Photodiode amplifier

A photodiode amplifier or photodiode amplifier is an electronic amplifier that converts the photocurrent of a photodiode into a proportional electrical voltage . It is a transimpedance amplifier with a photodiode connected on the input side as a signal-emitting current source. The impedance of the photodiode and the properties of the operational amplifier influence the frequency-dependent overall transmission behavior, which can be calculated analytically with the help of the linear circuit analysis.

Area of ​​application and importance

The photodiode amplifier is the standard procedure when it comes to measuring light intensities with the help of photodiodes. The circuit is relatively easy to implement in practice, since, in addition to a power supply and an operational amplifier, only a photodiode and a feedback impedance are required. The usable optical spectral range depends solely on the photodiode used . There are specialized photodiodes for the visible (VIS), the ultraviolet (UV) and the infrared (IR) range. The transmitted electronic frequency range depends on both the photodiode and the operational amplifier and the feedback impedance.

The photodiode amplifier has a wide, linear dynamic range . With a fixed feedback resistor , the dynamic range is approximately 10 ³ to 10 ⁴ , depending on the noise properties of the photodiode and operational amplifier. Typical values ​​for the feedback resistance are 100 Ω to 1 GΩ. A very large intensity range can be covered by changing the feedback resistance. It should be noted, however, that the greater the feedback resistance, the smaller the bandwidth . In addition, it must be taken into account that the dimensioning of the optimal feedback capacitance depends on the feedback resistance.

With photodiodes, with the exception of avalanche photodiodes , the high light sensitivities of photomultipliers are usually not achieved. The effort and costs for the electronics required are, however, considerably lower with the photodiode.

Electrical behavior

Circuit of a photodiode amplifier

Definition of relevant quantities in common units

The parameters defined below apply to all of the equations in this article. The associated SI units or derived SI units are also given.

• ${\displaystyle V(\omega )={\frac {U_{A}}{I_{Ph}}}}$ : Gain or transimpedance in V / A of the photodiode amplifier (is generally complex)
• ${\displaystyle I_{Ph}}$ : Photocurrent in A
• ${\displaystyle U_{A}}$ : Output voltage in V
• ${\displaystyle \omega =2\pi f}$ : Radial frequency in 1 / s
• ${\displaystyle f}$ : Vibration frequency in Hz
• ${\displaystyle j}$ : Imaginary unit
• ${\displaystyle R_{f}}$ : Feedback resistance or feedback resistance in Ω
• ${\displaystyle C_{f}}$ : Feedback capacity or feedback capacity in F
• ${\displaystyle A_{0}}$ : Maximum open-circuit voltage gain or open-loop voltage gain (with ) of the operational amplifier (OPV) ${\displaystyle \omega =0}$
• ${\displaystyle f_{T}=\omega _{T}/(2\pi )}$ : Transit frequency in Hz of the operational amplifier (at this frequency the open circuit voltage gain is equal to 1)
• ${\displaystyle f_{g}=\omega _{g}/(2\pi )}$ : Cutoff frequency (−3 dB) in Hz of the photodiode amplifier
• ${\displaystyle C_{Ph}}$ : Parallel capacitance of the photodiode in F
• ${\displaystyle R_{Ph}}$ : Parallel resistance of the photodiode in Ω

All of the equations shown below can in principle also be used for any other current sources than photodiodes, if they can be characterized by an impedance connected in parallel.

DC behavior

In the case of direct current, the gain (transimpedance) is:

${\displaystyle V(\omega =0)={\frac {U_{A}}{I_{Ph}}}=-R_{f}}$

In this case, the gain is determined solely by the feedback resistance. Because of the feedback to the negative input, the sign of the output voltage is inverted. The photodiode can also be turned around to obtain a positive output voltage. The photocurrent, which always flows in the diode reverse direction, must then also be drawn in the circuit diagram in reverse.

Transfer behavior assuming an ideal OPV

If an ideal OPV ( and ) is assumed, the gain (transimpedance) is: ${\displaystyle A_{0}=\infty }$ ${\displaystyle f_{T}=\infty }$

${\displaystyle V(\omega )={\frac {U_{A}(\omega )}{I_{Ph}(\omega )}}={\frac {-R_{f}}{1+j\omega R_{f}C_{f}}}}$

Since the imaginary unit appears on the right side of the equation , the transimpedance is complex valued, which means that the photodiode amplifier can cause a phase shift. With the ideal OPV, the amplifier has a simple first-order low-pass behavior. There are no excessive amplitudes. This behavior can be enforced by a sufficiently large feedback capacitance (see below) . H. the bandwidth, is reduced. The cutoff frequency (−3 dB) is here: ${\displaystyle j}$ ${\displaystyle C_{f}}$

${\displaystyle f_{g}={\frac {1}{2\pi R_{f}C_{f}}}}$

Transfer behavior assuming a real (compensated) OPV

Amplitude response (example)

Phase response (example)

Step response (example)

In general, the real properties of the operational amplifier must be taken into account in order to be able to describe the transmission behavior of the photodiode amplifier appropriately analytically, especially at higher frequencies. The gain (transimpedance) is a good approximation:

${\displaystyle V(\omega )={\frac {U_{A}(\omega )}{I_{Ph}(\omega )}}={\frac {-R_{f}}{1-{\frac {\omega ^{2}}{\omega _{T}}}R_{f}(C_{f}+C_{Ph})+j\omega \left({\frac {1}{\omega _{T}}}+R_{f}\left({\frac {1}{R_{Ph}\omega _{T}}}+C_{f}+{\frac {C_{Ph}}{A_{0}}}\right)\right)}}}$

In deriving this equation, it was assumed that the OPV is compensated; i.e., has a first-order low-pass behavior. The OPV can then be described by two parameters ( and ). Terms that contain or can be neglected because these two values ​​are usually very small. Since it appears in the denominator , the photodiode amplifier has a 2nd order low-pass behavior. Accordingly (depending on the component properties) an amplitude increase (see below) can occur in the vicinity of the cutoff frequency. ${\displaystyle A_{0}}$ ${\displaystyle f_{T}}$ ${\displaystyle 1/A_{0}}$ ${\displaystyle 1/R_{Ph}}$ ${\displaystyle \omega ^{2}}$

In order to keep the other equations clear, the following abbreviations are introduced:

${\displaystyle a={\frac {1}{\omega _{T}}}+R_{f}\left({\frac {1}{R_{Ph}\omega _{T}}}+C_{f}+{\frac {C_{Ph}}{A_{0}}}\right)}$

${\displaystyle b={\frac {R_{f}}{\omega _{T}}}(C_{f}+C_{Ph})}$

The cutoff frequency (−3 dB) then results from:

${\displaystyle f_{g}={\frac {1}{2\pi }}{\sqrt {{\frac {1}{2b^{2}}}\left(2b-a^{2}\right)+{\sqrt {{\frac {1}{4b^{4}}}\left(2b-a^{2}\right)^{2}+{\frac {1}{b^{2}}}}}}}}$

The amplitude response can be calculated using the following equation:

${\displaystyle |V(\omega )|={\frac {|U_{A}(\omega )|}{|I_{Ph}(\omega )|}}={\frac {R_{f}}{\sqrt {\left(1-\omega ^{2}b\right)^{2}+(\omega a)^{2}}}}}$

The amplitude response can be used to derive the frequency at which an amplitude increase (gain peak) occurs:

${\displaystyle f_{GP}={\frac {1}{2\pi }}{\sqrt {\frac {\omega _{T}}{R_{f}(C_{f}+C_{Ph})}}}}$

The phase response is described by the following equation:

${\displaystyle \phi (\omega )=-{\frac {180^{\circ }}{\pi }}\cdot \arctan \left({\frac {\omega a}{1-\omega ^{2}b}}\right)-180^{\circ }\cdot \sigma \left(\omega -\omega _{GP}\right)+180^{\circ }}$

The pole in the argument of the arctangent function is compensated by the phase jump of −180 ° at . The inversion of the input signal is taken into account through the frequency-independent addition of 180 °. ${\displaystyle \omega _{GP}=2\pi f_{GP}}$

The step response of the photodiode amplifier can be calculated with the help of the Laplace transform . The following case distinction must be made:

Case a)

${\displaystyle \ \ \ {k_{3}}^{2}={k_{2}}^{2}-{k_{1}}^{2}>0\ \ \ {\mbox{mit}}\ \ \ k_{1}={\frac {a}{2b}}\ ,\ \ \ k_{2}={\frac {1}{\sqrt {b}}}\ \ \ }$

There are overshoots in the step response:

${\displaystyle U_{A}(t)=-I_{Ph}R_{f}\left[1-e^{-k_{1}t}\left\{{\frac {k_{1}}{k_{3}}}\sin(k_{3}t)+\cos(k_{3}t)\right\}\right]}$

Case B)

${\displaystyle \ \ \ {k_{3}}^{2}={k_{1}}^{2}-{k_{2}}^{2}>0\ \ \ {\mbox{mit}}\ \ \ k_{1}={\frac {a}{2b}}\ ,\ \ \ k_{2}={\frac {1}{\sqrt {b}}}\ \ \ }$

There are no overshoots in the step response:

${\displaystyle U_{A}(t)=-I_{Ph}R_{f}\left[1-e^{-k_{1}t}\left\{{\frac {k_{1}}{k_{3}}}\sinh(k_{3}t)+\cosh(k_{3}t)\right\}\right]}$

Occasionally, a reverse bias is switched in series with the photodiode in order to obtain a steeper step response. This effect is based on the fact that the effective parallel capacitance of the photodiode decreases when a bias is used. The equations listed for the transfer behavior remain valid.

Dimensioning of the feedback capacitance to avoid excessive amplitudes

In order to avoid peaks in the amplitude response (gain peak) and thus also temporal overshoots without unnecessarily restricting the bandwidth of the photodiode amplifier, the feedback capacitance should be selected as follows:

${\displaystyle C_{f}={\sqrt {\frac {4\,C_{Ph}}{R_{f}\,\omega _{T}}}}}$

This equation applies assuming a compensated OPV and creates a critical damping (corresponds to a first-order low-pass behavior). When using uncompensated OPVs, a slightly larger selection may have to be selected in order to avoid excessive amplitudes. This equation makes it clear that the frequency-dependent and temporal transmission behavior of the photodiode amplifier depends largely on the parallel capacitance of the photodiode and on the transit frequency of the OPV. The influence of these two quantities must therefore not be neglected in particular if the photodiode has a large area (i.e. large ) and / or the transit frequency of the OPV is small. ${\displaystyle C_{f}}$ ${\displaystyle C_{Ph}}$ ${\displaystyle f_{T}}$ ${\displaystyle C_{Ph}}$

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1. ↑ Transfer behavior of analog circuits , P. Hoppe, Vieweg & Teubner Verlag 1994, Chapter 20, ISBN 978-3-322-94030-8 (eBook) and ISBN 978-3-519-06169-4 (softcover)