# Computing filtering capacitance for a transimpedance amplifier

### Abstract

#### For citation:

Trifonov A.V. Computing filtering capacitance for a transimpedance amplifier. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2018;(1):6-9.
https://doi.org/10.38013/2542-0542-2018-1-6-9

## Introduction

For amplification of photodiode current in photodetectors, an operational amplifier (OA) is often used as a preamplifier, connected according to the transimpedance amplifier (TIA) connection diagram [1, 2]. Fig. 1 shows one of the possible schematic diagram options of such TIA. Fig. 2 shows an AC equivalent diagram of TIA, where photodiode is represented in the form of an ideal current source I _{вх}, and capacitance C_{IN} is equal to the sum of photodiode capacitance, OA input capacitance, and circuit capacitance. The TIA transfer function is essentially a ratio between the output voltage and the input current, and has a resistance dimension.

**. Schematic diagram of transimpedance amplifier (TIA): C**

Fig. 1

Fig. 1

_{F}– balancing capacitance; R – load resistance; −U, +U – feed voltages; U

_{вых}– output voltage; VD1 – photodiode; DA1 – operational amplifier

**. Equivalent schematic diagram of TIA**

Fig. 2

Fig. 2

Due to capacitance C_{IN} in the OA negative feedback (NFB) loop, an additional phase shift occurs between the voltages at the OA inverting input and output. In the absence of filtering capacitor C_{F} it leads to reduced phase margin in OA operation and possible self-excitation [1, 2].

Given below are the design equations for the magnitude of C_{F}, offered by the leading manufacturers of integrated circuits:

- Texas Instruments

where GBW – OA gain bandwidth product (1/s);

- National Semiconductor

In deriving these equations, voltage transfer ratio from the OA output to its inverting input was selected equal to the transfer ratio of voltage divider formed by capacitance CIN and parallel connection of elements R and CF (see Fig. 2). However, it only holds true if the said voltage divider is not connected to the OA inverting input, i.e. the NFB circuit is open. If, on the contrary, the NFB circuit is closed, voltage transfer ratio from the OA output to its inverting input will be quite different. Therefore, the above equations have to be corrected.

## Deriving an equation for computing filtering capacitance

First, we shall apply the signal-flow graph method, briefly outlined in [3]. We shall use a single-pole model of the OA transfer function [4]:

where p = jω – Laplace operator;

A – OA gain bandwidth product (rad/s).

**. TIA signal-flow graph**

Fig. 3

Fig. 3

Fig. 3 shows a signal-flow graph corresponding to the equivalent schematic diagram of TIA given in Fig. 2. Building upon the signal-flow graph plotting rules, we can write down:

where G = 1/R – load resistor conductance.

Applying the Mason’s formula [3], we determine the TIA transfer ratio:

Having made simple transformations, we have:

It is known that the second-order low-frequency circuit has the following transfer function [4]:

where K – transfer ratio;

ω_{0} – frequency of transfer function poles;

Q_{0} – Q-factor of transfer function poles.

Having compared expressions (1) and (2), we can write down:

One of the main TIA advantages, as compared to a common voltage amplifier, is a possibility to considerably expand the bandpass. The bandpass is determined by the frequency of poles ω_{0}, therefore in practice, as can be seen from expression (3), to increase the value of ω_{0}, a typical condition to be satisfied is

C_{IN} >> C_{F} . (5)

For the same reason, as a rule, the following inequality is true

It is known that self-excitation occurs when amplifier loop gain factor is greater than unity and the output signal is supplied to the input in phase with the input signal. In that case the criterion of stability against self-excitation is expressed through phase margin, which must be no less than 45° [5]. It means that with a frequency at which the value of amplitude-frequency response (AFR) of loop transfer function is equal to unity, the phase-frequency response (PFR) value shall differ from –360° by 45°, amounting to –315°.

Proceeding from the data of Fig. 3, let us write the voltage transfer function from the inverting input to the OA output:

The voltage transfer function from the OA output to its inverting input is found in accordance with Fig. 3, using the Mason’s formula and making simple algebraic transformations:

The NFB loop transfer function (from the inverting input to the OA output and further from the OA output to its inverting input) looks as follows:

Setting modulus of the right-hand part of expression (9) equal to unity and making simplifications, with account of condition (5), we have the following equation:

Having solved this equation, with account of condition (6), we find frequency on which the AFR value of the loop transfer function is equal to unity:

In accordance with the rules of operations with complex numbers, and taking into account condition (5), PFR of the loop transfer function defined by expression (9) has the following view:

According to the information on the graphs of antitrigonometric functions given in mathematical handbooks, it can be inferred that for the considered case the following relationship holds true:

arctg х = 180° - arctg х.

Then, according to the criterion of ensuring phase margin, expression (11) can be written as follows:

where Ψ – magnitude of required phase margin (deg).

Substituting ω = ω_{1} from formula (10) in the obtained expression, we have:

where x, y – dimensionless variables;

х = ARC_{IN} ,^{ }y = C_{F} / C_{IN}.

Fig. 4 shows a functional dependence of variables х, y for the case Ψ = 45°.

The value of Q_{0}, defined by expression (4), can also be expressed by means of variables x, y with account of condition (5):

In this way, knowing the values of A, R, C_{IN}, from equation (13) for the set value of Ψ, it can be possible to determine the value of variable y and, accordingly, the value of CF . And if it is the Q-factor of TIA transfer function poles that matters in the first instance, then it is necessary to find from equation (14) for the given value of Q0 the magnitude of y. Substituting the found magnitude of y in equation (13), we obtain the value of phase margin Ψ.

**. Functional dependence of variables x, y at Ψ = 45°**

Fig. 4

Fig. 4

## Conclusion

The proposed method for computing filtering capacitance for a TIA makes it possible, together with finding its magnitude, to determine the magnitude of the Q-factor of TIA transfer function poles and the phase margin. Under conditions of serial production, it will enable to objectively estimate the allowance for scattering of the TIA output signal parameters occurring due to process scattering of radio element parameters and influence of destabilising factors, as well as estimate the magnitude of phase margin.

If the calculation formula given in [2] is used, then phase margin Ψ will be not 45°, as claimed in that document, but much greater (at least 109°). In that case the magnitude of Q-factor Q_{0} will pretty much be equal to 1, which is approximately 3 times as little as in case of Ψ = 45°.

## References

1. Wang T., Erhman B. Compensate transimpedance amplifiers intuitively. Texas Instruments: application report SBOA055A. March 1993. Revised March 2005. 3 р.

2. Design considerations for a transimpedance amplifier. National Semiconductor Corporation: National Semiconductor. Application Note 1803. Maithil Pahchigar. February 28, 2008. 4 р.

3. Мамонкин И.Г. Усилительные устройства. М.: Связь, 1977. 360 с.

4. Мошиц Г., Хорн П. Проектирование активных фильтров: пер. с англ. / под ред. И.Н. Теплюка. М.: Мир, 1984. 320 с.

5. Фолкенберри Л. Применения операционных усилителей и линейных ИС: пер. с англ. / под ред. М. В. Гальперина. М.: Мир, 1985. 572 с.

#### For citation:

Trifonov A.V. Computing filtering capacitance for a transimpedance amplifier. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2018;(1):6-9.
https://doi.org/10.38013/2542-0542-2018-1-6-9