May 06, 2012

More advanced discussion and design information including noise calculation is available in these articles:

- 1.6 MHz Bandwidth Transimpedance Photodiode Amplifier (ADA4627-1)
- 45 kHz High Gain Transimpedance Photodiode Amplifier (OPA606)
- Transimpedance Noise Calculator Bare Bones Calculator
- Transimpedance Cumulative Noise Calculation
- Transimpedance Full Noise Calculation
- Bias Offset Noise
- Transimpedance Reference Data
- Transimpedance f3db and Cf Design Curves for Maximally Flat Response
- Feedback Amplifier Stability
- Feedback Amplifier Stability Calculator
- Op-amp Output Capacitance Isolation

In this model, the actual frequency response of the photodiode, when terminated into a simple resistive load is implicitly modeled by its capacitance in parallel with a load resistance. This is a lumped-element simplification of the actual photodiode transport processes (junction drift and carrier diffusion) which determine the true high frequency response. When the photodiode is used with a feedback circuit such as the transimpedance amplifier discussed here, the frequency response will be determined by the feedback network as well as the amplifier characteristics (GBW etc.).

The photodiode is often reverse-biased to reduce the capacitance Cp, increase linearity particularly at higher optical signal levels, and increase Rp. For high-quality photodiodes, the leakage current resulting from reverse-bias will be very small (< 1 nA). The ideal photodiode would have zero Cp and infinite Rp along with the maximal possible responsivity (~ λ(um)/1.24 A/W). Real photodiodes have characteristics which depend on the intended usage. High speed photodiodes for fibre-optic systems designed for 100 Mb/s to 10 Gb/s require photodiodes with very low capacitances of <0.5 pF and low reverse leakage current. For low speed control and monitoring applications where light-collection efficiency and convenience is required, larger area photodiodes are used with capacitances ranging from 10 pF to 100 pF or higher and lower junction shunt resistances which become important as well as higher leakage current. The multifaceted design tradeoffs involved in photodiode/op-amp design, with particular emphasis on low noise for very high sensitivity applications have been discussed in the literature, for example: Photodiode Monitoring with OP AMPS.

This article will focus on photodiodes for intermediate bandwidth applications (BW from 1 MHz - 100 MHz) with Cp ~ 5 pF and Rp ~ 100Mohm. For the circuit discussed, the photodiode is essentially an ideal current source with a small shunt capacitance, large shunt resistance and low series resistance. This photodiode capacitance along with the amplifier feedback network will determine the overall amplifier bandwidth and stability. While noise minimization will be important in low level applications, this article will focus on frequency response and stability and assume higher optical power levels are used.

The model for the photodiode includes the effect of series wiring inductance which may be important at higher frequencies depending on the lead length from the photodiode. As a rough benchmark, for a total lead length of 2cm, the self inductance for 22 AWG (0.064 cm diam) leads is 16 nH. At 100 MHz the inductive reactance will be 10 ohm and will increase proportionally with frequency. The op-amp will be modeled as ideal except for a finite gain-bandwidth product with a single low-frequency pole, and input shunt resistance and capacitance as shown. Since in most cases the series resistance Rs of the photodiode is small (< 10 ohm) and the inductive reactance of L is usually negligible for f < 100 MHz, we will simplify the circuit by ignoring these. In this case, the shunt resistances and capacitances of the photodiode and op-amp are in parallel and can be treated as single parameters. This circuit has the nice property that the response has a 2nd order filter-response shape which provides better insight into the capacitance effects on bandwidth and stability. (The exact transimpedance transfer function for the detailed model is provided in Appendix A below) The resulting simplified circuit is shown below:

The total input shunt capacitance

- Ci = 15 pF (Cd = 5 pF + Camp = 10 pf)
- Cf = design variable
- Ri = 100 Mohm
- Rf = 10kohm
- OpAmp GBW = 100 MHz single pole response over entire open loop gain curve

- finite GBW with an idealized single pole 6dB/octave rolloff
- finite input capacitances
- finite input resistance

Tz(s) = Vo(s)/Ip = Zf(s) / (1 + 1/(Ao(s)*beta(s)))

where s is the complex radian frequency s = jw = j*2*pi*f

Zf(s) is the frequency dependent feedback network impedance:

Zf(s) = Rf || Cf = Rf / (1+s*Rf*Cf)

Ao(s) is the frequency dependent single pole open loop gain function for the op-amp:

Ao(s) = Ao/(1 + s/wb)

beta is the complex feedback fraction and NoiseGain(s) = 1/beta(s) = (Zf(s) + Zi(s))/Zi(s):

NoiseGain(s) = (Rf + Ri)/Ri *(1 + s*Rf*Ri/(Rf + Ri)*(Cf + Ci))/(1 + s*Rf*Cf)

which for very high Ri which usually applies to high speed photodiode/amplifier designs reduces to:

NoiseGain(s) = (1 + s*Rf*(Cf + Ci))/(1 + s*Rf*Cf)

This noise-gain function has a pole at:

fp = 1/(2*pi*Rf*Cf)

and a zero at:

fz = 1/(2*pi*Rf*Ri/(Rf+Ri)*(Cf + Ci))

or for Ri>>Rf, approximately:

fz = 1/(2*pi*Rf*(Cf + Ci))

In some discussions, the transimpedance amplifier response is simplified to:

Tz(s) ~= Zf(s)

which roughly predicts the high frequency rolloff, but doesn't include feedback phase shift effects which leads to response peaking as will be demonstrated in the example below. In an ideal transimpedance amplifier configuration, with negligible Ci and Cf, the transimpedance amplifier 3db bandwidth will be equal to the unity-gain bandwidth (or GBW) of the operation amplifier, since in this case the noise-gain will be frequency independent and unity. The frequency Fc at which the noise-gain curve intersects the open-loop op-amp gain curve determines the closed-loop circuit bandwidth. For a realistic circuit, with finite Ci, the noise-gain curve will have a zero which causes the noise-gain curve to rise with increasing frequency. This leads to a reduced intersection frequency, a corresponding lower transimpedance bandwidth and the potential for instability and oscillation due to feedback network phase-shift.

The frequency dependent

Cf_45 = Sqrt(Ci/(2*pi*Rf*GBW))

along with the corresponding 3dB frequency response BW which in this case is just the pole frequency at this value of Cf:

f3dB(Cf_45) = Sqrt(GBW/(2*pi*Rf*Ci))

These expressions are often used as a starting point for design. Often this value of Cf in fact leads to too much peaking in the frequency response and overshoot in the transient response. This is due to additional phase shift in real op-amps arising from additional high-frequency poles in the open-loop response. A more conservative starting point uses the "optimally flat" Cf value, which is approximately 40% higher in value than Cf_45. The "optimally flat" response is discussed in detail below..

Cf_45 = 1.55 pF

f3db = 1/(2*Pi*Rf*Cf) = 10.3 MHz

A numerical calculation using the complete transimpedance transfer function and NoiseGain expressions above and using Cf = 1.55 pf predicts:

Intersection Frequency: f(|Ao(s)beta(s)|=1) = 12.2 MHz

Phase Margin: 55 deg

f3db = 12.2 MHz

Peaking: 0.96 dB peaking at 6.5 MHz

showing that the pole frequency fp (10.3 MHz) is not quite at the intersection point (12.2 MHz) but is lower in frequency, providing more phase margin than predicted by the simple expression. Simulation plots showing the magnitudes of the transimpedance, open loop gain and the noise gain in dB are shown below. While the "phase margin" is strictly defined as the difference between 180 deg and the phase of Ao(s)beta(s) at |Ao(s)beta(s)|=1, the curve below displays this phase difference at all frequencies :

For comparison, solving the EXACT expression for Cf which places fp at the intersection point yields a Cf = 1.36 pf. Using this Cf value in the exact calculation predicts:

Intersection Frequency: f(|Ao(s)beta(s)|=1) = 11.7 MHz == fp

Phase Margin: 50 deg

f3db = 12.9 MHz

Peaking: 1.6 dB peaking at 7.4 MHz

demonstrating that even when fp is placed exactly at the intersection point, the phase margin is still greater than 45 deg.

These exact calculations on a fairly basic photodiode/transimpedance amplifier circuit demonstrate that the Cf result obtained using the well known simplified expressions is a conservative estimate providing somewhat greater phase margin and actually places the pole frequency below the intersection point of |Ao(s)| and |NoiseGain(s)| or |Ao(s)*beta(s)|=1. Greater phase margin, flatter response and somewhat lower transimpedance bandwidth is provided by selecting Cf to achieve an "optimally flat" response as discussed in the next section.

Tz(s) = Vo(s)/Ip = Rf*Ao/(Ao + 1) * wo^2/(s^2 + s*wo/Q + wo^2)

Since this is a 2nd order response, the transimpedance must eventually roll off at a rate of 40 dB/decade, as evident in the |Tz(f)| plot above. In this model, an "optimally flat" transfer function response is obtained if we choose Q = 1/Sqrt(2) = 0.7071 which leads to a 3dB bandwidth for Tz(s) of exactly F0. (The less damped Cf_45 case discussed above corresponds approximately to Q = 1.0). It is easy to solve for the corresponding value of Cf, given GBW, Ci and Rf, by using either a simple iterative approach which converges rapidly, or the exact quadratic solution for Cf can be used. Simplified well-known expressions which offer some insight can be easily obtained for Cf and f3db which are sufficiently accurate in most cases of interest assuming Cf < 0.2*Ci. These simplified equations give Cf results which are somewhat higher than the exact Cf values and f3db values somewhat lower than the exact values:

Cf_flat = Sqrt(Ci/(pi*Rf*GBW))

The optimally flat Cf value is Sqrt(2) times higher than the Cf_45 value above or 2.18 pF in this example. The phase margin is increased to ~ 67 deg. This "optimally flat" Cf value shows no frequency peaking and leads to a 3dB bandwidth of the transfer function equal to the natural frequency F0:

f3db_flat = F0 = Sqrt(GBW/(2*Pi*Rf*(Cf_flat + Ci)) .

It is noted that f3db_flat is the geometric mean of the noise-gain zero frequency (1/(2*pi*Rf*(Cf + Ci)) and the GBW product (or unity-gain frequency in the simplified single-pole Ao(s) model used here).

It should be noted that with lower Rf or GBW values, the simple expressions above can differ considerably from the true results. For example, with GBW=10MHz, Rf=1kohm and Cs=15pf, the simple expressions predict Cf= 22pf and f3db=6.6MHz but the exact results are Cf=15pf and f3db=7.3MHz.

Although for most practical cases of interest (Ci, Rf, GBW values) a solution for Cf exists, it should be noted that for a given target value of Q, there is a minimum value of Ci below which Cf is undefined. This minimum value (which corresponds to Cf=0) is:

Ci_min = Q^2/(2*pi*Rf*GBW)

which for the optimally flat response case, Q = 1/Sqrt(2) is:

Ci_min = 1/(4*pi*Rf*GBW)

This simply means that for lower values of Ci, this value of Q is not achieveable because the feedback zero fz = 1/(2*pi*Rf*(Ci+Cf), is very close to, or above the unity-gain frequency of the op-amp so that the feedback network phase shift is negligible and the rolloff is determined by the op-amp single-pole rolloff. In simple terms, the circuit components for the configuration create a damped response, incompatible with the requested Q value.

The results for the optimally flat response, and the resultant response curves for our sample case are:

Phase Margin: 67deg

f3db = 9.63 MHz

Peaking: no peaking

- op-amp: finite GBW with single-pole Ao(s); input capacitances (include in Ci); op-amp internal output resistance = 0
- feedback resistance Rf
- feedback total shunt capacitance Cf
- Ri = infinite in first model; Ri = finite in the second model
- input total shunt capacitance Ci at inverting input

Cf and f3dB Calculator for Optimally Flat Transimpedance Response | ||
---|---|---|

GBW (MHz): | Rf (ohms): | Ci (pf): |

Cf _simp (pf): | f3dB_simp (Hz): | |

Cf_exact (pf): | f3dB_exact (Hz): |

The second calculator computes detailed and exact transimpedance amplifier properties for given GBW, Rf, Cf, Ri and Ci. (With Ri included, the 2nd order transfer function Tz(s) above has an identical form with a slightly modified Q value involving Ri||Rf).

- exact transimpedance 3dB bandwidth and Q value
- intersection frequency Fc where |Ao(s)beta(s)|=1 and phase margin
- peaking frequency Fpeak (if defined) and Peaking value in dB
- the noise gain pole fp and zero fz

Apart from parasitic components, probably the total input capacitance Ci will be the least certain property since it depends on the photodiode capacitance (which is bias dependent), and the various op-amp input capacitance contributions which depend greatly on the op-amp design and which are not always specified in the manufacturers' specifications. Remember that

- Q>1 significant peaking and overshoot
- Q=1.000 Cf_45
- Q=0.707 optimally flat
- Q=0.500 critically damped
- Q<0.5 no peaking or overshoot

One usually refers to the damping of an LCR circuit in terms of a damping coefficient alpha = R/(2*L) which describes the exponential damping of the transient response. "Critical" damping occurs for alpha = wo which can be seen corresponds to Q = 0.5. In the example above, critical damping would correspond to R = 200 ohm. Therefore, the "optimally flat" condition with Q = 0.7071 corresponds to strong but not quite critically damping. The Q=1 case corresponds to R = 100 ohm. The modeled pulse response for this LCR circuit demonstrates the pulse-overshoot expected:

- Q = 1.0 with 16% overshoot
- Q = 0.7071 with 4% overshoot (optimally flat response)
- Q = 0.5 with 0% overshoot (critically damped)

This driver generates short optical pulses with fall-time of about 15 ns, consistent with the 25 MHz BW of this LED. Due to the high LED drive current, the rise time is shorter at 6 ns, limited by the oscilloscope. The optical pulse shape is shown below using a Thor Labs PDA10A photodiode with a bandwidth of 150 MHz. A Tektronix 60 MHz TDS 210 oscilloscope was used for all measurements below.

The transimpedance pulse responses using a 50 MHz photodiode and a National LM6171 100 MHz GBW op-amp is shown below. With a feedback capacitor of 1 pF and about 0.3 pF parasitic capacitance and with Rf = 10kohm, the pulse response shows a smooth fall time of about 32 ns (11 MHz) demonstrating almost perfect compensation. This is close to the modeled example above. By comparison, the result with just the 0.3 pF parasitic capacitance demonstrates an undercompensated configuration with response peaking leading to pulse overshoot/ringing:

With a much lower bandwidth op-amp, such as the 12 MHz GBW OPA606 op-amp, the lower loop-gain means that for a given nominal transimpedance gain, a greater compensation capacitor Cf must be used to eliminate gain peaking and pulse-overshoot. This of course means that the bandwidth will be lower. The results shown below for the OPA606 illustrate this. Again with an Rf of 10k, the feedback capacitor required to eliminate peaking is about 6 pF with a resultant 3dB bandwidth of about 3 MHz, almost 4 times lower than the bandwidth achievable with the 100 MHz op-amp. The ringing shown for the Cf ~ 0.3 pF corresponds to a very low phase margin of 25 deg. with an associated transimpedance gain peaking of over 7 dB.

The measured results above were obtained using simple breadboarding as shown below. This works reasonably well, and allows one to quickly compare active and passive devices and component layout criticality. However one must obviously quantify the parasitic capacitances inherent in breadboarding. For example, inter-track capacitance can vary from 0.3 pF to over 2 pF which is clearly important in transimpedance circuits. Nevertheless, it is possible to achieve good results in bandwidths of up to about 50 MHz with some care.

A fall time of about 15 nS or a bandwidth of about 23 MHz is observed as predicted by the simulation. Since the noise-gain asymptote at high-frequencies is (Cf + Ci)/Cf, this can be used to roughly estimate the noise-gain at the intersection. This indicates that since Ci is usually considerably greater than Cf, increasing Cf will LOWER the noise-gain at the intersection, which will eventually move the decompensated op-amp into instability. For the LT1222, a Cf > 1 pF leads to instability and oscillation is observed.

Vo(s) = Ip_n*Zf(s) / (1 + 1/(Ao(s)*beta(s)))

where

beta(s) = Zi_ef/(Zi_ef + Zf)

Zf = Rf/(1 + s*Rf*Cf)

The circuit reduction is shown below:

The Thevenin impedance Zi_ef is simply the impedance, looking back (to the left) into the network with Ip open-circuited. With the following convenient impedance combinations defined:

Zi = Ri/(1+s*Ri*Ci) parallel combination of Ri and Ci

Zs = Rs + sL series combination of Rs and L

Zoa = Roa/(1 + s*Roa*Coa) parallel combination of Roa and Coa

we find that the Thevenin effective impedance is:

Zi_ef = Zoa*(Zi + Zs)/(Zoa + Zi + Zs)

and the effective (short-circuit) Norton current is:

Ip_n = Zi/(Zi + Zs)*Ip

Therefore the transimpedance, related to the original (fixed) photodiode current source is:

Tz(s) = Vo(s)/Ip = Zi/(Zi + Zs)*Zf(s) / (1 + 1/(Ao(s)*Zi_ef/(Zi_ef + Zf) ))

This form can be easily calculated and plotted on any scientific calculator with complex math capability. Similar equivalent circuit reduction can be applied to more complex input networks.

**Compensating for the Effects of Input Capacitance ..**, Analog Devices MT-059**Design Considerations for a Transimpedance Amplifier**, National Semiconductor AN-1803**Texas Instruments: High Speed Analog Design and Applications Seminar****LMP7717 88 MHz, Precision, Low Noise Op-Amp**, National Semiconductor LMP7717 Data Sheet**Compensate Transimpedance Amplifiers Intuitively**, Texas Instruments Application Report SBOA055A**Understand and apply the transimpedance amplifier**, David Westerman, PlanetAnalog Document 2007**Photodiode Monitoring with OP AMPS**, TI Technical Document SBOA035 1995**Designing Photodiode Amplifier Circuits with OPA128**, TI Technical Document SBOA061 1994**OPA380 Precision High-Speed Transimpedance Amplifier**, OPA380 Data Sheet**OPA656 Wideband, Unity-Gain Stable, FET-Input OpAmp**, OPA656 Data Sheet**IC OP-AMP Cookbook**, Walter Jung, 2nd Edn. 1980 SAMS.**Microelectronic Circuits**, A. Sedra and K. Smith, 2nd Edn. Holt, Rinehart and Winston, 1987 p.724