Noise and Interference in Instrumentation — Part 3: Modelling Diode and BJT Noise

Parts 1–2 introduced noise sources and their spectral densities. This lecture shows how to build circuit-level noise models of real semiconductor devices: first the diode, then the bipolar junction transistor (BJT). Both are non-linear, so the analysis starts from a small-signal (linearised) model and attaches the physical noise mechanisms to it. The lecture ends with the subtle part — why a BJT’s input-referred noise needs a voltage and a current source, and why they cannot be treated as independent.

1. Small-signal model of a diode

A diode is a non-linear device described by the Shockley equation:

with I_S the reverse saturation current (order of 10⁻¹⁰ A), η the ideality factor (typically 1–2) and V_T the thermal voltage. The equation assumes steady state — it says nothing about how the diode responds to time-varying signals.

We study small signals because they are the ones most susceptible to noise. For a small signal the diode is usually forward biased, and in the small region around the operating point Q the curve can be replaced by its tangent:

This slope has units of conductance, so its inverse is a resistance — the dynamic resistance r_d = ηV_T/I_DQ. To capture time-varying behaviour the junction capacitance C_j and the diffusion capacitance C_d are placed in parallel with r_d.

2. Diode noise sources

A diode has three main noise sources, all attached to the small-signal model:

• Shot noise — the PN junction is a potential barrier that carriers must cross, so the diode current has shot noise, modelled as a parallel current source across the junction (hence across r_d): ī_ns² = 2qI_DQΔf, with I_DQ the quiescent current.
• Flicker noise — being a semiconductor device it has 1/f noise, also a parallel current source: ī_nF² = K_F I_DQ^α ln(f_H/f_L), where K_F and α are device-dependent constants.
• Thermal noise — the contacts and silicon substrate have a real resistance R_s, modelled as a resistor in series with r_d that generates a series voltage noise v̄_nRs² = 4k_BT R_sΔf.

3. Total diode noise voltage

The three sources sit in different branches, so they cannot simply be added. R_s’s thermal noise is a series voltage, whereas shot and flicker noise are parallel currents. The fix is to convert the parallel currents into an equivalent series (Thévenin) voltage across r_d, after which KVL applies and, with the sources uncorrelated, the mean-square values add:

Substituting I_DQ = ηV_T/r_d and V_T = k_BT/q collapses the shot-noise term and gives the compact form from the lecture:

For simplicity the capacitances are ignored here (open circuit), which is reasonable for frequencies up to the MHz range where their roll-off has not yet begun.

4. BJT small-signal model and current gain

There are several small-signal models for a BJT. The lecture groups them into physics-based models (the hybrid-π model, with simplified, low-frequency and high-frequency variants, and the T-model) and two-port / black-box models (h-parameters, built from measured terminal behaviour). The analysis uses the high-frequency hybrid-π model with g_m, r_π, r_o, C_π and C_μ (Fig. 2).

Linearising I_C ≈ I_Se^V_BE/V_T about the operating point gives the transconductance g_m = I_CQ/V_T (collector shorted so r_o does not matter), and the DC current gain follows directly:

With the collector shorted for AC, the base sees r_π in parallel with C_π and C_μ, so the AC current gain is a low-pass response:

This resembles an RC low-pass filter with corner f₋₃ = 1/[2πr_π(C_π+C_μ)]. As ω approaches ω_T the current gain falls to 1: beyond that the BJT no longer works as an amplifier, so ω_T is the maximum useful operating frequency. Because g_m = I_CQ/V_T, the unity-gain frequency is not a fixed property of the device — it changes with the quiescent collector current.

5. The six noise sources of a BJT

A BJT has six main noise sources: three thermal, two shot and one flicker.

• Thermal noise from the bulk resistances of the emitter, base and collector regions (R_E, R_B, R_C), added in series with the corresponding terminals. R_E and R_C are generally ignored: the emitter is heavily doped (minimal resistance), and the collector, although lightly doped, has a wide cross-section and a well-doped substrate to carry signals outward. R_B cannot be ignored: the base is a very narrow, only moderately doped region, so its resistance is comparatively high, and noise generated there is amplified by β.
• Shot noise: there are two current paths, and each crosses a potential barrier. Forward-biasing the base–emitter junction sends electrons from the highly doped emitter into the thin base; almost all diffuse across and are swept into the collector, so the collector current is a stream of carriers crossing a barrier — collector shot noise ī_nC² = 2qI_CQΔf, in parallel with the C–E junction. The small difference between I_E and I_C is the base current I_B (a few charges recombine in the base); it crosses the B–E junction and gives base shot noise ī_nB² = 2qI_BQΔf, in parallel with the B–E junction.
• Flicker noise: as with any semiconductor, trap states give 1/f noise, modelled as a current source in parallel with the B–E junction: ī_nF² = K I_BQ^α ln(f_H/f_L).

As with the diode, the dynamic resistances r_π and r_o are not real resistors and add no noise.

6. Referring BJT noise to the input

A BJT is a two-port, so its equivalent input-referred noise model needs a series voltage source and a parallel current source across the input port. Everything is transferred to the input side: base thermal noise is already in series at the input; base shot and flicker noise are already across the input; only the collector shot noise is still at the output.

To refer the collector shot noise to the input as a current, divide by the squared magnitude of the AC current gain; to refer it as a voltage, divide by the squared transconductance (an AC quantity):

Adding up the pieces, the complete input-referred noise model of the BJT is:

The additions above are valid only when the sources have zero mean and are uncorrelated, and when KVL/KCL permit the addition. In reality the collector and base noise are somewhat correlated, but for this course they are treated as uncorrelated.

7. Why both a voltage and a current source are needed

This is the conceptual heart of the lecture. The input-referred collector shot noise appears twice in the model — as ī_nc−ir and as v̄_nc−ir — but they are the same noise source, so they are not independent: when the total noise at the output is calculated, the contributions of these two terms cannot simply be added as squared quantities. Their correlation matters.

The calculation uses superposition for each source with the signal source replaced by its source resistance R_S (at low frequency, so C_π and C_μ drop out). The result for the output noise current of a common-emitter stage is:

The last term is the cross-correlation between the voltage and current forms of the collector shot noise. Two observations follow:

• Ideal source (R_S = 0): the expression collapses to ī_out² ≈ g_m²(v̄_nB² + v̄_nc−ir²), and it is the base thermal noise and the voltage form of the collector shot noise that dominate — the base shot and flicker noise vanish because R_B is small.
• Non-zero source resistance: the current form of the collector shot noise now matters too, because it flows through R_S. If only the current-source part were propagated to the output (or only the voltage part), the result would not equal the true collector noise current ī_nC; with both parts and the correlation term, it does.

8. Exercise and take-homes

Important take-homes from the lecture:

• Noise is fundamentally an AC quantity and a small signal in itself; begin with small-signal models for analysis.
• Dynamic resistances in small-signal models do not generate noise — they represent behaviour, not physical resistors. Bulk resistances and terminal resistances are physical and must be considered.
• The capacitances were mostly ignored during these derivations; think about how including them would change the noise characteristics (effective bandwidth).
• A diode has thermal, shot and flicker noise. A BJT has several intrinsic noise sources, of which four are important (base thermal, base shot, collector shot, flicker).
• You must be able to calculate the AC current gain β_AC and explain why it is needed; refer noise sources to the input so a generic noise model results; and understand why the 2-port model needs both a voltage and a current source at the input.
• Explain how source impedance changes which noise sources (voltage or current) dominate; use superposition to propagate individual sources, and recognise when input-referred sources are not independent.
• Derivations are not examinable in detail — but the concepts used (and why each simplification is valid) are.