8.2 Frontend - Preamplifier
8.2.1 Small Signal Model
To study the behaviour of the preamplifier a small signal model of the preamplifier (fig.
8.7) is plotted in fig. 8.3. The cascode configuration of the schematic has been dropped in the small signal model for simplicity;gm denotes the input transistor’s transconductance and Ctr its gate-source plus gate-drain capacitance;Cl is given by the sum of the Cdg’s and the Cdb’s of M3 and M4, respectively. Rl is the drain-source resistance rds of M4 (the resistance looking into M3 is much higher due to the cascode circuit). The feedback capacitor Cf b equals C1 and Rf b equals the on-resistance of M7 which operates in the triode (linear) region.
The straight forward calculation of the current-to-voltage transfer function is easily carried out (see also [CS91]) assuminggmRout1 with Rout=Rf bkRl and is given by
vout(s) =− gm
gm
Rf b +sgmCf b+s2CinCoutiin(s) (8.1) with Cout = Cf b+Cl, Cin = Cdet +Ctr+Cf b. Equation 8.1 is a second order (in s) transfer function (notation as in fig. 8.3). The values for the preamplifier described can be found in table 8.2.
Iin Cdet
Cfb
Vin
gm Rl Cl
in Ctr
Rfb
Vout
V
Figure 8.3: Small signal model of preamplifier; the values to be inserted are given in table 8.2.
By replacing s=2πjνin eq. (8.1) and taking the absolute value we obtain the frequency transfer functionvout/iin(ν) in fig. 8.4; in the range betweenνp1andνp2the preamplifier integrates current according to vout = iin/(sCf b) = qin/Cf b. At the dominant pole’s ωp1 origin the feedback resistor breaks with the feedback capacitance hence denoting the point of proper integration operation (at very low frequencies the amplifier looks like a transresistance amplifier).
Under the assumption that the poles are widely spread, the pole positions are given by ωp1 = −1
τ1 =− 1 Rf bCf b ωp2 = −1
τ2 =− gmCf b
CinCout =−GBWCf b
Cin (8.2)
The first pole is given by the time constant Rf bCf b of the feedback components deter-mining the continous reset time; the second pole is the result of the capacitive feedback (chapt. 7.1.3).
The achieved GBW (2.84 rad/ns or 453 MHz) is quite remarkable for CMOS; the fastest video opamp of AMS features a 50 MHz GBW product. However, the preamplifier open-loop cell is not unity-gain stable since the nondominant cascode pole at approx. 40 MHz lies (considerably) below 1.22·GBW. However, with a preamplifier “voltage gain” (given by Cin/Cf b, see chapt. 7.1) of 10...50, the phase margin amounts to a minimum of 89◦ and hence is more than sufficient.
By inverse Laplace transformation the time domain response can be obtained. The input current is approximated by a Dirac δ-pulseI(t) = Qδ(t) with an integrated area of Q.
Thus, the output signal in the time domain becomes vout(t)≈ Qτ1
Cf b(τ1−τ2)(e−τt1 −e−τt2) . (8.3) Since in all practical casesτ2τ1, this expression represents an exponential rising step with a slowly decaying tail as introduced by the DC feedback path. The output voltage
gm 6.28 mV/A νp1 15.5 kHz
Cdet 20 pF νp2 6.6 MHz
Ctr 2.92 pF νz1 200 kHz
Cf b 342 fF τ1 10.2µs
Rf b 30 MΩ τ2 23.9ns
Rl 2.06 MΩ tr 52.6ns
Cl 1.86 pF v0=Rlgm 12940
Cin 23.26 pF GBW 453 MHz
Cout 2.20 pF
Table 8.2: Small signal values of preamplifier; the values in the table are taken from SPECTRE model 2 equations.
in response to a charge of 1 fF is shown in fig. 8.5 withτ1= 10.2µs,τ2 = 23.9 ns as found in the preamplifier. It should be noticed here, that Cl includes the couple capacitance to the subsequent shaper stage; so no further loading takes place.
The rise time (10% to 90%) is determined by time constant τ2 and can be calculated from eq. (8.2) by
tr= 2.2·τ2 = 2.2· Cin
GBWCf b (8.4)
i. e. the rise time can be minimized by a small total input capacitances, a large feedback capacitance and a large gain-bandwidth amplifier.
The input impedance of the charge amplifier can be calculated to be (assuming s 1/(Rf bCout) and gm 1/Rout, sCf b)
Zin(s) = vin
iin(s) = 1 Rout
1 +sRoutCout gm
Rf b +sgmCf b+s2CinCout . (8.5) We observe a zero at
ωz =− 1 RoutCout
. (8.6)
The pole positions are identical to those obtained for the transfer functions and are given by
ωp1 = − 1 Cf bRf b ωp2 = − gmCf b
CinCout =−GBWCf b
Cin . (8.7)
The input impedance for frequencies belowνp1 is purely ohmic and equals Zinohm1= Rf b
gmRl = Rf b
v0 ≈13.2kΩ (8.8)
i. e. as in the case of a current amplifier the input impedance is given by feedback resistance divided by the open loop gain.
105 106 107
104 105 106 107
ν[Hz]
Vout/Iin[V/A]
νp1 νp2
Figure 8.4: Bode-plot of transfer function vout/iin(ν); in the range betweenνp1 and νp2 the preamp integrates current. The gain is given by vout=iin/(sCf b) =qin/Cf b.
0 0.5 1 1.5 2 2.5 3 3.5 4
0 50 100 150 200 250 300
τ[ns]
Vout[mV]
τ2
τ1
Figure 8.5: Preamplifier response to a δ-like current pulse of 1fF charge; the rise is determined by τ2. τ1 is the time constant of the exponential discharge of the feedback resistor Cf b via Rf b.
103 104
104 105 106 107
ν[Hz]
|Zin|[Ω]
νp1 νz1 νp2
Figure 8.6: Bode-plot of input impedance; in the range between νp1 and νz1 the input impedance is capacitive with Cin =v0Cf b. For signal frequencies between νz1 and νp2 the input impedance is ohmic: Rin=τ2/Cin.
The input impedance in the range between νp1 (15.5kHz) and νz1 (200 kHz) can be appoximated by
Zinint(s) = 1
sgmRlCf b = 1
sv0Cf b . (8.9)
Hence, the input impedance equals the feedback capacitance multiplied with the open loop gain. This feature is a very desired one since due tov0Cf b Cdet nearly no charge division takes place in this frequency range - all charge generated in the detector is
“drawn” to the amplifier feedback capacitance.
At ωz =−1/(RoutCout) the input impedance becomes again ohmic with Zinohm2 = Cout
gmCf b = τ2
Cin ≈1020Ω (8.10)
wich is due to the fact that the load impedance becomes capacitive and the voltage division between Cf b andCout becomes frequency independent. Zinohm2 is directly linked to the preamplifier risetime (=2.2τ2) via eq. (8.10). Hence input resistance and amplifier risetime are equivalent manifestations of the gain cell’s-GBW. As is know from the transfer function, the amplifier is in integrating mode in the second ohmic region.