency displaced by 120°, generated by three windings spaced 120° apart on the stator.

Its advantages over single-phase are a constant total instantaneous power (so machines run without torque pulsation and are self-starting), about 1.5 times the output from a machine of the same size, less conductor material for the same power transmitted, and the ability to produce a rotating magnetic field — which is why all substantial equipment, including X-ray generators (4.1), is three-phase. • Phase sequence (usually R-Y-B) determines the direction of rotation of a motor; interchanging any two lines reverses it. • Star (Y) connection: the three ends are joined to form a neutral.

Then VL = √3 Vph and IL = Iph — hence the familiar 400 V line / 230 V phase system.

A fourth (neutral) wire may be brought out. • Delta (Δ, mesh) connection: the windings form a closed loop with no neutral.

Then VL = Vph and IL = √3 Iph. • Power in both connections is the same expression:

P = √3 VLIL cos φ = 3 VphIph cos φ, with Q = √3 VLIL sin φ and S = √3 VLIL. • Balanced and unbalanced systems: in a balanced system the three loads are identical, the phasor sum of the three line currents is zero and the neutral carries no current — so in a balanced star load the neutral could in principle be omitted.

In an unbalanced system the neutral carries the resultant current, which is exactly why the three-phase four-wire system is used for distribution to mixed single-phase loads: the neutral holds each phase voltage at its proper value despite unequal loading, and must never be fused or disconnected. • Measurement of power: by three wattmeters (one per phase, needing access to the neutral); by one wattmeter (balanced loads only, with an artificial neutral, P = 3W); or, the standard method, by the two-wattmeter method, which measures total power in any three-wire system, balanced or not.

Here P = W₁ + W₂, and the power factor follows from tan φ = √3(W₁ − W₂)/(W₁ + W₂).

The readings behave characteristically: at unity power factor the two readings are equal; at power factor 0.5 one wattmeter reads zero; and below 0.5 one reading becomes negative and its connections must be reversed and the value subtracted.

7.4

Power Supplies and Voltage Regulators

ABmE0704

This section covers half-wave and full-wave rectifiers, capacitive, LC, RC and active filters, Zener diodes and Zener regulators, bandgap voltage references, constant current diodes and voltage-to-frequency converters.

The Regulated Power Supply Every mains-powered medical instrument contains the same chain: transformer (steps the voltage down and, crucially, provides isolation from the mains — see 6.6) → rectifier (converts AC to unidirectional DC) → filter (smooths the pulsating DC) → regulator (holds the output constant against changes of input and load) → load.

Performance is described by line regulation, load regulation, ripple, output impedance and efficiency.

Parameter Half-wave Full-wave (centre-tap or bridge) Output DC voltage Vdc = Vm/π = 0.318 Vm Vdc = 2Vm/π = 0.636 Vm RMS output Vm/2 Vm/√2 Ripple factor 1.21 0.482 Ripple frequency f (50 Hz) 2f (100 Hz) — easier to filter Rectification efficiency 40.6 % 81.2 % Peak inverse voltage Vm 2Vm (centre-tap);

Vm (bridge) Transformer utilisation factor 0.287 0.693 (centre-tap), 0.812 (bridge) Number of diodes 1 2 (centre-tap) or 4 (bridge) • Ripple factor is defined as the ratio of the RMS value of the AC (ripple) component to the DC value, and is the measure of how well the output approximates pure DC.

The bridge rectifier is the usual choice because it needs no centre-tapped transformer, has half the peak inverse voltage of the centre-tap circuit and the best transformer utilisation, at the cost of two diode drops in the path. • Three-phase rectifiers give a much smaller ripple at a higher frequency (six-pulse ripple at 300 Hz), which is why the X-ray generators of 4.1 use them.

Filters • Capacitor (shunt) filter — the commonest: a capacitor across the load charges to the peak of each pulse and discharges slowly through the load between peaks.

The ripple voltage is approximately Vr(pp) = Idc/(fC) for full-wave rectification and the ripple factor ≈ 1/(4√3·fCRL), so ripple falls as C, RL and the ripple frequency increase — the filter is good at light load and poor at heavy load.

Its drawback is a large repetitive surge current at switch-on and on each peak, which sets the diode rating. • Inductor (choke) filter: an inductor in series with the load opposes changes in current, so it performs better at heavy load and worse at light load — exactly the opposite of the capacitor filter. • LC (choke-input or L-section) filter: combines the two, giving good regulation at both light and heavy load and much lower ripple; the π (CLC) filter gives the lowest ripple of the passive types but poorer regulation and a high surge current. • RC filter: substitutes a resistor for the choke — cheap, small and light, but the resistor drops voltage and dissipates power, so it is suitable only for small load currents. • Active filters in the power-supply context means an amplifying/regulating element in place of a passive network, which is effectively what a series regulator does; in the signal context they are the op-amp filters of 7.2.

Zener Diodes and Voltage Regulators • A Zener diode is operated in reverse breakdown, where the voltage across it is nearly constant over a wide range of current.

Two mechanisms are involved: the Zener effect proper (below about 5 V, heavily doped, field emission, with a negative temperature coefficient) and the avalanche effect (above about 6 V, carrier multiplication, with a positive temperature coefficient) — so a diode of about 5.6-6 V has nearly zero temperature coefficient and makes the most stable reference.

It is characterised by its Zener voltage VZ, dynamic resistance rZ, and maximum power dissipation PZ(max) = VZIZ(max). • The shunt (Zener) regulator: a series resistor RS drops the excess voltage and the Zener shunts the surplus current.

The design condition is that the Zener current must stay between IZ(min) (to remain in breakdown) and IZ(max) (to stay within its power rating) for all combinations of input voltage and load current, with RS = (Vin − VZ)/(IZ + IL).

It is simple and cheap but inefficient (it wastes current when the load is light) and limited to small loads. • Series regulators: a pass transistor in series with the load is driven by an error amplifier comparing a sample of the output with a reference — the structure of every three-terminal regulator such as the 78xx (positive) and 79xx (negative) series and the adjustable LM317.

They give low ripple and noise and fast response, but are linear, so the efficiency is only Vout/Vin and the difference is dissipated as heat, requiring a heat sink.

They include short-circuit, overcurrent, thermal and sometimes crowbar (over-voltage) protection. • Switching regulators (SMPS): the pass element is switched on and off at high frequency (tens to hundreds of kilohertz) with pulse-width modulation, and an LC filter recovers the DC.

Efficiency is 80-95 %, the transformer and filter components are small, and step-up, step-down and inverting topologies are all possible; the penalties are switching noise and EMI (a real issue near sensitive biopotential amplifiers), greater complexity and a slower transient response.

Almost all modern medical equipment uses them, with careful screening and filtering.

Bandgap References and Constant Current Diodes • A bandgap voltage reference produces a stable voltage that is almost independent of temperature and supply, and is the reference inside virtually every modern regulator, ADC and DAC.

Its principle is elegant: the base-emitter voltage of a transistor has a negative temperature coefficient of about −2 mV/°C, while the difference between the base-emitter voltages of two transistors operating at different current densities, ΔVBE = VT·ln(n), is proportional to absolute temperature (PTAT) and therefore has a positive coefficient.

Adding a suitably scaled multiple of ΔVBE to VBE makes the two coefficients cancel, giving an output close to the bandgap voltage of silicon extrapolated to 0 K, about 1.22-1.25 V — hence the name.

The classic circuits are the Widlar, Brokaw and Kuijk references.

Its advantages over a Zener are that it works at low supply voltages, is quieter, and is easily fabricated on-chip, which is why it dominates in integrated circuits;

Zeners (especially buried Zeners) remain in use where the very lowest noise and drift are needed. • Constant current diode (current-regulating diode, CRD): a JFET with its gate connected to its source, packaged as a two-terminal device.

Because a JFET in pinch-off draws IDSS almost independently of the voltage across it, the device behaves as a constant current source over a wide voltage range — the dual of the Zener diode, which is a constant voltage device.

It is used to bias circuits, drive LEDs, charge capacitors linearly to generate a ramp, and supply the tail current of differential amplifiers (7.2); its limitations are a modest current range, tolerance and voltage compliance limits.

Voltage-to-Frequency Converters • A voltage-to-frequency converter (VFC) produces a pulse train whose frequency is linearly proportional to the input voltage; the inverse device is a frequency-to-voltage converter. • Charge-balance type (the accurate one): the input voltage is integrated; when the integrator output reaches a threshold, a comparator fires a precision one-shot that injects a fixed charge packet to reset it.

Since the charge injected per pulse is constant, the pulse rate must be proportional to the input current and hence to the input voltage, giving excellent linearity (0.01 %) and good noise rejection because the input is integrated.

Multivibrator (astable) type VFCs are simpler and cheaper but less linear. • Why it matters in biomedical engineering: a frequency is a digital-like quantity that can be transmitted, counted and isolated far more easily and accurately than an analogue voltage.

Hence VFCs are used for simple, inexpensive analogue-to-digital conversion by counting pulses in a fixed interval (an integrating ADC with excellent mains-noise rejection), for transmitting analogue signals through an optocoupler or transformer across a patient isolation barrier without loss of accuracy, for frequency-modulated biotelemetry (5.3), for long-distance transmission over noisy lines, and for driving counters, timers and simple displays.

The isolation application is the one worth remembering: an optocoupler is grossly non-linear for an analogue voltage but perfectly adequate for a digital pulse train, so V-F and F-V conversion is a standard way of getting a signal across an isolation barrier in patient-connected equipment.

7.5

DC Circuits and Circuit Analysis

ABmE0705

This section covers Ohm's law with its applications and limitations, circuit elements and sources, series and parallel connection of resistors and sources, Kirchhoff's laws, and the superposition, Thevenin and Norton theorems with matrix methods of analysis.

Ohm's Law, Circuit Elements and Sources • Ohm's law: at constant temperature, the current through a conductor is directly proportional to the potential difference across it — V = IR, with power P = VI = I²R = V²/R and resistance R = ρL/A.

Limitations: it applies only to linear (ohmic) conductors at constant temperature, and not to non-linear devices such as diodes, transistors and thermistors, nor to electrolytes, gas discharges, vacuum tubes or superconductors — and it is the temperature qualification that fails first in practice, since a resistor's value rises as it heats. • Circuit elements: active (sources, which supply energy) and passive (R, L, C, which absorb or store it); linear/non-linear; bilateral/unilateral; and lumped/distributed.

The passive elements store or dissipate:

R dissipates I²R, L stores ½LI² in its magnetic field, and C stores ½CV² in its electric field. • Sources: an ideal voltage source maintains a fixed voltage whatever the current, so it has zero internal resistance; an ideal current source supplies a fixed current whatever the voltage, so it has infinite internal resistance.

Practical sources have a finite internal resistance r, represented as a voltage source E in series with r or, equivalently, a current source E/r in parallel with r — and either may be converted into the other, which is the basis of source transformation. • Effect of internal resistance: the terminal voltage of a practical source falls as the load current rises — V = E − Ir, so a source is only 'stiff' when r ≪ RL.

This is why a weak battery still reads its nominal voltage on a high-impedance voltmeter but collapses under load.

By the maximum power transfer theorem, maximum power is delivered to the load when RL = r, but then the efficiency is only 50 % — which is why power systems are designed for RL ≫ r (high efficiency) and only signal and communication circuits are matched for maximum power.

Series and Parallel Connections • Resistors in series: the same current flows through each, voltages add, and Req = R₁ + R₂ + …, which is always larger than the largest.

The voltage divider rule gives V₁ = V·R₁/(R₁ + R₂). • Resistors in parallel: the same voltage appears across each, currents add, and 1/Req = 1/R₁ + 1/R₂ + …, which is always smaller than the smallest.

For two resistors Req = R₁R₂/(R₁ + R₂) ('product over sum'), and for n equal resistors Req = R/n.

The current divider rule for two branches gives I₁ = I·R₂/(R₁ + R₂) — note that the current divides in inverse proportion to the resistances. • Sources in series: emfs add if connected in the same sense (and subtract if opposed), and the internal resistances add — this increases the available voltage.

Sources in parallel: only sources of the same emf should be paralleled, and the arrangement reduces the effective internal resistance and so increases the current that can be supplied without excessive voltage drop.

Paralleling sources of unequal emf causes a damaging circulating current.

Kirchhoff's Laws • Kirchhoff's current law (KCL, the junction rule): the algebraic sum of the currents at any node is zero — what flows in must flow out.

It is a statement of the conservation of charge. • Kirchhoff's voltage law (KVL, the loop rule): the algebraic sum of the emfs and potential drops around any closed loop is zero.

It is a statement of the conservation of energy. • Sign convention: currents entering a node are taken as positive and leaving as negative (or the reverse, consistently); going round a loop, a rise in potential is positive and a drop negative.

An assumed current direction that turns out to be wrong simply gives a negative answer, which is self-correcting. • Systematic methods built on these laws: mesh (loop) analysis, which applies KVL to each independent loop and solves for loop currents — best when there are few loops and voltage sources predominate; and nodal analysis, which applies KCL at each node relative to a chosen reference (ground) and solves for node voltages — best when there are few nodes and current sources predominate.

The number of equations required is b − n + 1 for mesh analysis and n − 1 for nodal analysis, where b is the number of branches and n of nodes.

Network Theorems Theorem Statement and use Superposition In any linear network with more than one source, the response in any branch is the algebraic sum of the responses caused by each source acting alone, all other voltage sources being replaced by a short circuit and current sources by an open circuit (internal resistances retained).

It applies to current and voltage but NOT to power, because power is a square-law quantity — the classic examination trap Thevenin Any linear two-terminal network can be replaced, as seen from those terminals, by a single voltage source VTh in series with a single resistance RTh.

VTh is the open-circuit voltage at the terminals;

RTh is the resistance looking back into the network with all independent sources deactivated.

Its value is that the load can then be varied and recalculated in one line — the standard tool for analysing a transducer bridge feeding an amplifier Norton The dual: the same network is replaced by a current source IN in parallel with RN, where IN is the short-circuit current at the terminals and RN = RTh.

The two equivalents interconvert by VTh = INRTh Maximum power transfer Maximum power is delivered when RL = RTh (or, in AC, when ZL = ZTh*, the complex conjugate), the maximum power being VTh²/4RTh at an efficiency of 50 % Reciprocity, Millman, Tellegen, substitution Reciprocity: in a linear bilateral network a source and its response may be interchanged.

Millman's theorem combines several parallel voltage sources into one Matrix Methods • Mesh and nodal analysis produce a set of simultaneous linear equations that are naturally written in matrix form. • Mesh form: [R][I] = [V], where [R] is the resistance matrix, whose diagonal elements Rkk are the sum of all resistances in mesh k and whose off-diagonal elements Rjk are the negative of the resistance common to meshes j and k (negative because, with all loop currents taken in the same direction, they oppose in the shared branch); [V] is the vector of the algebraic sum of source emfs in each mesh.

The matrix of a network of passive bilateral elements is symmetric. • Nodal form: [G][V] = [I], where [G] is the conductance matrix, whose diagonal elements are the sum of the conductances at node k and whose off-diagonal elements are the negative of the conductance between nodes j and k, and [I] is the vector of currents injected into each node by sources. • Solution: by Cramer's rule (each unknown is the ratio of two determinants, Ik = Δk/Δ), by matrix inversion, [I] = [R]−1[V], or by Gaussian elimination, which is what computer packages use.

The great advantage of the matrix formulation is that it is systematic and programmable, which is the basis of SPICE and every other circuit simulator; the same structure extends to AC circuits simply by using complex impedances in place of resistances.

7.6

AC Operation of Magnetic Circuits, Transformers and DC Machines

ABmE0706

This section covers hysteresis and eddy current losses in magnetic circuits, the types and operation of transformers, and the basic principles of DC motors and generators.

Magnetic Circuits and Core Losses • A magnetic circuit is the closed path of magnetic flux, and it obeys a relation analogous to Ohm's law: flux Φ = mmf/reluctance = NI/S, where the reluctance S = l/(μ₀μrA).

The corresponding field quantities are magnetising force H = NI/l (ampere-turns per metre) and flux density B = μ₀μrH (tesla). • Hysteresis: when a ferromagnetic material is taken round a cycle of magnetisation, B lags behind H, tracing the hysteresis loop with its remanence (residual flux density at H = 0) and coercivity (the reverse H needed to reduce B to zero).

The area of the loop represents the energy lost per cycle per unit volume, dissipated as heat in realigning the magnetic domains.

Steinmetz's empirical law:

Ph = η·f·Bm x·V, where the Steinmetz exponent x is about 1.6-2.0 — so hysteresis loss is proportional to frequency and to roughly the square of the peak flux density.

It is reduced by choosing a material with a narrow loop — silicon steel for transformers (a 'soft' magnetic material), whereas permanent magnets require a wide loop (a 'hard' material). • Eddy currents: the alternating flux induces circulating currents in the conducting core itself, which dissipate I²R heat.

Pe = k·f²·Bm²·t²·V — so eddy current loss varies with the square of the frequency, the square of the flux density and, critically, the square of the lamination thickness t.

This is why cores are built from thin laminations (0.3-0.5 mm) insulated from one another with varnish or oxide, which breaks up the current paths, and why silicon is added to the steel to raise its resistivity.

At high frequencies, laminations are inadequate and ferrite (a non-conducting ceramic) or powdered iron cores are used — which is why the switching supplies of 7.4 and the RF circuits of 7.2 use ferrite. • Together hysteresis and eddy current losses are the 'iron' or 'core' losses, and because they depend on frequency and flux density rather than on load current, they are essentially constant at constant voltage and frequency — the 'constant losses' — whereas copper (I²R) losses vary with the square of the load.

Transformers • A transformer transfers electrical energy between two circuits by mutual induction at constant frequency, with no moving parts and no electrical connection between the windings.

Its operation depends on an alternating flux, so it cannot work on DC — indeed connecting a transformer to DC destroys it, since the winding then presents only its small resistance. • EMF equation:

E = 4.44·f·N·Φm = 4.44·f·N·Bm·A volts — the single most examinable transformer formula (the 4.44 arising as 4 × form factor 1.11).

It follows that for a given voltage and frequency the flux density is fixed, and that a higher frequency permits a smaller core — the principle exploited by switch-mode supplies. • Turns ratio:

V₁/V₂ = N₁/N₂ = I₂/I₁ = k for an ideal transformer, and impedance transforms as the square of the turns ratio, Z₁ = k²Z₂ — which is the basis of impedance matching. • Equivalent circuit: winding resistances R₁ and R₂, leakage reactances X₁ and X₂ (from flux that does not link both windings), and a shunt magnetising branch of X₀ (magnetising current) and R₀ (core loss).

Voltage regulation is the fall in secondary voltage from no load to full load, expressed as a percentage, and arises from these impedances.

Efficiency η = output/(output + copper loss + iron loss), and is maximum when the variable copper loss equals the constant iron loss — typically 96-99 % in power transformers.

Losses are measured by two standard tests: the open-circuit (no-load) test, performed on the low-voltage side, which gives the iron loss and the magnetising branch; and the short-circuit test, performed on the high-voltage side at reduced voltage, which gives the copper loss and the equivalent impedance. • Types: by construction — core type (windings surround the core) and shell type (core surrounds the windings); by function — step-up and step-down, power and distribution, isolation (1:1, used for patient safety, as discussed in 6.6), autotransformer (a single tapped winding — cheaper and more efficient but with no isolation, so it is unsuitable for patient circuits), instrument transformers (CT and PT) for measurement, audio and pulse transformers, and the high-tension transformer of an X-ray generator (4.1). • Three-phase transformers may be connected star-star, delta-delta, star-delta or delta-star, the last two also introducing a 30° phase shift.

DC Generators • A DC generator converts mechanical into electrical energy, working on Faraday's law, with the direction of the induced emf given by Fleming's right-hand rule. • Construction: the yoke (frame and magnetic path), field poles and field winding (producing the main flux), the armature — a laminated cylinder carrying the winding in which the emf is generated — the commutator, and the brushes.

The generated emf is inherently alternating; it is the commutator, a split-ring mechanical rectifier rotating with the armature, that converts it to unidirectional output at the brushes — this is the defining feature of a DC machine. • EMF equation:

E = ΦZNP/(60A) volts, where Φ is the flux per pole, Z the total number of armature conductors, N the speed in rpm, P the number of poles and A the number of parallel paths (A = 2 for a wave winding, A = P for a lap winding).

In short, E ∝ ΦN. • Types by excitation: separately excited, and self-excited — shunt (field in parallel with the armature; nearly constant voltage, used for general supply and battery charging), series (field in series, carrying the full load current; voltage rises steeply with load, used as a booster) and compound (both windings — cumulative compounding can be arranged to give a flat or rising characteristic). • Armature reaction — the distortion of the main field by the armature's own mmf — shifts the magnetic neutral axis and causes sparking; it is corrected by interpoles and compensating windings.

Losses are copper, iron, brush contact and mechanical (friction and windage).

DC Motors • A DC motor is the same machine run in reverse, converting electrical into mechanical energy by the force on a current-carrying conductor in a magnetic field, F = BIl, with the direction given by Fleming's left-hand rule. • Back emf: as the armature rotates it generates an emf opposing the applied voltage — Eb = V − IaRa.

Its significance is that the motor draws exactly the current it needs: as the load increases the speed falls, the back emf falls, and the armature current rises.

It also explains why the starting current is dangerously large, since at standstill Eb = 0 and the current is limited only by the small armature resistance — hence the need for a starter with a series resistance that is cut out as the motor accelerates. • Torque and speed:

T ∝ Φ·Ia and N ∝ Eb/Φ = (V − IaRa)/Φ.

Hence speed is controlled by (a) armature voltage or resistance control — giving speeds below normal — and (b) field (flux) control — giving speeds above normal, the standard Ward-Leonard scheme using the former. • Characteristics of the three types: the shunt motor runs at almost constant speed regardless of load, and is the general-purpose 'constant speed' machine (lathes, blowers, centrifuges); the series motor develops a very high starting torque, its speed varying inversely with load — and it must never be started on no load, because with almost no flux the speed rises dangerously and it may 'run away', so it is used only where the load is permanently coupled (traction, cranes, hoists); the compound motor combines a high starting torque with a safe no-load speed. • Relevance in medical equipment:

DC and brushless DC motors drive infusion and syringe pumps (5.4, 6.5), centrifuges, dialysis pumps, X-ray tube anodes (an induction motor), ventilator turbines, surgical drills, patient tables and imaging gantries, where precise speed control, smooth torque, low electrical noise and freedom from brush sparking matter; this is why brushless DC and stepper motors with electronic commutation have largely replaced brushed machines in modern devices.