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Transimpedance amplifiers - Application in optical sensing with photodiodes

In usual optical detectors a given flux of photons (the quantum of the electromagnetic field) are directly converted into a flux of electrons (quantum of electrical current). An associated practical problem is that our measurement instruments are often more appropriate to measure voltage, instead of currents. A transimpedance amplifier is a configuration which converts a current into a (possibly large) voltage with the help of an amplifier.

A brief recapitulation of amplifiers nomenclature

An amplifier is a device that receives some signal (current or voltage) at its input and outputs another signal (also a current or voltage). There are then 4 possible configurations. Suppose that at the input one has a voltage signal \(V_{in}\), and the output is another voltage \(V_{out}\). This amplifier can be characterized by its voltage gain \(A_v = \frac{V_{out}}{V_{in}}\). For an amplifier which inputs and outputs currents, we also can characterize the current gain as \(A_i = \frac{I_{out}}{I_{in}}\). Notice that \(A_v\) and \(A_i\) are unitless quantities. Now, if an amplifier accepts a current as an input and outputs a voltage, we can’t characterize any gain in terms of an unitless quantity. To remain the gain characterization in terms of signal out/signal in, we must have \(Z_{amp}=\frac{V_{out}}{I_{in}}\), which has units of Ohms. For a constant, DC signal, \(Z_{amp}\) would be a resistance, while in general for time varying signals \(Z_{amp}\) is called an impedance. Equivalently, if an amplifier has an input voltage and outputs a current, it’s called a transconductance amplifier, due to its gain \(G_{amp}=\frac{I_{out}}{V_{in}}\) (expressed in Siemens, or \(\Omega^{-1}\))

Design of a transimpedance amplifier

Some characteristics of photodiodes

Here we will comment some physical aspects of sensing light with photodiodes. At the end of this section there are some photodiode typical characteristics and the circuit design starts in the following section.

Currents in a photodiode

A photodiode is mainly characterized by the PN junction, which is an interface between positively (P) and negatively (N) doped semiconductors. Because of the spatial distribution of dopants at such interface, there is intrinsic tendency of charges to flow (diffusion process) from the P-doped region to the N-doped region and vice-versa. However, this spatial charge diffusion must cease for a PN junction in equilibrium, where there is no external input power.

The spatial charge diffusion means that close to the PN junction there are some negative charges at the P-doped region, while there are also positive charges at the N-doped region. The electric field established after the charge redistribution must be such that no extra charges can flow through the junction. In other words, the field must establish a barrier to the charge flow, and is oriented from the N-region to the P-region.

Considering the PN junction in equilibrium, there are basically two ways to conduct an electric current across this junction:

1. Increase (decrease) the electric potential at the P(N)-doped region. The extra energy can be used to overcome the barrier after some threshold, and an electric current may occur. The current flows the P-doped to the N-doped region.
2. If a photon hits the semiconductor, it may transfer it’s total energy to a charge carrier (either an electron or a hole). If these charge carriers are produced in the PN junction, they will be accelerated by the existing field due to the diffused spatial charges. The current flows the N-doped to the P-doped region. This current is also called reverse current.

The current associated with a voltage \(V_D\) across the diode terminals is well approximated by the Shockley diode equation: \[I(V_D) = I_D \left(e^{V_D/V_T}-1\right)\] \(I_D\) is the dark, or saturation current, and \(V_T=kT/q\) for an ideal diode (\(k\) is the Boltzmann constant, \(T\) the temperature and \(q\) is the electrical charge). The dark current depends on several parameters as doping, temperature and PN junction area, such that it’s a parameter that varies significantly from diode to diode.

The total current across a diode includes also the reverse current, and can be expressed as

\[I = I_D \left(e^{V_D/V_T}-1\right) - I_{rev}\]

We could also include other contributions to the current, as those due to thermal noise, shot noise and 1/f noise. For these contributions, see for example this Honeywell’s Application Note on Photodiodes.

Under zero voltage bias (\(V_D=0\)), one has that \(I=-I_{rev}\) and only the photogenerated current is relevant. Thus, photodiodes are very useful to measure light. How can we relate \(I_{rev}\) to the properties of the electromagnetic field?

Photogenerated current

Suppose that \(N\) photons/s are impinging the photodiode active surface. If all photons are used by charge carriers to produce a current, then \(I_{rev}=qN\). However, this photon-flux to electron-flux conversion process cannot occur with 100% of efficiency due to several reasons, as surface reflection of photons at the photodiode surface and carrier recombination in the semiconductor. To consider the imperfections, we introduce the quantum efficiency \(\eta\), and define \[I_{rev} = \eta q N\]

Often we are not directly interested in the number of photons/s, given that it’s difficult to work in the quantum optics regime. It’s more usual to be interested in the relation between the optical power to the generated current. The energy per photon is \(E=\frac{hc}{\lambda}=\frac{1240\text{ eV.nm}}{\lambda}\), and \(N\) photons/s impinge the photodiode. Thus, the relation between current and power can be expressed through the Responsivity \[\text{Resp}(\lambda) = \eta \frac{\lambda}{1240\text{ nm}} \text{ A/W}\]

Thus for a monochromatic source, one has that \(I_{rev}= R P\). If the light source has a known power spectrum \(P(\lambda)\) and \(\text{Resp}(\lambda)\) is also known, then \(I_{rev}= \int P(\lambda) \left|\frac{d\text{Resp}(\lambda)}{d\lambda}\right| d\lambda\).

The responsivity does not grow indefinitely with \(\lambda\). Indeed, light can efficiently excite single carriers only at wavelengths larger than the material’s bandgap. For silicon at ambient temperature, that means wavelengths around \(1100\text{ nm}\). Thus, the typical responsivity is a curve like the one below. Notice that photodiodes as the FDS100 or BPW34 can reach responsivities quite close to the ideal case (\(\eta=1\)) in the near infrared, while the overall quantum efficiency reduces at smaller wavelengths.

%pylab inline

# Represent the ideal photodiode response
wavelengths = linspace(350, 1150, 100) # UV to near IR
bandgap = 1100 # Silicon bandgap at room temperature
responsivity = wavelengths/1240 # Ideal case responsivity
responsivity[wavelengths>bandgap] = zeros(wavelengths.shape)[wavelengths>bandgap]

# Consider the data for Thorlabs's FDS100 and BPW34 as a comparison
fds100 = loadtxt(r'assets\TIA_FDS100_Responsivity_data.csv', skiprows=1, unpack=True)

# Data retrieved directly from the datasheet using the getdata graph digitizer
bpw34 = loadtxt(r'assets\TIA_bpw34_spectral_response.csv', unpack=True)
i = absolute(bpw34[0]-950).argmin() # Which index corresponds to 950 nm?
bpw34[1] = bpw34[1] * 0.66 / bpw34[1,i] # Adjust the normalized response by the responsivity

plot(wavelengths,responsivity, label="Ideal case")
plot(fds100[0], fds100[1], label="FDS100")
plot(bpw34[0], bpw34[1], label="BPW34")

grid()
xlabel(r"Wavelength (nm)")
ylabel(r"Responsivity (A/W)")
title(r"Responsitivity vs Wavelength")
legend()
savefig(r'assets\TIA_ideal_responsivity.png')

As a final note, we are considering here ordinary photodiodes, where each impinging photon generates a single electron. There is another kind of detector where the carriers lye in a very unstable state, such that a photon may produce an avalanche of carriers, the so-called avalanche photodiode (APD). The physics and electronics behind APD’s is completely different from the case studied here.

Photodiode Norton equivalent

Now that some relevant aspects of the physics behind a photodiode were discussed, we may discuss on how to represent it electrically. Given that it’s quite natural to speak about the currents in a photodiode, it’s often preferable to discuss its properties in terms of its Norton equivalent, instead of Thévenin’s. In summary, we may represent an arbitrary DC circuit by a current source \(I\) in parallel with an equivalent resistor \(R_{Shunt}\). We have already talked about the charge distribution near the PN junction, and given that there are charges separated by a distance over a given area, we may easily accept that there should be an effective capacitance \(C_J\) in parallel to the current flow. This would be a model for the semiconductor itself, while there may be also a resistance associated with the electrical contacts in series with the semiconductor, \(R_{Series}\). If we consider all of these contributions, an effective model for the photodiode is given below.

The value of \(R_{Series}\) is often negligible. Meanwhile, \(R_{Shunt}\) can be estimated from the small-signal conductance \(g=\frac{dI}{dV_D} = \frac{I_D}{V_T} e^{V_D/V_T}\). At zero bias voltage \(V_D=0\), and \(g = \frac{I_D}{V_T} = 1/R_{Shunt}\). Given that \(V_T\approx 25\,\text{mV}\) and \(I_D\approx\) pA-nA range, \(R_{Shunt}\gtrsim 1\,\text{G}\Omega\) and can be safely neglected. These simplifications are also valid under reverse bias (\(V_D<0\)). The last effective component is the junction capacitance \(C_J\). \(C_J\) depends on several internal characteristics (as junction area, doping profile, temperature, bias voltage…) and is very important in the determination of the AC properties of the transimpedance amplifier. We may thus simplify the photodiode circuit to the model below.

Typical photodiode specs

1. Thorlabs FDS100

• Optical characteristics
  • Responsivity: 0.65 A/W (@ 980 nm)
  • Active area: 3.6mm x 3.6mm
• Electrical characteristics
  • Dark current: 1 nA (Typ.)
  • Internal capacitance: 237 pF (@ 0V Reverse bias)

2. Vishay BPW34

• Optical characteristics
  • Responsivity: 0.66 A/W (@ 950 nm)
  • Active area: 7.5 mm\(^2\)
• Electrical characteristics
  • Dark current: 2 nA (Typ.)
  • Reverse light current: 50 \(\mu\)A (Typ.)
  • Internal capacitance: 70 pF (@ 0V Reverse bias)

# Conversion of BPW34 datasheet parameters to responsivity @ 950 nm
I_revCurrent = 50e-6 # A
irradiance = 1 * 1e-3/(1e-2)**2 # W/m^2
area = 7.5 * (1e-3)**2 # m^2
R = I_revCurrent / (irradiance * area)

Direct signal measurement

We may now use the simplified model to consider what we will measure by connecting a photodiode directly to a voltmeter or an oscilloscope. The circuit is essentially a resistor \(R\) across the diode terminals, and a voltmeter that measures \(V_D\). Instead of \(C_J\) only, we may also consider that the measurement instrument also contains some effective capacitance across it’s input terminals, \(C_{in}\), and the total capacitance is \(C = C_J + C_{in}\).

DC analysis

Let us consider that the light source has a constant optical power \(P\). Given that a current \(I\) flows across the photodiode, \[V_D= -R I = R I_{rev} - R I_D \left(e^{V_D/V_T}-1\right).\]

The previous expression is transcendental in \(V_D\), such that we’re not able to express \(V_D\) in terms of \(I_{rev}\) using elementary functions. However, we may gain understanding with a graphical solution. Let’s consider what happens for the BPW34 photodiode when connected across a 1 M\(\Omega\) resistor (typical value for oscilloscopes). Using the dark current given above, \(R I_D = 2\,\text{mV}\). Also, for improved clarity, let’s consider \(V_{rev}=R I_{rev}\) as the voltage contribution to \(V_D\) due to light only. Ideally, we would be interested in having \(V_D=V_{rev}\). However, the increase in \(V_D\) leads to a nonlinear relation between \(V_{D}\) and \(V_{rev}\). When does the deviation from the linear behavior becomes significant? The figure below shows the measured voltage \(V_D\) versus the voltage due to photodetection only, \(V_{rev}\).

%pylab inline
V_D = linspace(0, 100, 100) # Voltage across diode, in mV

R = 1e6     # 1MOhm resistor
I_D = 2e-9  # Dark current (in A)
V_T = 25    # Carrier thermal energy at room temperature (in mV)

v_rev_0nA = V_D
# the 1e3 factor below converts the voltage to mV
v_rev_2nA = V_D + (R * I_D * 1e3) * (exp(V_D/V_T)-1)

# Approximate a mean linear relation between V_D and v_rev
i = abs(V_D-60).argmin() # Which V_D index is closer to V_D==60 mV?
V_mean_slope = (v_rev_2nA[i]-v_rev_2nA[0])/(V_D[i]-V_D[0])
v_linearized = V_mean_slope * V_D

plot(v_rev_0nA, V_D, label=r'$I_D=0$')
plot(v_rev_2nA, V_D, label=r'$I_D=2$ nA')
plot(v_linearized, V_D, label=r'Linearized relation')
legend()
ylabel(r'$V_D$ (mV)')
xlabel(r'$V_{rev}=V_D + R I_D \left(e^{V_D/V_T}-1\right)$ (mV)')
grid()
savefig(r'assets\TIA_diode_saturation.png')

Notice that the dark current is very important to the photodiode linearity. The deviation grows exponentially with an increase in the \(V_D/V_T\) ratio, what indicates that the upper voltage limit to a linear detector using this circuit is some small multiple of \(V_T\). However, if we consider only small \(V_D\) voltages, its possible to obtain an approximate linear relation. Assuming that one calibrates the photodiode voltage at \(V_D=60\,\text{mV}\), the deviation between the linearized relation and the actual relation between \(V_D\) and \(V_{rev}\) is relatively small for \(0 \leq V_D \leq 60\,\text{mV}\). This is an important limitation when using photodiodes directly, given that such small voltages are difficult to measure using oscilloscopes (small signals may be easily lost when the photodiode operates in this linearized range). Another remark is that the photodiode response can be brought to the linear regime with the reduction of the load resistor \(R\), at the cost of reduced sensitivity.

To sum up the discussion, there are a few worthy comments. Supposing that we’re in the linearized representation of the PD, we may state that \(V_{rev}=\alpha V_{D}\). Also, the typical currents necessary to maintain the diode in the linearized region occur for \(V_D\approx 50\,\text{mV}\), or when \(I_{rev}\approx 50\,\text{nA}\) (Assuming \(R=1\,\text{M}\Omega\)).

AC analysis

Now we will consider what happens to the signal when the light beam pulses. This is specially important, considering that photodiodes often used to measure light from ultrafast lasers, where the light pulses have durations ranging from ns (\(10^{-9}\) s) to fs (\(10^{-15}\) s). Given that the currents and voltages are dynamic, the effective capacitor \(C\) becomes relevant. Using Kirchhoff laws for the voltages and currents, we may check that the voltage applied to all elements is \(V_D\), whereas the current through the diode is divided between the capacitor \(C\) and the resistor \(R\).

\[-I = I_{rev} - I_D \left(e^{V_D/V_T}-1\right) = C \frac{dV_D}{dt} + \frac{V_D}{R}\]

\[RC \frac{dV_D}{dt} + \left[V_D + R I_D \left(e^{V_D/V_T}-1\right)\right] = RI_{rev}\]

Again, an exact solution to the above equation is considerably difficult. However, if we consider that the photodiode operates in the linearized region for simplicity (\(V_D/V_T\) less than a factor of 2 or 3), we can approximate \(\left[V_D + R I_D \left(e^{V_D/V_T}-1\right)\right] \approx \alpha V_D\). Just to have an idea of the order of magnitude, the DC analysis above indicated that \(\alpha\approx0.75\). Now it becomes much simpler to understand the behavior of \(V_D\),

\[RC \frac{dV_D}{dt} + \alpha V_D = RI_{rev}.\]

The above expression is similar to the case where a capacitor \(C\) discharges through a resistor \(R\). There is a source term (\(RI_{rev}\)) which is due to the impinging light beam. We may then consider the transfer function between the light source and \(V_D\). Assuming \(V_D=V_{D\omega}e^{j\omega t}\), and \(I_{rev} = \text{Resp}(\lambda) P_\omega(\lambda) e^{j\omega t}\), and

\[V_{D\omega} = \frac{R}{\alpha + j \omega RC} \text{Resp}(\lambda) P_\omega(\lambda)\]

Therefore, if the beam power contains terms oscillating at a high frequency (\(\omega \gg \alpha/RC\)), then these terms will not be present in \(V_{D\omega}\). Thus, for fast signals one must use photodiodes with small capacitances. Fast photodiodes can be easily recognized for their small active region area.

Another relevant condition is the circuit response due to an ultrashort pulse. In this case the pulse is so fast that the associated current occurs almost instantly. The beam power can be expressed in terms of the Dirac delta impulse, \(P(t)=E_{single-pulse} \delta(t)\), where \(E_{single-pulse}\) is the energy contained in the optical pulse and the Dirac Delta satisfies the following properties: \(\delta(t)=0\) for \(t\neq 0\) and \(\int_{-\epsilon}^{\epsilon} \delta(t) dt = 1\) for infinitesimal \(\epsilon\). For the short pulse, and considering that \(V_D=0\) before the pulse, we have that \[\frac{dV_D}{dt} = \frac{1}{C} \text{Resp}(\lambda) E_{single-pulse} \delta(t) - \frac{\alpha}{RC} V_D.\]

Integrating both sides only from \(-\epsilon\) to \(\epsilon\), where \(\epsilon\) is infinitesimal, we have that

\[V_D(\epsilon) - V_D(-\epsilon) = \frac{1}{C} \text{Resp}(\lambda) E_{single-pulse} - \frac{\alpha}{RC} \int_{-\epsilon}^{\epsilon} V_D(t) dt.\]

\(V_D(-\epsilon)=0\), because it’s \(V_D\) before the pulse, while \(\int_{-\epsilon}^{\epsilon} V_D(t) dt \leq \text{Maximum}\left[V_D(-\epsilon<t<\epsilon)\right]2\epsilon \rightarrow 0\) in the limit where \(\epsilon \rightarrow 0\). Thus, the photodiode current produced by an ultrafast pulse is equivalent to consider an instantaneous charging of the effective capacitor \(C\) with a charge \(\text{Resp}(\lambda) E_{single-pulse}\), since

\[V_D(t=0^+) = \frac{1}{C} \text{Resp}(\lambda) E_{single-pulse}\]

To verify the behavior of \(V_D\) after the pulse, it simply an issue of considering \(V_D(t=0^+)\) as the initial condition. This circuit discharges as an RC circuit with a time constant \(RC/\alpha\).

\[V_D(t>0) = \frac{ \text{Resp}(\lambda) E_{single-pulse}}{C} e^{-\alpha t/RC}\]

\(V_D\) can rise very fast (and the limitations in the rise time are beyond this simple model), and decays according to the photodiode capacitor \(C\) and the measurement instrument resistance \(R\). The input resistance of an oscilloscope is typically around \(1\,\text{M}\Omega\), thus if one needs to reduce the circuit response time to observe fast features in the signal, one may connect a smaller resistance in parallel to the photodiode, to reduce the decay time.

Transimpedance amplifier design

While it’s possible to use diodes within their almost linear range, the associated voltages (~50mV) are quite small. It’s much easier to measure voltages in the 1-5V range, where there are various tools available to measure voltages. For instance, one may use an arduino board to perform several measurements. Thus, an amplified signal may bring in several interesting aspects. Here we will consider some basics of an important configuration for photodiodes: the transimpedance amplifier, where the photodiode current is converted into a voltage. The design of these amplifiers is also nicely covered in the Maxim Application Note 5129, and in the Texas Instruments Reference Design 176.

DC analysis

We will consider the ideal case where the photodiode is connected with to the inverting input and the ground, and a single resistor \(R_F\) is connected in the feedback path. We will consider an ideal opamp, where no current flows through the input terminals.

For an opamp under ordinary working circumstances, the output voltage \(V\) is related to the inverting and non-inverting input voltages \(v_-\), \(v_+\) is

\[V = A (v_+ - v_-)\]

The open loop gain \(A\) often has very large values (\(10^{10}-10^{12}\)), such that \(v_+ - v_- = V/A \approx 0\). The condition \(v_+ - v_-\) is also called the virtual short constraint. Given that \(v_+\) is grounded, \(v_-=v_+=v=0\). A nice feature of this configuration is that the voltage across the diode is \(0\,\text{V}\). Then, the dark current contribution becomes negligible and only \(I_{rev}\) contributes to the current through the diode. Given that \(I_{rev}\) must also flow through \(R_F\), \[V=R_F I_{rev}.\]

Notice that we the photodiode was biased \((v_+=v\neq0)\), the dark current would add to \(I_{rev}\), such that \(I_D \left(e^{-v/V_T}-1\right)\approx-I_D\). When considering the reverse bias voltage below, we assume the simplifying assumption that either the bias voltage is zero, such that the dark current is negligible \((I_D=0)\), or that the reverse bias is large enough (\(v \gg V_T\)) that the full dark current is a constant factor (\(I_D \left(e^{-v/V_T}-1\right)\approx-I_D\)). Also, the real opamp draws some bias current \(I_B\) at its inputs, and there is some offset voltage between \(v_+\) and \(v_-\), \(v_{os}=v_+ - v_-\neq 0\). Typical figures for opamps with BJT input transistors are \(I_B~100\,\text{nA}\), while FET or CMOS at the opamp input stage lowers to \(I_B<100\,\text{pA}\). The input offset voltage is often around a few mV or less, and can be trimmed to zero by following the opamp manufacturer suggestions. These are the most important opamp features at DC for the present application, and we will consider these effects below.

Consideration of the currents at the opamp inverting input gives

\[I_{rev} + I_{D} = \frac{V-(v-v_{os})}{R_F} - I_B,\]

\[V = v-v_{os} + R_F(I_{rev} + I_{D} + I_B)\]

where it should be remembered that \(I_D=0\) for the unbiased photodiode (\(v=0\)), while \(I_D\neq0\) in the biased case (\(v\neq0\)).

Unbiased photodiode

In this case \(v=0\) and \(I_{D}=0\). So, the offset voltage gives an error in the order of mV, while the base current \(I_B\) is subject to the same transimpedance gain as the reverse current. Thus, \(I_B\) limits the maximum achievable gain, and it’s interesting to keep it small by selecting FET input opamps. By knowing the output voltage range of interest, at a given reverse current, one may select the \(R_F\) value. For instance, the arduino ADC accepts voltages in the range of 0-5 V. For the BPW34, which has a typical \(I_{rev}=50\,\mu\text{A}\), we may select \(R_F=\frac{5}{50 \cdot 10^{-6}}=100\,\text{k}\Omega\). For higher sensitivities (smaller \(I_{rev}\)), the associated \(R_F\) must be larger.

Biased photodiode

In this case \(v\neq0\) and \(I_{D}\) is characteristic of the photodiode. The \(v_{os}\) represents again a small contribution to \(V\), and our major concerns are \(I_B\) and \(I_D\). A FET input opamp will have \(I_B<100\,\text{pA}\), while \(I_D\approx2\,\text{nA}\). Let’s consider that \(I_B\) is negligible. Then \(V\) has a minimum value of \(v+R_FI_D\) and \(R_FI_{rev}\) indicates the output variation due to the incident light. Increasing \(R\) increases the signal gain, but decreases the maximum output voltage range. If \(I_D\) is much smaller than the typical \(I_{rev}\), than we select \(R_F\) as before. For example, for the BPW34, \(I_D\approx2\,\text{nA}\), while \(I_{rev}\approx50\,\mu\text{A}\). Thus, selecting a \(100\,\text{k}\) resistor implies that the dark current offsets the output by \(200\,\mu\text{V}\). However, if \(I_{rev}\) and \(I_D\) have the same order of magnitude, they will be amplified together and the output offset due to \(I_D\) will become of the order of the output voltage. Thus, for very small light powers, the unbiased configuration is more easily amplified, while the biased photodiode is adequate for larger signals. The biased version will show itself specially useful when we consider the AC response of these circuits.

As a final remark, it’s possible to make \(v=0\) while maintaining the diode reverse-biased by connecting the photodiode to a battery instead of ground.

AC analysis - Oscillations

The DC analysis neglects that the opamp output needs some time to change it’s output value. Indeed, it can be seen in the datasheets that the gain effectively drops with a frequency increase. This characteristic is represented by the unit Gain-Bandwidth-Product (\(\omega_{GBP}\)) frequency, which indicates the frequency where the voltage gain becomes \(1\). To estimate the effect of opamp limited bandwidth in our analysis, we consider that the voltage gain becomes \(A \rightarrow A/\left(1+j\omega/\omega_0\right)\), to simulate the effect of a low pass filter between the input voltage difference and the output voltage. \(\omega_0\) is the filter low-pass 3dB frequency, but it’s more interesting to express it in terms of \(\omega_{GBP}\).

When \(\omega=\omega_{GBP}\), the gain factor becomes 1, \(|A/\left(1+j\omega_{GBP}/\omega_0\right)|=1\), or \(\omega_0 \approx \omega_{GBP}/A\). We can than verify how this affects our virtual short opamp simplification, given that now

\[V = \frac{A}{1+jA\omega/\omega_{GBP}} (v_+ - v_-)\]

In the limit \(A \rightarrow \infty\), we can state approximately that

\[v_+ - v_- = j\frac{\omega}{\omega_{GBP}} V \equiv \frac{1}{\omega_{GBP}} \frac{dV}{dt}\]

Thus, the opamp inputs will be kept at the same voltage only at signal frequencies much smaller than \(\omega_{GBP}\). When the signal frequency increases, the difference can become as large as the output voltage. This is an important feature in the design of transimpedance amplifiers because this opamp behavior will lead to oscillations in \(V\).

Consider again the currents at the opamp inverting input. We have that \(v_-=-\frac{1}{\omega_{GBP}} \frac{dV}{dt}\), and the currents must satisfy

\[ I_{rev} + I_D = - C \frac{dv_-}{dt} + \frac{V-v_-}{R_F} \]

\[ R_F(I_{rev} + I_D) = \frac{R_FC}{\omega_{GBP}} \frac{d^2V}{dt^2} + \frac{1}{\omega_{GBP}} \frac{dV}{dt} + V\]

Notice that the above equation for \(V\) is of second order in time, as in the harmonic oscillator equation. These oscillations, if present, aren’t related with the input light signal, and are undesirable features in the output signal.

To study these natural oscillations, we consider how \(V\) behaves without any external inputs (\(I_{rev}=I_D=0\)). Then,

\[\frac{d^2V}{dt^2} + \frac{1}{R_FC} \frac{dV}{dt} +\frac{\omega_{GBP}}{R_FC}V = 0\]

\(V = V_0 e^{j\lambda t}\) gives the characteristic equation

\[-\lambda^2 +j\lambda \frac{1}{R_FC} +\frac{\omega_{GBP}}{R_FC} = 0\]

\[\lambda = \frac{1}{2R_FC}\left(j \pm \sqrt{4\omega_{GBP}R_FC - 1}\right)\]

Thus, if \(4 \omega_{GBP}R_FC < 1\), then \(\lambda\) will be purely imaginary and \(V\) always decays exponentially. The oscillations are overdamped. The oscillations are called as critically damped when \(4 \omega_{GBP}R_FC=1\), and underdamped when \(4 \omega_{GBP}R_FC>1\). In the underdamped regime, \(V_0\) can oscillate several times before settling to its final value. This can be a problem specially when using pulsed lasers, where the optical pulse is much shorter than the circuit response times. If we consider the BPW34 typical figures given in the DC analysis, when using a 1 MHz GBP opamp, we have \(4 \omega_{GBP}R_FC=176\gg1\). Notice that we cannot modify \(C\), which is an intrinsic feature of our detection. To remove oscillations from the circuit above either we must decrease the opamp bandwidth or decrease the transimpedance gain through \(R_F\).

C = 70e-12
R_F = 1e5
GBP = 2 * pi * 1e6
4 * GBP * R_F * C # 175.92918860102841

What can be done to remove this misbehavior? The critical condition for the onset of oscillations is \(\omega_{GBP}R_FC=1\). If we want to maintain our transimpedance gain, we have to modify the loop bandwidth somehow. The cause of these oscillations is that \(v_-\propto\frac{dV}{dt}\), specially at higher frequencies. If we modify the feedback loop such that it becomes a follower at higher frequencies, the oscillations are not possible anymore. To decrease the loop impedance at higher frequencies, we may add a capacitor \(C_F\) in parallel to \(R_F\). The loop impedance becomes \(Z_L = \frac{R_F}{1+j\omega R_F C_F}\). We can apply this impedance to the equation for \(V\) in frequency representation and look for purely imaginary eigensolutions for \(\omega\), which therefore will not include oscillations.

After some algebra, it can be verified that as long as \(1 - 2 C_F R_F \omega_{GBP} - 4 C R_F \omega_{GBP} + C_F^2 R_F^2 \omega_{GBP}^2>0\), the solution will not oscillate. Solving for \(C_F\), it is obtained that given

\[C_F > \frac{1 + 2 \sqrt{\omega_{GBP} R_F C}}{\omega_{GBP} R_F}\]

the opamp output voltage will not oscillate. The solution above is slightly different than those proposed in Maxim Application Note 5129 or Texas Instruments Reference Design 176, but the numerical results are quite close. Indeed, the present equation is slightly more conservative. This is interesting, because the value found for \(C_F\) depend strongly on the opamp bandwidth \(\omega_{GBP}\), but this is another parameter which can vary significantly (\(\pm40\%\) according to Maxim’s AN), since it’s not a trimmed parameter. Thus, the present equation can be used without having to underestimate \(\omega_{GBP}\).

C = 100e-12
GBP = 2*pi* 1.0e6

R_F = logspace(4,7,100)
our_eq   = (1+sqrt(   4 * GBP * R_F * C))/  (GBP * R_F)
maxim_eq = (1+sqrt(1+ 8 * GBP * R_F * C))/(2*GBP * R_F)
ti_eq    = (1+sqrt(1+ 4 * GBP * R_F * C))/(2*GBP * R_F)

semilogx(R_F,   our_eq*1e12, label='This')
semilogx(R_F, maxim_eq*1e12, label='Maxim AN 5129')
semilogx(R_F,    ti_eq*1e12, label='TI RD 176')
grid(); legend(); xlabel('$R_F$ ($\Omega$)'); ylabel('$C_F$ (pF)')
title(r'Feedback capacitor selection - $C=100$ pF, $f_{GBP}=1$ MHz')
savefig(r'assets\TIA_feedback_capacitor_selection.png')

Therefore, if we return to our BPW34 design attempt, the feedback loop capacitor must have at least \(C_F=23\,\text{pF}\) for a \(100\,\text{k}\Omega\) feedback resistor, and \(\omega_{GBP}=2\pi\cdot 1.0\,\text{MHz}\).

C = 70e-12
R_F = 1e5
GBP = 2*pi* 1.0e6

C_F = (1+2*sqrt(GBP * R_F * C))/(GBP * R_F) # 23 pF
C_F

Transimpedance amplifier design summary

Some opamp specs

Below are shown some specs for some ordinary opamps.

TIA - Version 1

TIA - Version 2 - Noise considerations

Simulation

Experimental results

Complete python code