**Circuits and Systems** Vol.4 No.4(2013), Article ID:35753,8 pages DOI:10.4236/cs.2013.44048

Design of a Switched Capacitor Negative Feedback Circuit for a Very Low Level DC Current Amplifier

Faculty of Engineering, University of the Ryukyus, Okinawa, Japan

Email: hrhiga@eve.u-ryukyu.ac.jp

Copyright © 2013 Hiroki Higa, Naoki Nakamura. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received May 2, 2013; revised June 2, 2013; accepted June 9, 2013

**Keywords:** DC Amplifier; Small Current Measurement; Switched Capacitor

ABSTRACT

To miniaturize a very low level dc current amplifier and to improve its output response speed, the switched capacitor negative feedback circuit (SCNF), instead of the conventionally used high-ohmage resistor, is presented in this paper. In our system, a switched capacitor filter (SCF) and an offset controller are also used to decrease vibrations and offset voltage at the output of the amplifier using SCNF. The theoretical output voltage of the very low level dc current amplifier using SCNF is obtained. The experimental results show that the unnecessary components of the amplifier’s output are much decreased, and that the response speed of the amplifier with both the SCNF and SCF is faster than that using high-ohmage resistor.

1. Introduction

Response speeds of the measuring instruments are limited by those of very low level dc current amplifiers [1,2], when very small currents are measured by mass spectroscopes and radiation detectors. This implies that the amplifiers are required to observe rapid transient phenomena. The very low level dc current amplifier for measuring small currents generally consists of an amplifier having high input impedance and a high-ohmage negative feedback resistor. The amplifier with high-ohmage resistor has unavoidable effects of the stray capacitances across its terminals. This factor causes the amplifier to have a complicated frequency characteristic, which results in poor responses of the very low level dc current amplifier [1-3]. Some shielding techniques [4-6] have been reported for the purpose of decreasing these capacitive components. In spite of the fact that these methods have been employed, it was difficult to realize drastic improvements of the response speeds of the very low level dc current amplifier. Neither are the amplifiers with shielding methods appropriate for miniaturization. A positive feedback circuit [7] had also been used as another approach to decrease the stray capacitances. The amplifier however was unstable and began to oscillate in this case. The resultant high speed response of the amplifier has not been achieved so far.

To explore the optimum solution to the above problems, the switched capacitor negative feedback circuit (SCNF) has been developed. The switched capacitor (SC) circuit is equivalent to a resistor and is suitable for miniaturization. A theoretical output voltage of the very low level dc current amplifier using SCNF with a switched capacitor filter (SCF) was investigated in this paper. Furthermore, the output voltage of the amplifier was experimentally demonstrated.

2. Circuit Description

Figure 1 summarizes a very low level dc current amplifier, including SCF and a small current source. C_{g}, R_{g} and K are the input capacitance, input resistance, and amplification factor of the amplifier having a high input resistance, respectively. In the experiment, we utilized a triangular wave voltage produced by the function generator V_{g} and the differentiating capacitor C_{s} (reactance attenuator) to obtain a square wave current I_{s} with a high output impedance as an input signal to the amplifier. C_{o} is the output capacitance to the ground of C_{s}.

The SCNF and SCF are shown in Figure 2. The SCNF is composed of several analog switches (S_{1}, S_{2},···, S_{7}) and capacitors. These switches are controlled by two nonoverlapping clock signals. S_{1} is synchronous with S_{2} and S_{3}, and S_{4} is synchronous with S_{5}, S_{6} and S_{7}, respectively. An output voltage of the very low level dc current amplifier has vibrations due to charge and discharge actions of

Figure 1. Circuit configuration of very low level dc current amplifier. SCNF and SCF stand for switched capacitor negative feedback circuit and switched capacitor filter, respectively.

Figure 2. SCNF and SCF.

the SC circuit. Thus, as shown in Figure 2, the SCF was connected to the output of the amplifier. The switches S_{8} and S_{9} are synchronous with S_{1} and S_{4}, respectively.

3. Circuit Analysis

This section describes an equivalent resistance of SCNF and theoretical output voltage of the very low level dc current amplifier. In the following analysis, it is assumed that T_{1} is the time when S_{1}, S_{2}, and S_{3} are closed (S_{4}, S_{5}, S_{6}, and S_{7} are opened), and T_{2} is the time when S_{1}, S_{2}, and S_{3} are opened (S_{4}, S_{5}, S_{6}, and S_{7} are closed). Further, it is assumed that a small current I_{s} is dc current because the clock frequency of the SC circuit, f_{s}, is much higher than the frequency of I_{s}.

3.1. Equivalent Resistance of SCNF

From Figure 2, the voltage at node b, V_{b}, is represented by

, (1)

and an electric charge q_{1} at C_{1} is

.

From Equation (1) and the relationship that V_{o} = –KV_{i}, the electric charge q_{1} at C_{1} can be rewritten as

, (2)

for. Because the electric charge q_{1} at C_{1} during T_{2} is totally discharged, the quantity of the charge that is transported from node a to node b is equivalent to q_{1}. Thus, a current, I, flowing from node a into node b during one clock cycle T_{s} is

. (3)

Since the current to be measured in the very low level dc current amplifier I_{s} flows into the SC circuit, I_{s} = I. From the relationship that V_{o} = R_{feq}I_{s} [1], the equivalent resistance of SCNF R_{feq} is given by

, (4)

while the equivalent resistance of the SC circuit [8] R_{sc} is represented by

, (5)

where f_{s} is the clock frequency. The attenuation factor of the attenuator X [9] becomes

. (6)

Hence, from Equations (4) to (6), R_{feq} can be obtained as

. (7)

From Equation (6), it is observed that X is dependent on the ratio of capacitances of C_{2} and C_{3}.

3.2. Theoretical Output Voltage of the Amplifier

Output voltage of the very low level dc current amplifier using SCNF has vibrations due to charge and discharge actions of the SC circuit. In this section, theoretical output voltage of the very low level dc current amplifier is discussed.

The equivalent circuit of the SC circuit is shown in Figure 3(a). It is found from Equations (5) and (7) that the SCNF is equivalent to the capacitor of XC_{1} and four switches S_{1}, S_{3}, S_{4}, and S_{7}. The equivalent circuit of the very low level dc current amplifier is shown in Figure 3(b). The box labeled “*” in Figure 3(b) stands for the SCNF and this figure shows that the equivalent circuit of the SCNF shown in Figure 3(a) is connected with the equivalent circuit of the amplifier at the terminals between nodes a and c.

Millman’s theorem is applied to Figure 3(b), and the voltage V_{i} is represented by

,

(a)(b)(c)

Figure 3. Equivalent circuits of (a) SCNF and (b) very low level dc current amplifier. Figure 3(c) shows simplified input equivalent circuit of Figure 3(b).

where ω is the angular frequency. The input admittance of the very low level dc current amplifier Y_{in} is

, (8)

for. Using Equation (8), a simplified input equivalent circuit of the very low level dc current amplifier using SCNF can be drawn as shown in Figure 3(c).

An enlarged input voltage waveform of the amplifier at the positive final steady-state, V_{i}, is illustrated with the help of clock waveforms as shown in Figure 4. T_{s} is the clock cycle of the switches. T_{1} and T_{2} are (1 − α)T_{s} and αT_{s}, respectively. Let the input voltage of the amplifier at t = nT_{s} be V_{i}(n). Subscript symbols “+” and “−” mean just after and just before the time event occurs, respectively. For example, V_{o}(n + 1)_{−} means the voltage just before t = (n + 1)T_{s}. The amplitude of the input voltage for a cycle, V_{m}, is

(9)

Since electric charges of the SC circuit are conserved just before and after t = nT_{s}, the following equation is obtained

.

The input voltage just before t = nT_{s}_{ }is

. (10)

From Figure 4 and Equation (10), the voltage V_{m} is

. (11)

From Equations (9) and (11), the input voltage just after t = nT_{s} can be obtained as

Figure 4. Relationship between enlarged input voltage and clock waveforms.

. (12)

Thus the peak voltage T_{ip}_{1} during T_{1} is

(13)

Substituting Equation (12) into Equation (13) gives the following equation:

. (14)

Therefore, the peak output voltage of the amplifier during T_{1}, V_{op}_{1}, can be written as

. (15)

It is seen from Equation (15) that the theoretical output voltage of the very low level dc current amplifier using SCNF can be obtained by sampling V_{op}_{1}, and that the equivalent resistance, R_{feq}, is independent of duty ratio of the clock signal.

In this paper, the SCF was used to sample V_{op}_{1} from the output voltage of the very low level dc current amplifier using SCNF for the following reasons. Using a sample-and-hold circuit generally requires a clock generator that completely differs from two non-overlapping clock signals utilized by the SCNF, and using a low-pass filter provides for not theoretical output voltage, but approximately half amplitude of output voltage of the amplifier at a final steady-state. On the other hand, using the SCF allows for sharing the two non-overlapping clock signals. Both the SCF and SCNF can be also manufacturable by the same process. Therefore, the SCF is useful from the viewpoint of miniaturization.

4. Simulation Results

Using the electronic circuit simulator PSpice (Cadence Design System, Inc.), transient analyses of the very low level dc current amplifier using SCNF were carried out. K and C_{g} were set to 1300 and 17 pF, respectively. The equivalent resistance of the SC circuit R_{RC} was set to 1 MΩ by using C_{1} of 10 pF and f_{s} of 100 kHz. The attenuation factor X of 1/100 was also set by using both C_{2} of 1000 pF and C_{3} of 9.3 pF. Thus, the total equivalent resistance R_{feq} of the SCNF was 100 MΩ. The duty ratio of the clock cycle, α, was also set to 0.5. To evaluate response speeds of the very low level dc current amplifier, a square wave current I_{s} with a time period of 5 ms and an amplitude of 10 nA was input to the amplifier. From Equation (15), output voltage of 1 V should be obtained as the theoretical output voltage of the amplifier. A switch model [10] used in the computer simulation is shown in Figure 5. The symbols G, D, and S stand for gate, drain and source of a MOS-FET. Each analog switch is composed of a combination of an nMOS and pMOS, as shown in Figure 5(a). Assuming that parasitic capacitances between two terminals exist, as shown in Figures 5(b) and (c), the transient analyses by computer simulation were conducted.

First, it is assumed that nMOS has exactly the same parasitic capacitances as pMOS has. The parasitic capacitance values are shown in Table 1. These values were determined by trial and error. Figures 6(a) and (b) show the simulation results in the case of using the parasitic capacitances shown in Table 1. It can be seen that the output voltage of the amplifier has vibrations due to charge and discharge actions of the SCNF, and that the vibrations of output of the amplifier causes black areas in the output waveform (see Figures 6(a) and (b)).Thus, it is difficult to measure an input current from the output waveform of the amplifier. Average values of V_{op}_{1} from 10 ms to 12.5 ms and from 12.5 ms to 15 ms in Figure 6(a) were obtained as +1.0 V and −1.0 V, respectively. The simulation result in the case of using the SCF at the output of the amplifier is shown in Figure 6(c). In this case, the peaks of the output voltage during T_{1} were sampled by the SCF. The rise time of the output waveform of the SCF is 20.0 μs. As defined in general, the rise time is the time required for the output waveform to rise from 10% to 90% of its final steady-state value. It is clear from Figure 6(c) that using the SCF drastically reduces vibrations as well as unnecessary components, and that the input current I_{s} can be obtained by measuring the amplitude of its output voltage.

Secondly, on the assumption that nMOS and pMOS have totally different parasitic capacitances, as shown in Table 2, computer simulations of transient analyses of the very low-level dc current amplifier using SCNF were also performed. The parasitic capacitance values indicated

(a)(b)(c)

Figure 5. Switch model used in PSpice simulation. (a) Configuration of CMOS switch, (b) nMOS and (c) pMOS switch models with parasitic capacitances.

Table 1. Values of parasitic capacitances in the case of assumption that nMOS and pMOS have the same parasitic capacitive components.

(a)(b)(c)

Figure 6. Simulation results in the case of using parasitic capacitances shown in Table 1. (a) Output waveform of very low level dc current amplifier using SCNF, (b) its enlarged waveform at a positive final steady-state, and (c) output waveform of SCF. The rise time of output waveform in Figure 6(c) is 20.0 μs.

Table 2. Values of parasitic capacitances in the case of assumption that nMOS and pMOS have totally different parasitic capacitive components.

in Table 2 were also determined by trial and error. Figures 7(a) and (b) show the simulation results in the case of using the parasitic capacitances shown in Table 2. It can be found from Figure 7(a) that the output voltage of the amplifier has also vibrations, as shown in Figure 6(a), and that the amplitudes of the black painted areas are roughly 5 times larger than those in Figure 6(a). Further, it is seen that the output waveform of the amplifier has rapid changes in output voltage from T_{1} to T_{2}, and that the rapid change V_{c} is −25.7 V (see Figure 7(b)). The simulation result in the case of using the SCF at the output of the amplifier is shown in Figure 7(c). The rise time of the output waveform of the SCF is also 20.0 μs. It is obvious from Figure 7(c) that the amplitude of the output voltage of the amplifier becomes 1 V, and that the output waveform has the offset voltage of 2.04 V. Comparing Figure 7 with Figure 6, it is thought that a clock feed through generated by totally different parasitic capacitive components leads to a generation of the offset and the rapid changes in output voltage of the very low level dc current amplifier using SCNF. From other simulation results, it was found that using different parasitic capacitances of C_{dg-n} and C_{dg-p}, and those of C_{gs-n} and

(a)(b)(c)

Figure 7. Simulation results in the case of using parasitic capacitances shown in Table 2. (a) Output waveform of very low level dc current amplifier using SCNF, (b) its enlarged waveform at a positive final steady-state, and (c) output waveform of SCF. The rise time of output waveform in Figure 7(c) is 20.0 μs.

C_{gs-p} resulted in the generation of offset voltage of the amplifier, and that using larger parasitic capacitances of C_{dsub-n}, C_{dsub-p}, C_{ssub-n} and C_{ssub-p} led to an error of the equivalent resistance R_{feq}. On the other hand, parasitic capacitances of C_{gsub-n} and C_{gsub-p} did not have an effect on the output voltage of the amplifier.

5. Experimental Results and Discussion

First, the very low level dc current amplifier using SCNF shown in Figure 1 was made and its output responses were observed. The amplifier having high input resistance was composed of two differential amplification stages and an emitter follower stage, and the first stage had two JFETs to obtain high input resistance of the amplifier. The amplification factor K of the amplifier was 62 dB (From DC to 1.2 MHz), and output voltage waveform was observed using an oscilloscope. Since the triangular wave voltage, which had a time period of 5 ms and an amplitude of 10 V, was differentiated by the differentiating capacitor C_{s} of 1.25 pF, a square wave current with a time period of 5 ms and an amplitude of 10 nA was obtained as an input current I_{s} to the amplifier. As switches of the SC circuit, we used CMOS analog switches (MAX326, MAXIM Integrated Products, Inc.) having the maximum leakage current of 10 pA. Further, variable capacitors C_{1} and C_{3} were utilized. Parasitic capacitances of analog switches have a little effect on equivalent resistance of the SCNF R_{feq}, which causes errors in R_{feq} of the amplifier. Thus, the equivalent resistance R_{sc} of the SC circuit with the clock frequency f_{s} of 100 kHz, was set to 1 MΩ by adjusting capacitance of C_{1}, and then the attenuation factor of the feedback rate attenuator X was set to 1/100 by adjusting capacitance of C_{3}. Referring to Equation (7), the total equivalent resistance of the SCNF became 100 MΩ. An offset voltage controller, which was connected to the input of the amplifier and had a gain of unity, was also used to cancel the offset voltage in the experiment. The input stage of the offset controller was composed of a JFET which had much higher input impedance than the negative feedbackcircuit had, and its voltage drift was very small (several μV). Therefore, the offset controller did not have much effect on the current detection sensitivity of the amplifier.

Figures 8(a) and (b) show the output voltage waveform of the very low level dc current amplifier using

(a)(b)(c)

Figure 8. Experimental results. (a) Output waveform (scale H: 1 ms/div, V: 0.5 V/div), (b) its enlarged waveform (scale H: 5 μs/div, V: 0.5 V/div) of very low level dc current amplifier using SCNF, and (c) output waveform (scale H: 1 ms/div, V: 0.5 V/div) of SCF, consisting of C_{t} of 470 pF and C_{h} of 220 pF. The rise time of output waveform in Figure 8 (c) is 23.8 μs.

SCNF and its enlarged waveform at a positive final steady-state, respectively. It is seen from Figure 8(a) that the experimental output waveform does not extensively have the black area as shown in Figure 6(a) and 7(a). This is because rapid changes in output voltage of the amplifier do not appear on a CRT display of the oscilloscope due to its resolution, and as a matter of fact vibrations due to charge and discharge actions of the SCNF can be found at any enlarged waveforms in Figure 8(b). From this figure, it is clear that the output waveform of the amplifier has vibrations and rapid changes as observed in Figure 7(b). Because the amplifier in the PSpice simulation does have an amplitude limiter, in the case of simulation result in Figure 7(b), the minimum output voltage of the amplifier during T_{2} was indicated −22.6 V, while in the case of experimental result, the output voltage of the amplifier during T_{2} was saturated at −13 V due to the power supply of the negative voltage of the amplifier.

Peak values of the output voltage during T_{1} were sampled by the SCF. The experimental result is shown in Figure 8(c). It is clear from Figure 8(c) that using the SCF drastically reduces vibrations of the output of the amplifier, and that the theoretical input current I_{s} can be obtained by measuring the amplitude value of its output voltage. Thus it is possible to measure an input current of the very low level dc current amplifier using SCNF with the SCF. With respect to the Figure 8(c), the rise time of the output waveform of the SCF is 23.8 μs. On the other hand, that of the output waveform of the very low level dc current amplifier using conventionally used high-ohmage resistor is 92.0 μs. We are easily available to miniaturize SC circuits using IC-compatible techniques. Therefore, using the SCNF is an effective way to obtain both a faster output response and miniaturization of a very low level dc current amplifier.

Finally, a relationship between the equivalent resistance of the SCNF R_{feq} and the clock frequency f_{s} was investigated. It is clear from Equation (7) that R_{feq} can be set using the clock frequency and capacitances of C_{1}, C_{2}, and C_{3}. In this experiment, we first set to f_{s} of 100 kHz and C_{2} of 1000 pF, adjusted both capacitances of C_{1} and C_{3} to precisely obtain R_{feq} of 100 MΩ, and then changed f_{s} ranging from 50 kHz to 200 kHz. The equivalent resistance of SCNF was obtained by measuring output voltage of the SCF. The relationship between R_{feq} and f_{s} and error rate of R_{feq}, in the case of X of 1/100, are shown in Figure 9. Theoretical values of R_{feq} in Figure 9(a) were calculated using Equation (7) with C_{1} of 10.0 pF, C_{2} of 1000 pF, and C_{3} of 9.32 pF. It is obvious from Figures 9(a) and (b) that the experimental curve agrees well with the theoretical one, and that the error rate of R_{feq} is within 0.86 % for the theoretical values. From this experimental result, it is thought that R_{feq} ranging from 1.1 MΩ with X of 9/10 to 1.0 GΩ with X of 1/500 will be settable.

6. Conclusion

A theoretical output voltage of a very low level dc current amplifier using SCNF with SCF was derived from circuit analysis. It was experimentally demonstrated that a small current of 10 nA could be measured using the amplifier, and that response speed of the amplifier using SCNF was faster than that of the amplifier using conventionally used high-ohmage resistor. A consideration of a

(a)(b)

Figure 9. Experimental result: (a) relationship between R_{feq} and f_{s} and (b) error rate of R_{feq}.

very low level dc current amplifier using SCNF with SCF will be needed for practical applications as our future work.

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