Current Amplifier

4.3.1 Resistive and Capacitive Source Impedance

A widely used method to measure currents uses the voltage drop of the current measured across a resistor (often called shunt). However, the drawback of this method is the dependency on the value of the source impedance (in generalR_in||C_in), being the larger, the bigger the used shunt is. Therefore, shunts larger than 50 Ω are hardly ever used.

Fig. 4.5 shows a current amplifier (also called transconductance amplifier), which fea-tures (given an ideal open loop amplifier) an optimum input impedance of 0 Ω. Parallel impedances are then shortend and the entire signal current flows into the current ampli-fier being guided away over the feedback resistor. The source is modelled as a current source with the (parallel) resistive and capacitive source impedances R_in and C_in (the dashed encircled noise current sourcei²_sndenotes the shot noise of a photodiode’s leakage current; this will be dicussed in the example later).

The current amplification is simply given by

Vout =−R_{f b}Iin . (4.18)

The input equivalent noise is given according to the amplifier’s nature as a current noise (notations refer to fig. 4.5):

i²_ineq = 4kT 1

R_{f b}||R_in +i²_p+ v²_s

|(R_{f b}||R_in||_jωC¹_in)|² [in A²/Hz] . (4.19) The first term denominates the thermal noise of the resistances, the following terms describe the noise caused by the amplifier noise sources at the input. The reader should

R_in i_in2 i_sn2

i_fb

Rfb 2

A = v

i

0 8

p2 s2

I_in C_in

source

current amplifier

Figure 4.5: Current amplifier; the source is modelled as an ideal current source with a resistive and capacitive output impedance in parallel. i²_sn denominates the shot noise of a photodiode’s leakage current.

+ +

A =0 ⁸ A =₀ ⁸

DC

R R

AC

CC

Figure 4.6: DC/AC-coupled photodiodes

note, that the input capacitance does not lower the noise by filtering, but that the vs term increases with frequency due to the denominator which diverges for ν → ∞, because _jωC¹

in dominates the resulting value of the parallel impedances. In practice, this effect can often be neglected due to limitied bandwith of the used opamp (stability!).

4.3.2 Example: Photodiodes

Photodiodes are generally read out with current amplifiers [Grae95, Rein93, Rein96], since the generated photocurrent is directly proportional to the number of incident (bet-ter: absorbed) photons. The mechanism of photocurrent generation in photodiodes equals the one in silicon strip detectors; a detailed discussion has been given in chap-ter 2. In fast sensor applications the circuits depicted in fig. 4.6 are mostly employed.

Using DC-coupling the photocurrent flows directly into the amplifier; when applying AC-coupling the current flows over a couple capacitor. In the case of DC-coupling the resistor Rin in fig. 4.5 is infinite, in case of AC-coupling Rin equals the bias resistor.

Thus, the DC-coupling is the implementation with a lower noise, but leakage currents cause amplifier offsets with the danger of a limitation of the dynamic range.

The small signal equivalent circuit of the photodiode exhibits an additional parallel noise

R_in i_in2 i_sn2

Figure 4.7: Current amplifier with subsequent integrating stage

current source i²_sn (fig. 4.5) for the shot noise of the diode leakage current (also called dark current).

For DC-coupling the input equivalent current noise equals i²_ineq= 4kT 1

R_{f b} +i²_sn+i²_p+ v_s²

|(R_{f b}||_jωC¹_in)|² [in A²/Hz] (4.20) with

i²sn= 2qIshotnoise of the diode’s leakage currentI

As mentioned before, the diode capacitance C_in is of big importance since the last term in eq. (4.20) increases with frequency. Therefore, low noise voltage opamps should be employed in general for use with high capacitance diodes.

4.3.3 Charge Measurement with a Current Amplifier

A current amplifier can be used in order to measure charge by using a second voltage integrating stage behind the transimpedance amplifier (fig. 4.7).

The output voltage behind both stages is given by V_out=−R_{f b}

RC Z

I_indt=−R_{f b}

RCQ_in (4.21)

and is independent of the input capacitances (this is again due to the zero input impe-dance). The input equivalent noise assuming the integrating stage to be noiseless can be calculated to be

q_ineq² = 4kT 1

(R_in||R_{f b})|jω|² + i²_p

|jω|² + v_s²

R_{f b}² |jω|² +v_s²C_in² [in C²/Hz] . (4.22) When comparing eq. (4.22) to eq. (4.24) we notice that the termv_s²C_{f b}² has been replaced by _R2^v2s

f b|jω|², i. e. a serial noise contribution in case of the pure charge amplifier has become a parallel one. However, this term usually can be neglected w. r. t. the first term (thermal noise ofR_in and R_{f b}). On the other hand, due to the gain constraintR_{f b} can often not

R_in i_in2 i_sn2

C_fb

A = v

i

0 8

p2 s2

I_in C_in

i_fb

Rfb 2 charge amplifier

source

Figure 4.8: Idealized charge amplifier with source (capacitive and resistive output impedance); in a real world circuit the encircled noise sources have to be added.

be chosen as big as usual parallel resistances are, so that the parallel noise increases as compared to a pure charge amplifier solution.

Hence no obvious advantage of this concept as compared to the conventional charge am-plifier (sect. 4.4) can be encountered. Current amam-plifiers used to measure charge have lately been presented in high energy physics [Dab94, Jar96, RD2-93] - often in combina-tion with bipolar input stages; bipolar transistors exhibit a larger transconductance g_m causing lower values of the noise voltage sourcev_s; the base current shot noise which has to be taken into account as a parallel noise current source is overlayed by the noise of the feedback resistor. The reason is that for very small shaping times ofτ < 20 ns (see next section) the noise penalty due to the parallel noise is small, but the space consumption of a bipolar input transistor is considerably less than that of an optimum matched MOS transistor.

A detailed investigation of current amplifiers together with bipolar input transistors is given in chapt. 7.