Electronics

• transducer (sensor): converts a biological signal into an electrical signal

• analog processing: conditions the electrical signal (offset, amplify, filter) to prep it for A/D conversion

• A/D conversion: converts analog → digital (sampled + quantised) for digital signal processing

what are the components of a measurement system (sensing chain)

Digital signal → D/A conversion → Analog signal → Analog processing

→ Electrical signal → Transducer (actuator) → Biological signal

• D/A conversion: digital → analog

• analog processing: conditions the analog signal for actuation

• transducer (actuation): converts electrical signal back into a biological signal/effect

what are the components of an actuation system (reverse chain)

1. understand the physiology (physics) of the system

2. know the variables to be measured

3. know the function and limitations of the components used

what are they keys to a successful bioinstrument design

• a property governing how particles are affected by EM fields

• conserved — net charge in an isolated system is constant

• quantised — integer multiples of elementary charge e = 1.602176634×10^-19 C

• proton: +e, electron: -e

what is charge

• energy required to move a positive charge between two points (potential difference)

• moving + charge from low potential to high potential → high potential consumes energy; high → low releases energy

• 1 joule moves 1 columb through 1 volt

what is voltage

• rate of flow of charge past a point

• 1 ampere = 1 coulomb/second

• by convention, current flows from high → low potential (opposite to actual electron flow, since electrons are negative)

• voltage exists between two points; current flows through a device

• voltage sources: batteries (electrochemical), generators (electromechanical), solar cells (photovoltaic)

what is current

• DC (direct current): constant with time

• AC (alternating current): varies with time (typically sinusoidal)

DC vs AC

• RMS current/voltage = the equivalent DC value producing the same average power dissipation in a resistive load:

  Paverage​=I_rms²​R=V_rms²/R​​

• General definition:

  y_rms=sqrt(1/T₂−T₁∫^T1_T2 y(t)² dt​

• For sampled data:

  y_rms=sqrt(1/n ∑_n=1ⁿ y_i²)

• For a zero-mean sine wave y(t)=asin⁡(ωt):

  y_rms=a/sqrt(2)=y_pp/2sqrt(2)​​

  • y_pp - peak to peak amplitude, distance from top of waveform to bottom (y_pp = 2a)

RMS (root mean square)

Signal	Equation	RMS
DC	y = a	|a|
Sine	y = a sin(2πft)	a/√2
DC-shifted sine	y = a sin(2πft) + b	√(a²/2 + b²)
Square	±a	a
Triangle	—	a/√3
Sawtooth	—	a/√3

RMS of common waveforms

• component: device with terminals affecting electrons/fields

• circuit: components connected by conductive wires (traces)

• circuit diagram: standard symbols (IEC 60617, IEEE 315-1975)

• traces = ideal wires: no voltage drop; all points on the same trace share the same voltage

• diagram conventions: trace connection (3-terminal), trace junction (4-terminal), trace crossing (unconnected)

Circuit basics

• fixed voltage regardless of current drawn

• current supplied is determined by the external circuit

• symbols: general (time-varying), DC, AC (sinusoidal)

ideal voltage source

• fixed current regardless of voltage across it

• common misconception: it does NOT have zero voltage across it — voltage depends on the connected circuit

ideal current source

Factor	Name	Symbol		Factor	Name	Symbol
10²⁴	yotta	Y		10⁻²⁴	yocto	y
10²¹	zetta	Z		10⁻²¹	zepto	z
10¹⁸	exa	E		10⁻¹⁸	atto	a
10¹⁵	peta	P		10⁻¹⁵	femto	f
10¹²	tera	T		10⁻¹²	pico	p
10⁹	giga	G		10⁻⁹	nano	n
10⁶	mega	M		10⁻⁶	micro	µ
10³	kilo	k		10⁻³	milli	m

SI prefixes

• V = IR → I = V/R → R = V/I (V in volts, I in amps, R in Ohms)

• energy delivered to a resistor is dissipated immediately as heat — no energy stored

• made from conducting material (carbon, thin metal/carbon film, poorly conducting metal)

• have power ratings limiting max V and I

Ohm’s Law

• E3, E6, E12 series — logarithmically distributed standard resistor values

• more digits in the series name = more values available per decade (E12 has 12 values per decade)

E-series (standard values)

• 4-band: 1st digit, 2nd digit, multiplier, tolerance

• 5-band: 1st, 2nd, 3rd digit, multiplier, tolerance

• 6-band: adds temperature coefficient (ppm/°C)

• Example: 56 × multiplier gives value (e.g. 56 × 1kΩ = 56kΩ)

resistor colour code

• P=IV

• P=I²R

• P=RV^2​

(Units: P in Watts = J/s; I in Amps = C/s; V in Volts = J/C)

Power

Series:

R=R1+R2 (always larger)

Parallel (2 resistors):

R=R1R2/R1+R2=1/(1/R1+1/R2) (always smaller)

Parallel (N resistors):

R=1/∑_n=1^N 1/R_n​

• Voltage is the same across parallel resistors

• Current through resistor m: I_m=I⋅(1/Rm)/(∑_n=1^N 1/Rn)

Resistors in series and parallel

• I = V_in/R1 + R2

• V_out = IR2 = (R2/R1+R2)xV_in

• application: resistance strain gauge

  • resistance changes when the guage is stretched

  • bonded to a material → resistance change gives a measure of material strain

voltage divider

Kirchoff’s Current Law (KCL)

• sum of currents into a node = sum of currents out (conservation of charge)

• _n=1∑^N​ I_n​ = 0

  • e.g. I₁ + I₂ - I₃ - I₄ = 0

Kirchoff’s Voltage Law (KVL)

• sum of voltages around any closed loop = 0

• consequence: components in parallel have the same voltage drop

Kirchoff’s Circuit Laws

1. choose the minimum number of closed loops so every component appears in at least one loop

2. write a KVL equation for each loop (careful with current direction)

3. arrange into matrix form [R][I] = [V] and solve for the currents [I]

4. once currents are known, use Ohm’s law to find voltage drops across each resistor and hence node voltages

outline the method for solving circuits (mesh analysis)

• two terminal device storing energy in an electric field

  • Q = CV

    (Q in coulombs, V in volts, C in farads)

• differentiating (C constant):

  • I = C dV/dt

• 1 A into 1 F capacitor → voltage rises at 1 V/s

physical structure

• two conductive plates separated by a dielectric (insulator)

• blocks DC current flow → acts as an open circuit at DC

capacitors in series/parallel

• parallel: C = C1 + C2 + … + C_n

• series: C = 1/(1/C1 + 1/C2 + … 1/C_n)

RC circuit (discharging)

• charged capacitor across a resistor: from KCL,
  C dV/dt = I = -V/R

• solution: exponential decay

  V = V_ie^-t/RC

• time constant τ = RC — time for voltage to fall to e^-1 = 0.368 of initial value

Capacitors

• two-terminal device storing energy in a magnetic field when current flows; releases energy when current is interrupted

• typically an insulated wire coiled around a core
  V = L dI/dt

  • L in henries — 1 V across 1 H → current rises at 1 A/s

Inductors in series/parallel:

• parallel: L = 1/(1/L₁ + 1/L₂ + … + 1/L_n)

• series: L = L1 + L2 + … + L_n

Transformers:

• two closely coupled coils (primary/secondary)

• AC voltage on primary → voltage on secondary scaled by turns ratio; current scaled inversely (power conserved)

• uses: changing AC line voltage; electrically isolating circuits

Inductors

• periodic signal: x(t) = x(t + nT), T = period

• sinusoid: x(t) = asin(ωt + ϕ)

  • a = amplitude, ω = angular frequency (rad/s) = 2πf = 2π/T, φ = phase (rad)

• representative periodic waveforms: sine, square, triangle, sawtooth

Time-varying signals

Why complex numbers:

• linear circuit driven by a sine wave → output is a sine wave, same frequency, different amplitude/phase

• Ohm’s law generalises: resistance → impedance (complex: real resistive part + imaginary reactive part)

• convention note: engineers use j = sqrt(-1) (since i means current), but physicists/mathematicians use i

Complex Number Forms:

• cartesian: z = x + iy, where x = ℜ(z), y=ℑ(z)

• polar: z = re^iθ = r(cosθ + isinθ)

  ∣z∣=r= sqrt(x²+y²) ∠z=θ=tan⁻¹(y/x)

Representing Voltages/Currents as complex numbers:

• V₀cos(ωt + ϕ) is represented by complex number V₀e^iϕ

• actual signal recovered via: ℜ(e^iωt⋅V₀​e^iϕ)=V₀​cos(ωt+ϕ)

Impedance of Basic Components:

Z_R​=R Z_C​=1/iωC​= -i/ωC ​Z_L​=iωL

• Generalised Ohm's law: I=V/ZI=V/Z or V=IZV=IZ

• Series/parallel impedance combination rules are the same as for resistance

Impedence on the Complex Plane:

• resistor impedance lies on the real axis

• inductor impedence lies on the positive imaginary axis (+iωL)

• capacitor impedence lies on the negative imaginary axis (-i/ωC)

• Why bother with complex representation? Differential equations in the time domain become algebraic equations on the complex plane

• Time can be visualised as a third axis: eiωteiωt traces a helix combining sin⁡(ωt)sin(ωt) and cos⁡(ωt)cos(ωt)

Impedance and Complex Numbers

Component	Voltage vs Current
Resistor	Voltage in phase with current
Capacitor	Voltage lags current by 90°
Inductor	Voltage leads current by 90°

Phase relationships

• replace resistors with impedances:

  I = Vin/Ztotal, Ztotal = Z1 + Z2, Vout = Z₂I = Vin x Z2/Z1+Z2

• combining R with C (or L) → frequency-dependent voltage divider (a filter), because Zc and Z_L depend on ω

Series Combinations
Resistor + Inductor:
Z=R+iωL, ∣Z∣=sqrt(R²+ω²L²​), θ=tan⁻¹(ωL/R​)

Resistor + Capacitor:
Z = R - i/ωC, |Z| = sqrt (R² + 1/ω²C²), θ = tan^-1(-1/RωC)

Generalised Voltage Divider (Frequency-Dependent)

High-Pass Filter (Capacitor in series, output across resistor)

• Circuit: Vin → C → R → ground; Vout taken across R

  Vout = Vin(R + i/ωC)R/(R² + 1/ω²C²)

Magnitude:
∣Vout​∣= 2πfRC/sqrt(1+(2πfRC)²​)​∣Vin​∣

Phase:
∠Vout​=tan⁻¹(1/ωRC​)

• Behaviour: passes high frequencies (ω » 1/RC), blocks low frequencies (ω « 1/RC)

• Cutoff frequency: ω_c = 1/RC (i.e. τ = RC is the time constant, ω_c its reciprocal)

• At cutoff, output amplitude = 1/sqrt(2) of input (power = ½ of input) → this is the -3dB point

  • 20log₁₀(1/sqrt(2)) = 10log₁₀(1/2) ≈ -3dB

• At DC (ω = 0): capacitor blocks all current → Vout = 0

Low-Pass Filter (Resistor and Capacitor swapped)

• Circuit: Vin → R → C → ground; Vout taken across C
  Vout = (1/sqrt(1 + ω²R²C²)) x Vin

• Same cutoff: ω_c = 1/RC, f_c = 1/2πRC

• At DC (ω = 0): capcaitor blocks current → no drop across R → Vout = Vin (opposite behaviour to high-pass)​

Bode Plots (Both Filters)

• x-axis: frequency (log scale)

• Magnitude in dB, Phase in degrees

• High-pass: magnitude rises with frequency at +20 dB/decade below cutoff, flattens above; phase goes from 90° → 0° (45° at cutoff)

• Low-pass: magnitude falls with frequency at −20 dB/decade above cutoff, flat below; phase goes from 0° → −90° (−45° at cutoff)

• Both show the characteristic −3 dB point exactly at the cutoff frequency

Inductors vs Capacitors in Filters

• Inductors could replace capacitors in RL filters

• Rarely used in practice: bulky, expensive, poorer electrical characteristics than capacitors

• Practical inductor use: ferrite beads/chokes to raise impedance at very high frequencies (suppress oscillations) — this damps unwanted high-frequency noise/ringing (e.g. EMI, circuit instability) that could otherwise corrupt sensitive signals or cause a device to fail EMC compliance testing

Filters