Photodiode Transimpedance Amplifier — Design & Noise Analysis

Design tool for Butterworth-optimized photodiode transimpedance amplifiers (TIAs), with focus on balanced (differential) detection applications.

Application

Two identical photodiodes are connected to two identical TIA channels. The small difference between their output voltages is extracted as the signal. This requires:

• Low and stable $I_b$ (JFET-input opamp): inter-channel DC offset $= R_f \times \Delta I_b$
• Low temperature drift of $I_b$: bipolar opamps produce unacceptable offset drift for this application
• Noise performance is secondary to $I_b$ stability

Features

• Butterworth-optimized $C_f$ — exact solution (no approximation of $\omega_c R_f C_f \gg 1$)
• 2nd-order Butterworth NEB — $\pi\sqrt{2}/4 \cdot f_{-3\text{dB}}$, not the single-pole approximation
• Piecewise voltage noise integration — accounts for noise gain shape (zero, rising, plateau, rolloff regions)
• Opamp comparison table — compare all opamps in the database with one function call
• Parameter sweep & plotting — explore $R_f$ vs SNR/bandwidth/shot-noise-limit trade-offs
• CSV-based component database — add new opamps or photodiodes by editing a CSV file

Project structure

OPAMP_circuit_design/
├── tia_design.py                  # Calculation engine (functions & dataclasses)
├── Photodiode_TIA_design.ipynb    # Interactive notebook
├── docs/theory_note.tex           # Full derivations of TIA formulas (LaTeX source)
├── docs/theory_note.pdf           # Compiled PDF
├── data/
│   ├── opamps.csv                 # Opamp parameter database
│   └── photodiodes.csv            # Photodiode parameter database
└── README.md

Quick start

from tia_design import *

opamps = load_opamps()
pd = load_photodiodes()['PIN-6D']

# Single design
result = design_tia(opamps['OPA827'], pd, R_f=20e3, Vb=12, E_photo=0.15e-3)
print_result(result)

# Compare all opamps
compare_opamps(opamps, pd, R_f=20e3, Vb=12, E_photo=0.15e-3)

# Compare JFET-input opamps only
jfet = {k: v for k, v in opamps.items() if v.input_type == 'JFET'}
compare_opamps(jfet, pd, R_f=20e3, Vb=12, E_photo=0.15e-3)

Example output

=== TIA Design: OPA827 ===
  C_d  = 55.0 pF,  C_i = 73.0 pF
  C_f  = 7.26 pF  (Butterworth optimum)
  BW   = 1.48 MHz,  NEB = 1.64 MHz
  Noise gain peak = 11.1

  Noise breakdown (output-referred RMS):
    E_noe  (opamp voltage) = 79.7 uV
    E_noR  (Rf thermal)    = 23.2 uV
    E_no_bg    (total BG)  = 83.0 uV
    E_noi_sig  (sig shot)  = 131.6 uV

  Figures of merit:
    Signal         = 1650.00 mV
    SNR            = 10604  (80.5 dB)
    bg/shot ratio  = 0.63  (shot-noise limited)
    Offset/ch      = 60.0 nV

Adding a new opamp

Add one row to data/opamps.csv. Each column corresponds to a specific datasheet parameter:

CSV column	Datasheet parameter	Unit	Where to find in datasheet
name	Part number	—	Front page
f_c	Gain Bandwidth Product (GBW or GBP)	Hz	"Electrical Characteristics" table, or "Open-Loop Gain vs Frequency" plot (frequency at 0 dB)
I_b	Input Bias Current	A	"Electrical Characteristics" table, typical value
f_f	1/f noise corner frequency	Hz	"Input Voltage Noise Spectral Density vs Frequency" plot — the frequency where the 1/f slope meets the white-noise floor
e_nif	Input Voltage Noise Density (flat/white region)	V/sqrt(Hz)	Same plot as above — the flat floor level at high frequency (typically > 1 kHz)
i_nif	Input Current Noise Density (flat/white region)	A/sqrt(Hz)	"Input Current Noise Spectral Density vs Frequency" plot — the flat floor level. For JFET opamps this is often listed only in the characteristics table
C_icm	Common-Mode Input Capacitance	F	"Electrical Characteristics" table or "Typical Application" section
C_id	Differential Input Capacitance	F	Same as above. If not listed separately, set to 0
input_type	Input stage type	—	JFET, bipolar, or CMOS — from product description or "Features" section
notes	Free-form description	—	—

Example row:

OPA828,45e6,1e-12,200,3.4e-9,4.5e-15,5.2e-12,3e-12,JFET,New low-noise JFET opamp

Photodiode parameters

Photodiode parameters are stored in data/photodiodes.csv:

CSV column	Datasheet parameter	Unit	Description
name	Part number	—	—
Cd_spec	Junction Capacitance at specified reverse bias	F	"Electrical Characteristics" table, at Vb_spec
Vb_spec	Reverse bias voltage for Cd_spec	V	The bias condition under which Cd_spec is specified
phi_b	Built-in voltage (contact potential)	V	Typically 0.2-0.6 V for Si. Used in the abrupt-junction model: C_d(Vb) = Cd0 / sqrt(1 + Vb/phi_b)
Id_spec	Dark current at specified reverse bias	A	"Electrical Characteristics" table, at Vb_spec
r_phi	Responsivity	A/W	"Responsivity vs Wavelength" plot, at operating wavelength
wl	Operating wavelength	m	Wavelength at which r_phi is specified

Operating conditions

The design_tia() function takes the following operating condition parameters:

Parameter	Unit	Description
R_f	Ohm	Feedback resistance of the TIA. Determines transimpedance gain (V_out = R_f x I_p). Larger R_f gives higher gain and lower relative thermal noise, but reduces bandwidth
Vb	V	Reverse bias voltage applied to the photodiode. Affects junction capacitance C_d (higher Vb reduces C_d) and dark current I_d
E_photo	W	Incident optical power on the photodiode. The signal photocurrent is I_p = E_photo x r_phi
T	K	Operating temperature (default 296 K = 23 C). Affects Johnson noise of R_f
C_s	F	Stray capacitance in parallel with C_f (e.g. PCB parasitic). Default 0
C_d_override	F	Optional override for photodiode capacitance. Use when the measured C_d differs from the abrupt-junction model

Theory

Full step-by-step derivations are given in docs/theory_note.pdf.

Closed-loop transfer function

The TIA uses an opamp with a single-pole open-loop gain $A(s) = \omega_c / s$, where $\omega_c = 2\pi f_c$ (gain-bandwidth product in rad/s). Applying feedback analysis to the inverting configuration with feedback impedance $Z_f = R_f \parallel C_f$ and input capacitance $C_i = C_d + C_{icm} + C_{id}$, the closed-loop transimpedance is:

$$Z_{TI}(s) = \frac{-R_f \, \omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$$

where the natural frequency and damping ratio are:

$$\omega_n^2 = \frac{\omega_c}{R_f (C_i + C_f + C_s)}, \qquad 2\zeta\omega_n = \frac{1 + \omega_c R_f C_f}{R_f (C_i + C_f + C_s)}.$$

Butterworth design

The Butterworth (maximally-flat) condition $Q = 1/\sqrt{2}$ (i.e. $\zeta = 1/\sqrt{2}$) eliminates gain peaking. Substituting into the damping equation and using $C_c = 1/(\omega_c R_f)$:

$$(1 + \omega_c R_f C_f)^2 = 2\,\omega_c R_f (C_i + C_f + C_s)$$ $$\Rightarrow \quad C_f = \sqrt{C_c \bigl(2(C_i + C_s) - C_c\bigr)}, \qquad C_c = \frac{1}{2\pi f_c R_f}.$$

This is the exact solution without the common approximation $\omega_c R_f C_f \gg 1$. See Graeme 1996.

Bandwidth and noise equivalent bandwidth

For the Butterworth response, the $-3,\text{dB}$ bandwidth equals the natural frequency:

$$f_{-3\text{dB}} = \frac{\omega_n}{2\pi} = \sqrt{\frac{f_c}{2\pi R_f (C_i + C_f + C_s)}}.$$

The noise equivalent bandwidth for an $n$-th order Butterworth filter is $\text{NEB} = f_{-3\text{dB}} \cdot \frac{\pi}{2n} / \sin!\left(\frac{\pi}{2n}\right)$. For $n = 2$:

$$\text{NEB} = f_{-3\text{dB}} \cdot \frac{\pi\sqrt{2}}{4} \approx 1.11 \, f_{-3\text{dB}}.$$

This replaces the single-pole approximation $\text{NEB} = (\pi/2) f_{-3\text{dB}}$.

References

• Graeme, J. (1996). Photodiode Amplifiers: Op Amp Solutions. McGraw-Hill.
• Hobbs, P. C. D. (2009). Building Electro-Optical Systems: Making It All Work. 2nd ed., Wiley.

Requirements

Python >= 3.10. Install dependencies:

pip install -r requirements.txt