Aug 04, 2012

To focus on the effects of this simplified feedback network, the operational amplifier will be idealized with zero output impedance, input impedances that can be lumped into Ri, Ci and a single pole

Irrespective of the actual circuit configuration (non-inverting amplifier, inverting amplifier, transimpedance amplifier etc.), the circuit stability (against oscillation) is determined solely by the properties of this complex voltage-divider feedback network between the operational amplifier output and the inverting (-) input of the amplifier.

The circuit will be stable against oscillation provided that the

The frequency-dependent

where the feedback-network impedances, Zf and Zi for this specific configuration of parallel R, C components are:

The initial factor (Rf + Ri)/Ri is the DC noise gain. The complex frequency is:

At high frequency, the noise gain approaches (1 + Ci/Cf), independent of the resistors in the divider network. The noise gain contains one zero and one pole as can be seen when rewritten as:

or

The noise-gain zero frequency is given by:

and the noise-gain pole frequency is given by:

The point of intersection in frequency of the noise gain (magnitude) with the open-loop gain (magnitude) is given by a simple quadratic equation for fc^2:

where

The phase angle of the loop gain, Ao(fc)*beta(fc)) or equivalently Ao(fc)/Noise-Gain(fc)),

As the phase angle of the loop gain (phase shift from the output to the inverting input) approaches -180 degrees, the feedback will become positive and the circuit will oscillate. The

The second example below shows a configuration which would apply typically to a moderately high gain (~ 30dB) non-inverting amplifier configuration. Note that in this case, the order of the noise-gain zero and pole are reversed compared to the first example and also the intersection point fc (14 MHz) occurs at a higher frequency than either Fz or Fp. In this example, Fp and Fz are fairly close together in frequency and since the pole Fp occurs at lower frequency, the noise gain decreases initially until the zero at Fz cancels the phase shift accumulation of the pole and flattens out the noise-gain curve reasonably well before the intersection point. In this case the phase margin at fc is 116 degrees, a very stable configuration. Increasing Cf will move Fp to lower frequency resulting in a greater drop in noise gain before Fz can flatten the gain. This is typically used to lower the gain of amplifiers where extra bandwidth is not required. Note however that the noise gain (magnitude) will always be greater than unity (0 dB). Also, if the operational amplifier is NOT unity gain stable (the single pole Ao(s) open loop gain curve discussed here is always unity gain stable), then the noise gain must not be lowered below the minimum stable gain value at the intersection point fc.

An example for a GBW=12 MHz transimpedance amplifier demonstrates how the noise gain, transimpedance and phase margin change as the feedback capacitance is continuously changed.

- 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
- input total shunt resistance Ri at inverting input
- input total shunt capacitance Ci at inverting input

- noise-gain zero
**Fz**and pole**Fp**frequencies - intersection frequency
**Fc**where |Ao(s)beta(s)|=1 - phase
**Margin**at Fc - noise gain value
**|NG(Fc)|**at intersection frequency

Total input-referred noise current is calculated (see discussion below) for the Tz configuration for the conditions described below.

For the noninverting (NI) circuit configuration, the f3db bandwidth, peaking frequency and values are also calculated (different then Tz/Inv configurations). The text area below the calculator summarizes results cumulatively which can be conveniently cut/pasted for future reference. These three circuits are shown below along with the transfer functions for the specific feedback network discussed above.

The default parameters for the calculator demonstrate a transimpedance amplifier example with a high gain (5Mohm) and total input capacitance (20 pF total) with Cf chosen to be close to the maximally flat value.. This example would apply for example to the 16 MHz OPA627 DiFet op-amp used with a 5 pF small area photodiode such as the Vishay Semiconductors BPW24R Si photodiode:

To use the calculator, enter the values for GBW (in MHz), Rf, Ri (in kohms), Cf, Ci (in pF) and spectral densities for the op-amp voltage noise

The same calculator is also available separately on a simplified page.

A alternate calculator with more detailed noise results is also available.

Fz: Noise gain zero frequency

Fp: Noise gain pole frequency

Fc: Intersection frequency of noise gain and open loop gain

Margin: Phase margin of loop gain at intersection frequency (stable for margin >45deg)

NG(Fc): Noise gain at intersection frequency Fc

f3dB: 3dB bandwidth relative to DC value

Q: Quality factor of resonant peaking

Fpeak: Frequency of resonant peaking

Peak: Resonant peaking value in dB

Cf(flat): Cf value for maximally flat (Q=1/Sqrt(2)) response for Tz and INV circuits, if possible

(A more detailed noise calculator provides total noise (over all frequency) and cumulative noise up to any frequency.)

The last term in the In_eq expression is a result of the noise-gain zero which gains-up the op-amp noise voltage at frequencies higher than the noise-gain zero. Note that this current-density term is EXPLICITLY bandwidth dependent and can in many cases dominate the other noise current components, particularly in high-bandwidth circuits with large input capacitance and high gain (Rf). The op-amp input noise current term and the thermal noise term of Rf don't experience the same enhanced noise effect and their noise contributions start rolling off at about the pole frequency in a similar way to the transimpedance output signal voltage rolloff.

Generally speaking, to maximize transimpedance circuit sensitivity, it is desirable to use as high a gain (Rf) as possible provided there is enough signal bandwidth for the intended application and to choose an op-amp with very low input voltage noise en and input capacitance to lower the gaining-up effect. For maximum sensitivity, the goal is to select an op-amp with low en, in and Cin to achieve a noise performance in the Tz circuit which is limited only by the intrinsic thermal noise of Rf, for the particular photodiode and with the required bandwidth. The diagram below which corresponds to the default parameters in the calculator above, shows how the noise-gain curve starts to increase (gaining up the op-amps voltage noise en at a rate of 20 dB/decade) above the noise-gain's zero frequency ~ 1.7 kHz and continues to increase, raising the noise, and finally levelling off at the pole frequency 106 kHz and finally rolling off as limited by the op-amp GBW limit:

For a more detailed discussion of noise considerations for transimpedance circuits, see:

**Photodiode Amplifiers Op Amp Solutions**, J. Graeme, McGraw Hill, 1996, p. 87.

- Photodiode Monitoring with OP AMPS, TI Technical Document SBOA035 1995

- OPA656 Wideband, Unity-Gain Stable, FET-Input OpAmp, OPA656 Data Sheet.

For example, for In_tot = 27 pA, and with Responsivity = 0.5A/W, the minimum detectable optical power in this Tz circuit with the specified bandwidth would be 54 pW.

Photodiodes are typically reverse biased to reduce junction capacitance. However a biased photodiode, like any reverse biased PN junction exhibits leakage ("dark") current. This dark current introduces extra shot noise to the circuit characterized by a constant spectral density:

or, using convenient units:

Evaluation of this expression from the photodiode specifications will indicate if it contributes significantly to input-referred current noise. If so, it should be added to the RMS noise-current summation above. Generally smaller photodiodes will have less dark current. For example the commercial Vishay Semiconductors BPW24R Si photodiode exhibits a dark current of ~ 3nA at reverse voltages of less than 10 V. This corresponds to a shot-noise current density of ~ 31 fA/Sqrt(Hz) (or 0.03 pA/Sqrt(Hz). In the default calculator example, this is comparable to the final noise-gain term.

The photodiode dark current will also contribute to an output DC offset voltage by an amount Id*Rf or for the 5 Mohm case above and a 3nA dark current, an offset of 15 mV.

Transimpedance (Tz) Amplifier Circuit Configuration:

Inverting (INV) Amplifier Circuit:

Non-inverting (NI) Amplifier Circuit:

In this general case the

- Transimpedance Photodiode Amplifier
**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