Electronics Guide

Resistor Selection

All resistors generate thermal noise (Johnson-Nyquist noise) with a voltage spectral density of:

en = sqrt(4kTR) V/sqrt(Hz)

where k is Boltzmann's constant, T is absolute temperature, and R is resistance. This fundamental noise cannot be eliminated, only minimized by using the lowest practical resistance values and, where possible, operating at reduced temperatures.

Beyond thermal noise, resistors can exhibit excess noise:

• Carbon composition resistors: Generate significant excess noise, especially under DC bias, due to the granular contact structure. Avoid in low-noise applications
• Carbon film resistors: Better than composition but still exhibit measurable excess noise
• Metal film resistors: Excellent choice for low-noise designs with minimal excess noise
• Wire-wound resistors: Lowest excess noise but introduce inductance; suitable for DC and low-frequency applications
• Thin-film resistors: Outstanding noise performance for integrated and precision applications

The noise index, specified in dB, indicates excess noise above thermal. A noise index of 0 dB means the resistor generates 1 microvolt RMS per volt of DC drop per decade of bandwidth. Metal film resistors typically achieve noise indices of -20 dB or better.

Capacitor Selection

While ideal capacitors generate no thermal noise (having no resistive element), real capacitors have parasitic resistances that contribute noise:

• Equivalent Series Resistance (ESR): The resistive component generates thermal noise; lower ESR is preferred
• Leakage resistance: High-value resistances appear in parallel; generally negligible for noise
• Dielectric absorption: Can cause memory effects but not direct noise contribution

For coupling and bypass applications in sensitive circuits:

• Film capacitors: Polypropylene and polystyrene offer excellent noise performance
• Ceramic capacitors (C0G/NP0): Stable, low loss, suitable for precision circuits
• Avoid high-K ceramics: X7R and Y5V types exhibit piezoelectric effects that can convert mechanical vibration to electrical noise

Active Device Selection

Active devices are characterized by equivalent input noise voltage (en) and noise current (in), both specified in spectral density units:

• Bipolar transistors: Offer lowest voltage noise at moderate source impedances; shot noise in base current creates current noise that dominates at high source impedances
• JFETs: Very low current noise due to high input impedance; voltage noise higher than bipolar but excellent for high-impedance sources
• MOSFETs: Higher flicker noise than JFETs due to surface-channel conduction; suitable for integrated circuits but generally not the lowest-noise discrete option
• Operational amplifiers: Specified with en and in; choose based on source impedance and frequency range

The crossover impedance, where voltage noise and current noise contribute equally, is Rcross = en/in. For minimum total noise, the source impedance should match this value, though practical constraints often prevent achieving this optimum.

Noise Model of Amplifiers

Any amplifier can be modeled with an equivalent input noise voltage en in series with the input and an equivalent input noise current in flowing into the input. For a source with resistance Rs, the total input-referred noise is:

en_total = sqrt(en^2 + (in x Rs)^2 + 4kTRs)

The three terms represent amplifier voltage noise, amplifier current noise multiplied by source resistance, and thermal noise of the source resistance itself.

Optimum Source Resistance

Minimizing the total noise yields an optimum source resistance:

Ropt = en / in

At this source resistance, the voltage and current noise contributions are equal. However, this optimum assumes the source resistance can be freely chosen, which is rarely the case in practice.

When the source impedance is fixed:

• Low source impedance (Rs less than Ropt): Voltage noise dominates; select amplifier with lowest en
• High source impedance (Rs greater than Ropt): Current noise dominates; select amplifier with lowest in
• Matched source impedance (Rs approximately equal to Ropt): Both contribute; overall noise figure is optimized

Impedance Transformation

When the source impedance is not optimal, transformation techniques can improve noise performance:

• Transformer coupling: Transforms source impedance while adding minimal noise; most effective for low-frequency applications
• Reactive matching networks: LC networks can transform impedance at specific frequencies; add negligible noise
• Resistive matching: Never appropriate for low-noise design as resistors add thermal noise

Noise Reduction by Paralleling

When n identical devices operate in parallel:

• Voltage noise: Reduces by sqrt(n) since uncorrelated noise sources add in quadrature
• Current noise: Increases by sqrt(n) as n uncorrelated current sources combine
• Optimum source resistance: Decreases by factor of n

For an amplifier with original en and in, paralleling n units yields:

en_parallel = en / sqrt(n)

in_parallel = in x sqrt(n)

Practical Implementation

Paralleling is most effective when:

• Source impedance is lower than the single-device optimum
• Voltage noise is the dominant limitation
• Power consumption and cost increases are acceptable

Implementation considerations:

• Output combination: Sum output currents or average output voltages
• Bias matching: Ensure uniform current sharing among parallel devices
• Thermal coupling: Mount devices on common heatsink for matched temperature
• Layout symmetry: Equal trace lengths to maintain balanced operation

Diminishing Returns

Each doubling of parallel devices reduces voltage noise by only 3 dB. Practical limits include:

• Power consumption increases linearly with device count
• At some point, other noise sources (source resistance, interference) dominate
• Matching and bias complexity increases
• Beyond 4-8 devices, alternative approaches often prove more practical

Impedance Transformation

A transformer with turns ratio N:1 transforms impedances by N^2:

• A source impedance Rs appears as Rs x N^2 at the secondary
• An amplifier input impedance Rin appears as Rin / N^2 at the primary

This allows matching a low-impedance source to an amplifier's optimum source impedance. For example, a 50 ohm source can be transformed to 5000 ohms with a 1:10 step-up transformer, matching an amplifier with Ropt of 5000 ohms.

Noise Contribution of Transformers

Real transformers add noise through:

• Winding resistance: Thermal noise from primary and secondary resistances
• Core losses: Equivalent resistance representing hysteresis and eddy current losses
• Coupled noise: External interference magnetically coupled through the core

High-quality audio and instrumentation transformers minimize these effects through:

• Low-resistance windings using heavy gauge wire
• High-permeability, low-loss core materials
• Electrostatic shielding between windings
• Magnetic shielding enclosures

Common-Mode Rejection

Transformers inherently reject common-mode signals that appear equally on both input terminals:

• Balanced primary windings: Center-tapped or bifilar windings cancel common-mode signals
• Electrostatic shields: Grounded Faraday shields between windings block capacitive coupling
• Isolation: Galvanic isolation breaks ground loops that cause common-mode interference

Frequency and Bandwidth Limitations

Transformer performance is inherently band-limited:

• Low-frequency limit: Set by primary inductance; adequate inductance requires large cores for low frequencies
• High-frequency limit: Limited by leakage inductance and winding capacitance
• Optimal bandwidth: Typically 2-3 decades for well-designed audio transformers

Transformers are most practical for audio frequencies (20 Hz to 20 kHz) and certain RF bands. DC and very low frequencies require alternative coupling methods.

Electrostatic Shielding

Electrostatic shields protect against capacitively coupled electric field interference:

• Principle: A conductive enclosure intercepts electric field lines, preventing them from reaching the enclosed circuit
• Grounding: The shield must be connected to the circuit's reference potential
• Effectiveness: Shielding effectiveness depends on enclosure conductivity, thickness, and lack of apertures

Practical implementations:

• Metal enclosures for complete circuits
• Shielded cables for signal interconnections
• Ground planes on PCBs
• Guard traces around sensitive nodes

Magnetic Shielding

Magnetic fields require different shielding approaches:

• High-permeability materials: Mu-metal and similar alloys redirect magnetic flux around the protected region
• Thickness requirements: Multiple thin layers often outperform single thick shields
• Saturation concerns: Strong fields can saturate high-permeability materials, reducing effectiveness
• Frequency dependence: High-permeability shields are most effective at low frequencies; eddy current shielding in conductive materials works at higher frequencies

Guard Rings and Driven Guards

Guard techniques protect high-impedance nodes from leakage currents:

• Guard rings: A conductive trace surrounding the sensitive node, driven to the same potential, intercepts surface leakage currents
• Driven shields: Cable shields driven by a buffer following the signal voltage eliminate capacitive loading and leakage
• Triaxial cables: Inner shield guards the center conductor; outer shield provides electrostatic protection

Guard effectiveness depends on maintaining the guard at exactly the same potential as the protected node. Any voltage difference causes leakage current to flow. Buffer amplifiers driving guards must have extremely low offset and noise.

PCB Layout for Shielding

Printed circuit board layout significantly affects noise immunity:

• Ground planes: Solid copper pour beneath sensitive circuits provides shielding and low-impedance return paths
• Component placement: Keep sensitive amplifier inputs away from switching circuits, digital logic, and power stages
• Trace routing: Avoid running noisy signals near sensitive traces; use perpendicular crossings when necessary
• Via stitching: Multiple vias connecting top and bottom ground planes improve high-frequency shielding

Twisted Pair Fundamentals

Twisting two conductors together provides several benefits:

• Magnetic field rejection: Equal and opposite voltages induced in each half-twist cancel when summed
• Electric field rejection: Both conductors receive equal capacitive coupling; differential receiver rejects the common-mode component
• Controlled impedance: Twist geometry determines characteristic impedance, typically 100-120 ohms for common cables

The tightness of the twist (twists per unit length) determines rejection effectiveness. Tighter twists provide better rejection but increase capacitance and attenuation.

Differential Signaling Advantages

In differential transmission, the signal is encoded as the difference between two complementary waveforms:

• Common-mode rejection: Interference affecting both conductors equally is rejected by the differential receiver
• Doubled signal swing: The differential signal is twice the single-ended amplitude, improving signal-to-noise ratio by 6 dB
• Ground independence: The signal depends on the difference between conductors, not their absolute potential relative to ground
• Reduced EMI emission: Equal and opposite currents in closely spaced conductors produce minimal net radiation

Common-Mode Rejection Ratio

The effectiveness of differential signaling depends on receiver CMRR:

Vout = Ad x Vdiff + Acm x Vcm

where Ad is differential gain, Acm is common-mode gain, Vdiff is the differential signal, and Vcm is the common-mode interference.

CMRR = Ad / Acm (often expressed in dB)

Factors limiting CMRR:

• Cable imbalance: Differences in conductor length, capacitance, or resistance
• Source imbalance: Unequal source impedances on the two signal lines
• Receiver imbalance: Mismatched input impedances or gain paths in the receiver
• Frequency effects: CMRR typically degrades at higher frequencies

Implementation Considerations

Achieving maximum benefit from differential signaling requires attention to detail:

• Source termination: Match source impedance to cable impedance for minimum reflections
• Receiver termination: Terminate the cable at the receiver to absorb incident energy
• Ground connections: Ground cable shield at one end only to avoid ground loops, or use isolation
• Balanced drivers: Use drivers that produce truly balanced, complementary outputs
• Instrumentation amplifiers: For precision measurements, use high-CMRR instrumentation amplifiers as receivers

Bandwidth Limiting

The total noise voltage in a system is:

Vn = en x sqrt(BW)

where en is noise spectral density and BW is noise bandwidth. Reducing bandwidth from 1 MHz to 1 kHz reduces total noise by a factor of approximately 32 (30 dB).

Optimal bandwidth equals the signal bandwidth plus any necessary margin for:

• Component tolerances
• Temperature variations
• Frequency drift
• Filter transition band

Low-Pass Filtering

Low-pass filters are the most common noise-limiting technique:

• First-order filters: Simple RC circuits provide 20 dB/decade roll-off; noise bandwidth equals 1.57 times the -3 dB frequency
• Higher-order filters: Steeper roll-off approaches noise bandwidth equal to -3 dB frequency
• Active filters: Provide gain and sharp cutoffs but add noise from active devices
• Switched-capacitor filters: Precise cutoff frequencies but add clock noise; include anti-aliasing filter

Band-Pass Filtering

When the signal occupies a limited frequency band, band-pass filtering rejects noise both above and below the signal band:

• LC filters: Low noise, high Q achievable, but large at low frequencies
• Crystal filters: Extremely narrow bandwidth for specific applications
• Active band-pass filters: Flexible center frequency and bandwidth but add noise
• Synchronous detection: Equivalent to extremely narrow band-pass centered on reference frequency

Notch Filtering

Notch filters remove specific interference frequencies:

• Power line rejection: 50/60 Hz notch filters remove power frequency interference
• Twin-T networks: Passive notch with adjustable depth
• State-variable filters: Active topology providing precise notch frequency
• Adaptive notch: Automatically tracks interfering frequency

Notch filters are most effective when interference is at a known, stable frequency well separated from signal frequencies.

Power Supply Filtering

Power supply noise directly affects amplifier output. Effective supply filtering includes:

• Bulk capacitance: Large electrolytics provide low-frequency energy storage
• High-frequency bypass: Ceramic capacitors close to IC power pins block high-frequency noise
• RC or LC filtering: Additional stages for sensitive circuits
• Separate regulators: Dedicate low-noise regulators to sensitive analog sections

Operating Principle

The chopper stabilization process involves three steps:

1. Modulation: The input signal is multiplied by a square wave at the chopping frequency, translating it to odd harmonics of the chopping frequency
2. Amplification: The modulated signal is amplified at frequencies where amplifier noise is predominantly thermal (white)
3. Demodulation: Synchronous demodulation with the same square wave returns the signal to baseband

Amplifier offset and 1/f noise, which are DC and low-frequency, are modulated to the chopping frequency and filtered out, not appearing at the output.

Noise Reduction Mechanism

Chopper stabilization is effective because:

• Offset elimination: DC offset becomes AC at the chopping frequency and is removed by output filtering
• 1/f noise rejection: Low-frequency noise is similarly modulated away from the signal band
• White noise floor: The effective input noise approaches the amplifier's white noise level

Chopper amplifiers routinely achieve equivalent input noise of less than 10 nV/sqrt(Hz) down to DC, whereas conventional amplifiers exhibit rising noise at low frequencies.

Chopping Artifacts

Practical chopper amplifiers exhibit several artifacts:

• Clock feedthrough: Switching transients couple to the output; minimized by careful layout and timing
• Intermodulation: Input signals near the chopping frequency alias back to baseband
• Residual offset: Charge injection from switches creates small residual offsets
• Bandwidth limitation: Input bandwidth limited to less than chopping frequency

Auto-Zero Techniques

Auto-zeroing is a related technique that periodically measures and subtracts amplifier offset:

• Ping-pong architecture: Two amplifiers alternate between amplifying and calibrating
• Sample and hold correction: Offset is sampled during null phase and subtracted during amplification
• Digital correction: Offset measured and corrected in the digital domain

Auto-zero amplifiers achieve similar low offset and 1/f noise rejection as choppers but may introduce different artifact characteristics.

Combined Architectures

Modern precision amplifiers often combine chopping with other techniques:

• Chopper plus auto-zero: Reduces both 1/f noise and chopping artifacts
• Nested chopper: Multiple chopping stages at different frequencies for extended low-frequency performance
• Chopper plus continuous-time path: Parallel paths optimize both DC precision and high-frequency performance

Single-Point Grounding

In single-point (star) grounding, all ground returns meet at one point:

• Advantages: Eliminates ground loops; prevents high-current return paths from affecting sensitive circuits
• Limitations: Impractical at high frequencies where ground lead inductance matters
• Best for: Audio frequencies and precision DC measurements

Multi-Point Grounding

Multi-point grounding uses a low-impedance ground plane:

• Advantages: Low inductance; effective at high frequencies
• Limitations: Ground currents may flow through sensitive areas
• Best for: RF and high-speed digital circuits

Hybrid Grounding

Most practical systems use hybrid approaches:

• Star grounding for low-frequency analog signals
• Ground plane for digital and high-frequency sections
• Single connection point between analog and digital grounds
• Careful routing to prevent return current interaction

Ground Loops

Ground loops occur when multiple ground connections create current paths:

• Mechanism: Voltage drops in ground conductors appear as error signals
• Sources: Power frequency currents, transient ground disturbances
• Solutions: Single-point grounding, isolation, differential signaling

Gain Distribution

The Friis equation for cascaded noise figure shows that early stages dominate overall noise:

Ftotal = F1 + (F2-1)/G1 + (F3-1)/(G1 x G2) + ...

This leads to key design principles:

• Maximize gain in the first stage (consistent with bandwidth and stability)
• Use the lowest-noise devices where signals are weakest
• Accept higher noise in later stages where signal level is higher

Analog and Digital Partitioning

Digital circuits generate substantial interference that can couple into sensitive analog sections:

• Physical separation: Place analog and digital sections in different areas of the PCB
• Separate power supplies: Use independent regulators with filtered or isolated power feeds
• Ground management: Connect analog and digital grounds at one point near the power supply
• Clock management: Use the slowest clock rates adequate for the application; avoid clock harmonics in sensitive analog bands

Shielded Enclosures

Complete system shielding may be necessary for extremely sensitive measurements:

• Welded or gasketed metal enclosures provide high attenuation
• Filtered feedthrough capacitors on power and signal lines
• Shielded connectors with proper termination
• Attention to ventilation openings, which can compromise shielding

Low-Noise Preamplifier Design

A typical low-noise preamplifier for audio or instrumentation applications:

• Input stage: JFET or selected low-noise bipolar transistor matched to source impedance
• Gain: 20-40 dB in first stage to overcome following stage noise
• Feedback: Carefully chosen to minimize noise contribution
• Power supply: Well-regulated with extensive filtering
• Layout: Ground plane, short signal paths, separated power and signal routing

Precision Measurement System

For microvolt-level DC measurements:

• Chopper-stabilized or auto-zero amplifier for offset and 1/f noise elimination
• Guarded inputs with driven shields for high-impedance sources
• Differential input with high CMRR instrumentation amplifier
• Heavy filtering to minimize bandwidth to measurement requirements
• Shielded, temperature-controlled enclosure for critical components

RF Low-Noise Amplifier

For radio frequency applications:

• Device selection based on noise figure at operating frequency
• Input matching network optimized for noise rather than power transfer
• Output matching for stability and power transfer
• Shielded module construction to prevent feedback
• Appropriate grounding for RF frequencies