C.1 Operational amplifier

The symbol for an operational amplifier is given in Fig. C.1A. Usually the power supply connections are left out (which will be done hereafter). The model in Fig. C.1B shows the major deviations from the ideal behavior.

The meaning of these circuit elements is as follows:

Further, the common mode rejection ratio (CMRR) accounts for the suppression of pure common mode signals relative to differential mode signals. Finally, amplifier noise is specified in terms of (spectral) voltage noise and current noise. For white noise these error sources are specified in terms of V/√Hz and A/√Hz, respectively. In Fig. C.1, the noise voltage can be modeled with a voltage source in series with the offset voltage, and the noise current by a current source in parallel to the bias current. For an ideal operational amplifier A, Z_c, Z_d and H are infinite, and V_off, I_b and Z_o are all zero. Any deviation of these ideal values is reflected in the transfer properties of interface circuits built up with operational amplifiers.

There are numerous types of operational amplifiers on the market with widely divergent specifications, and for a broad spectrum of applications. Table C.1 shows the major specifications for three types of operational amplifiers: types I and II are general purpose low-cost operational amplifiers, one with bipolar transistors at the input side (type I) and the other with JFETs at the input (type II). Type III is designed for high-frequency applications, with MOSFET transistors at the input. Also included are the temperature coefficient (tc) of the offset voltage, the CMRR, the supply voltage rejection ratio (SVRR), the slew rate, the DC gain A₀ and the unity gain bandwidth f_t.

In the analysis that follows we distinguish two types of errors: additive and multiplicative errors. Offset and bias currents cause additive errors. Multiplicative errors are caused by the finite gain and the first-order frequency dependence of the gain; in the analysis of the frequency dependence it is assumed that the DC loop gain is much larger than unity: A₀β1.

For a number of interface circuits we give, without any further derivation, their performance in terms of transfer function and input resistance, taking into account the nonideal behavior of the operational amplifier. Knowing the input impedance of an interface circuit, the error due to loading the sensor can easily be estimated.

C.2 Current-to-voltage converter

Most AD converter types require an input voltage with a range from 0 to the ADC’s reference voltage. When the ADC converts the output of a sensor with current output (such as a photo detector), that current must first be converted into a proper voltage that matches the ADC’s input range. The basic configuration that accomplishes this task is given in Fig. C.2A.

Assuming the operational amplifier has ideal properties, the transfer of this I–V converter is simply V_o=−I_iR and the input resistance is zero. The bias current of the operational amplifier sets a lower limit to the input range, the supply voltages an upper limit. The transfer including additive errors results in:

$V_{\text{o}}=-I_{\text{i}}R+V_{\text{off}}+I_{b1}R$ (C.1)

(C.1)

To reduce the effect of bias current, a current compensation resistance with value R is inserted in series with the noninverting input terminal (Fig. C.2B). The transfer becomes:

$V_{\text{o}}=-I_{\text{i}}R+V_{\text{off}}+I_{\text{off}}R$ (C.2)

(C.2)

which shows that current compensation reduces the error due to the bias current by about a factor of 5 (see Table C.1). The transfer including multiplicative errors due to finite gain A₀ is:

$V_{\text{o}}=-I_{\text{i}}R\frac{A_0}{1+A_0}\approx -I_{\text{i}}R\left(1-\frac{1}{A_0}\right)$ (C.3)

(C.3)

which means that the scale error is about the inverse of the DC gain of the amplifier. The first-order approximation of the transfer of an operational amplifier is:

$A\left(\omega \right)=\frac{A_0}{1+j\omega {\tau }_{\text{A}}}$ (C.4)

(C.4)

with τ_A the first-order time constant of the amplifier. Amplifiers with this transfer have a constant gain-bandwidth product: in feedback mode, the overall gain decreases as much as the bandwidth increases. At unity gain the bandwidth equals the “unity gain bandwidth” which is the value specified by the manufacturer, as in Table C.1. Apparently f_t=ω_t/2π=A₀/2πτ_A. Taking this into account the transfer of the I–V converter is:

$V_{\text{o}}=-I_{\text{i}}R\frac{1}{1+j\omega {\tau }_A/A_0}$ (C.5)

(C.5)

Note that the bandwidth of the converter equals the unity gain bandwidth of the operational amplifier. The input resistance in terms of input admittance results in:

$G_{\text{in}}=\frac{1}{R_{\text{i}}}+\frac{A_0}{R}$ (C.6)

(C.6)

In general, 1/R_iA₀/R so the input impedance R_in=1/G_in is about R/A₀; in most cases a sufficiently low value to guarantee that the whole input current flows through the feedback resistor.

C.3 Noninverting amplifier

With a voltage amplifier the output voltage range of a sensor can be matched to the input range of an ADC. Voltage amplification is accomplished by either an inverting or a noninverting amplifier configuration, depending on the required signal polarity. Fig. C.3A shows a noninverting amplifier with gain 1+R₂/R₁ and infinite input resistance. Hence the gain can be arbitrarily chosen by the two resistances, and the sensor experiences no electrical load. For R₂/R₁=0, the configuration is called a buffer (Fig. C.3B).

In the following formulas the feedback portion R₁/(R₁+R₂) is abbreviated by β according to the notation in Chapter 3, Uncertainty aspects. For the buffer amplifier, β=1 (unity feedback). The transfer of the noninverting amplifier including additive errors equals:

$V_0=\frac{V_{\mathrm{i}}}{\beta }+\frac{V_{\mathrm{off}}}{\beta }+I_{\mathrm{b}}^{-}{R}_2$ (C.7)

(C.7)

The lower limit of the signal voltage range is set mainly by the offset voltage and to a lesser degree the bias current. After nullifying the offset, it is the temperature dependency of the offset that limits the range. The transfer shows a scale error (multiplicative error) due to the finite gain:

$\frac{V_{\text{o}}}{V_{\text{i}}}=\frac{1}{\beta }·\frac{A_0\beta }{1+A_0\beta }\approx \frac{1}{\beta }·\left(1-\frac{1}{A_0\beta }\right)$ (C.8)

(C.8)

The scale error is the inverse of the loop gain A₀β. Taking into account the frequency-dependent gain as in Eq. (C.4), the transfer of the noninverting amplifier is:

$\frac{V_{\text{o}}}{V_{\text{i}}}=\frac{1/\beta }{1+j\omega {\tau }_{\text{A}}/A_0\beta }$ (C.9)

(C.9)

Obviously the bandwidth is proportional to β, and the transfer is inversely proportional to β. Hence the product of gain and bandwidth has a fixed value: the gain-bandwidth product. This relationship is shown in Fig. C.4.

Numerical example. A₀=10⁴, β=0.01, so A₀β=100; unity gain bandwidth A₀/τ_A=10⁶ rad/s, so the operational amplifier’s cutoff frequency is 1/τ_A=100 rad/s and the noninverting amplifier bandwidth is 10⁴ rad/s.

Fig. C.5 shows a simulation¹ of the transfer characteristic using a general purpose, low-cost operational amplifier for three gain factors: 1 (buffer), 100 and 10,000. Clearly the product of gain and bandwidth is constant. At higher frequencies the second-order behavior of the operational amplifier becomes apparent.

Finally, the input resistance of the noninverting amplifier is found to be:

$R_{\text{in}}=R_{\text{i}}\left(1+A_0\beta \right)+\beta R_2\approx A_0\beta R_{\text{i}}$ (C.10)

(C.10)

which is sufficiently high in most applications.

C.4 Inverting amplifier

Fig. C.6 shows the inverting amplifier configuration. Its voltage transfer is –R₂/R₁ and the input resistance is R₁, a much lower value than in the case of the noninverting amplifier.

The transfer including additive errors amounts:

$V_{\mathrm{o}}=-V_{\mathrm{i}}\frac{R_2}{R_1}+V_{\mathrm{off}}\left(1+\frac{R_2}{R_1}\right)+I_{\mathrm{b}}^{-}{R}_2$ (C.11)

(C.11)

The contribution of the bias current I_b⁻ can be reduced by inserting a resistance R₃ in series with the noninverting input terminal, with a value equal to R₁//R₂, that is the parallel combination of R₁ and R₂. With this bias current compensation, the term I_b⁻ in Eq. (C.11) is replaced by I_off.

The transfer including multiplicative errors due to a finite gain is:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-\frac{R_{\text{2}}}{R_1}·\frac{A_0\beta }{1+A_0\beta }\approx -\frac{R_{\text{2}}}{R_1}\left(1-\frac{1}{A_0\beta }\right)$ (C.12)

(C.12)

which, again, introduces a scale error equal to the inverse of the loop gain. The frequency-dependent gain is given by

$\frac{V_{\text{o}}}{V_{\text{i}}}=-\frac{R_{\text{2}}}{R_1}·\frac{1}{1+j\omega {\tau }_A/A_0\beta }$ (C.13)

(C.13)

Here too, the product of gain and bandwidth is constant. The frequency characteristics are similar to those in Fig. C.4. The input resistance becomes:

$R_{\text{in}}=R_1\left(1+\frac{R_2}{A_0{R}_{\text{i}}}\right)$ (C.14)

(C.14)

a value that is almost equal to R₁. Note that this value can be rather low, resulting in an unfavorable load on the transducer.

C.5 Comparator and schmitttrigger

C.5.1 Comparator

A voltage comparator (or short comparator) responds to a change in the polarity of an applied differential voltage. The circuit has two inputs and one output (Fig. C.7A). The output has just two levels: high or low, all depending on the polarity of the voltage between the input terminals. The comparator is frequently used to determine the polarity in relation to a reference voltage.

It is possible to use an operational amplifier without feedback as a comparator. The high gain makes the output either maximally positive or maximally negative, depending on the input signal. However, an operational amplifier is rather slow, in particular when it has to return from the saturation state. Table C.2 gives the specifications for two different types of comparators, a fast type and an accurate type.

Table C.2

Specifications for two types of comparators: fast (type I) and accurate (type II)

		Type I	Type II
Voltage gain	A	–	2×105
Voltage input offset	Voff	±2 mV±8 µV/K	±1 mV, max. ±4 mV
Input bias current	Ibias	5 µA	25 nA, max. 300 nA
Input offset current	Ioff	0.5 µA±7 nA/K	3 nA, max. 100 nA
Input resistance	Ri	17 kΩ	–
Output resistance	Ro	100 Ω	80 Ω
Response time	tr	2 ns	1.3 µs
Output voltage, high	vo,h	3 V	–
Output voltage, low	vo,l	0.25 V	–

Purpose-designed comparators have a much faster recovery time with response times as low as 10 ns. They have an output level that is compatible with the levels used in digital electronics (0 and +5 V). Their other properties correspond to a normal operational amplifier and the circuit symbol resembles that of the operational amplifier.

C.6 Integrator and differentiator

Velocity information can be derived from an acceleration sensor (for instance a piezoelectric transducer) and displacement from velocity (for instance from an induction-type sensor) by integrating the sensor signals. The reverse is also possible, by differentiation, but this approach is not recommended because of an increased noise level. Basic integrator and differentiator circuits are shown in Fig. C.8. Ideally the transfers in time and frequency domains are given by:

$\begin{array}{cc}{V}_{\text{o}}=-\frac{1}{\mathit{RC}}\int V_{\text{i}}\text{d}t& V_{\text{o}}=-\mathit{RC}\frac{\text{d}{V}_{\text{i}}}{\text{d}t}\end{array}$ (C.15)

(C.15)

$\begin{array}{cc}{V}_{\text{o}}=-\frac{1}{j\omega \mathit{RC}}{V}_{\text{i}}& V_{\text{o}}=-j\omega \mathit{RC}{V}_{\text{i}}\end{array}$ (C.16)

(C.16)

However, these circuits will not work properly. The integrator runs into overload because the constant offset voltage and bias current will be integrated too. This is prevented by limiting the integration range, simply by adding a feedback resistor in parallel to the capacitance C. The differentiator is unstable because of the (parasitic) second-order behavior of the operational amplifier. This too is prevented by adding a resistor, in series with the capacitance C, resulting in a limitation of the differentiation range.

C.6.1 Integrator

Fig. C.9 shows the modified or “tamed” integrator. Taking into account the finite gain of the operational amplifier A, the transfer function of this integrator is:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-\frac{A}{1+\left(1+A\right)\left(R/R_{\text{f}}\right)+j\omega \left(1+A\right)\mathit{RC}}$ (C.17)

(C.17)

Assuming a frequency-independent amplifier gain: A=A₀, the transfer function becomes:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-\frac{A_0}{1+\left(1+A_0\right)\left(R/R_{\text{f}}\right)+j\omega \left(1+A_0\right)\mathit{RC}}\approx -\frac{R}{R_{\text{f}}}·\frac{1}{1+j\omega R_{\text{f}}C}$ (C.18)

(C.18)

The condition for the last approximation is valid for A₀1 (which is always the case) and A₀RR_f (which is usually the case). The inclusion of resistance R_f limits the LF transfer to R_f/R, whereas the lower limit of the integration range is set to f_L=1/(2πR_fC).

Numerical example. Assume that a lower limit of the integration range of 100 Hz and a DC gain of 100 are required. The component values should satisfy R_f/R=10² and R_fC=1/200π, accomplished by, for instance R_f=100R=100 kΩ and C=16 nF. Fig. C.10 shows the frequency characteristic for this design, based on a simulation with an ideal and a real operational amplifier. Clearly the higher limit of the integration range is set by the frequency-dependent gain of the operational amplifier.

C.6.2 Differentiator

The transfer of the basic differentiator, taking into account the finite gain of the operational amplifier, is given by:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-j\omega \mathit{RC}·\frac{A}{1+A+j\omega \mathit{RC}}$ (C.19)

(C.19)

For A=A₀ and A₀1 this is:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-j\omega \mathit{RC}·\frac{A_0}{1+A_0+j\omega \mathit{RC}}=-A_{\text{0}}\frac{j\omega \mathit{RC}/A_0}{1+j\omega \mathit{RC}/A_0}$ (C.20)

(C.20)

This corresponds to a first-order high-pass characteristic with cutoff frequency at ω=A₀/RC. The differentiation range is limited to this frequency. For much lower frequencies, the transfer function is just −jωRC and corresponds to that of an ideal differentiator.

However, when the frequency dependence of the operational amplifier gain according to Eq. (C.4) is taken into account, the transfer function becomes:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-j\omega \mathit{RC}·\frac{A_0}{A_0+j\omega \left({\tau }_{\text{A}}+\mathit{RC}\right)-{\omega }^2{\tau }_{\text{A}}\mathit{RC}}$ (C.21)

(C.21)

This characteristic shows a sharp peak at ω²τ_ARC≈A₀, for which value the transfer is about A₀RC/(RC+τ_A).

Numerical example. The required gain is 0.1 at 1 rad/s. So the component values R and C should satisfy 1/RC=10 rad/s. Suppose the amplifier properties are: A₀=10⁵, unity gain bandwidth 10⁵ rad/s. Fig. C.11 presents the calculated characteristic for this example. The transfer shows a peak of ≈10⁴ at the frequency 10³ rad/s.

Fig. C.12 shows a simulation for a general purpose operational amplifier with frequency compensation (i.e. it behaves almost as a first-order system). Component values in this simulation are R=100 kΩ, C=1 μF. Differences compared to Fig. C.11 are due to different amplifier parameters A₀ and τ_A of the real amplifier.

If the operational amplifier has a second-order cutoff frequency (and most do have), instability may easily occur. Therefore the differentiating range should be limited to some upper frequency, well below the cutoff frequency of the operational amplifier. This is accomplished by inserting a resistor R_s in series with the capacitor C, as shown in Fig. C.13.

With the “taming resistor” R_s the transfer becomes:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-j\omega \tau ·\frac{A}{1+A+j\omega \left\{\mathit{RC}+\left(1+A\right)R_{\text{s}}C\right\}}$ (C.22)

(C.22)

Taking into consideration the finite and frequency-dependent gain of the operational amplifier according to Eq. (C.4) and assuming A₀1, the transfer is modified to:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-j\omega \mathit{RC}·\frac{1}{1+j\omega \left\{R_{\text{s}}C+\left(\mathit{RC}+{\tau }_{\text{A}}\right)/A_0\right\}-{\omega }^2{\tau }_{\text{A}}\left(R_{\text{s}}C+\mathit{RC}\right)/A_0}$ (C.23)

(C.23)

For RR_s and R_sC(RC+τ_A)/A₀ this can be approximated by:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-j\omega \mathit{RC}·\frac{1}{1+j\omega R_{\text{s}}C-{\omega }^2{\tau }_{\text{A}}\mathit{RC}/A_0}$ (C.24)

(C.24)

Numerical example. Same values as for the uncompensated differentiator and R_s=R/100. The transfer is calculated from Eq. (C.24) and displayed in Fig. C.14. It shows that the peak is reduced by inserting resistance R_s. The differentiation range runs from very low frequencies up to about 1000 rad/s.

The simulation in Fig. C.15 shows the effect of a taming resistor, here 100 Ω. Note that the sharp peak has completely disappeared. Further, note the big difference between an ideal operational amplifier and a real type.

C.7 Filters

Removing unwanted signal components (noise, interference, intermodulation products, etc.) can be accomplished by (analogue) filters. Simple filters consist of passive components only. However, loading such filters (at both input and output) may easily affect the performance. In such cases active filters (with operational amplifiers) are preferred, providing independent choices for various filter parameters (input and output impedance, cutoff frequency, gain). The first-order integrator and differentiator of the preceding section are essentially a low-pass and a high-pass filter, respectively. Fig. C.16 shows an active band-pass filter.

Assuming ideal operational amplifier characteristics the transfer function is:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-\frac{Z_2}{Z_1}=-\frac{j\omega R_2{C}_1}{\left(1+j\omega R_1{C}_1\right)\left(1+j\omega R_2{R}_2\right)}=-\frac{R_2}{R_1}·\frac{j\omega R_1{C}_1}{1+j\omega R_1{C}_1}·\frac{1}{1+j\omega R_2{C}_2}$ (C.25)

(C.25)

so this is is a combination of gain (−R₂/R₁), low-pass filter (f_l=1/2πR₂C₂) and high-pass filter (f_h=1/2πR₁C₁). Fig. C.17 shows the transfer function (frequency characteristic) obtained by simulation with an ideal and a low-cost operational amplifier. In this design, the pass-band gain is set to 100 and the high-pass and low-pass frequencies are set at 100 Hz and 1 kHz, respectively, a combination that can be obtained with the component values R₁=1 kΩ, R₂=100 kΩ, C₁=1600 nF and C₂=1.6 nF. Both curves are almost identical, since the unity gain bandwidth of the (real) operational amplifier is much higher than the filter gain over the whole frequency range.

Fig. C.18 shows a second-order low-pass filter of the “Sallen-Key” configuration.

Its transfer function equals:

$\frac{V_{\text{o}}}{V_{\text{i}}}=-\frac{1}{1+j\omega \left(R_1{+R}_2\right)C_2-{\omega }^2{R}_1{R}_2{C}_1{C}_2}$ (C.26)

(C.26)

Under particular conditions for the resistance and capacitance values this filter has a Butterworth characteristic, which means a maximally flat amplitude characteristic in the pass-band. The general transfer function of a Butterworth filter of order 2n is:

$\left|\frac{V_{\text{o}}}{V_{\text{i}}}\right|=\frac{1}{\sqrt{1+{\left(\omega /{\omega }_{\text{c}}\right)}^{2n}}}$ (C.26)

(C.26)

For n=1 the Butterworth condition is C₁/C₂=(R₁+R₂)²/2R₁R₂. When R₁=R₂=R, then C₁=2C₂. The cutoff frequency is f_c=1/2πRC₁. Fig. C.19 depicts a simulation of such a filter, with a cutoff frequency of 1 kHz (by making R₁=1 kΩ and C₁=160 nF).

Here, the influence of the finite unity gain bandwidth of the operational amplifier is clearly visible and limits a proper filter function to about 20 kHz in this example.

The slope of the stop-band characteristic of a first-order low-pass filter is −6 dB/octave or −20 dB/decade. To obtain better selectivity, higher order filters can be obtained by simply cascading (putting in series) filters of lower order. The slope of the characteristic in the stop-band results in −6n dB/octave for a low-pass filter of order n. Note that the attenuation in the pass-band is also n times more compared to that of a first-order filter. For instance, cascading three second-order Butterworth filters of the kind shown in Fig. C.18 results in a slope of −36 dB/octave.