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6.6Some Applications of Op Amps in Biomedicine

6.6.1Introduction

Op amps are extremely useful in the design of all sorts of analog electronic instrumentation systems. As demonstrated in Chapter 7 and Chapter 8, op amps find great application in the design of analog active filters and as building blocks for instrumentation amplifier (differential) amplifiers and other linear and nonlinear signal conditioning systems. As the preceding sections have shown, there are a number of specialized types of op amps for specific signal processing applications. The following sections focus on four applications that often have important roles in biomedical signal processing: integrators; differentiators; charge amplifiers; and an isolated two-op amp instrumentation DA useful for measuring ECG. This amplifier also includes a band-pass filter and an analog photo-optic coupler for galvanic isolation.

6.6.2Analog Integrators and Differentiators

A simple op amp integrator is shown in Figure 6.16. An op amp integrator is hardly ever used as a stand-alone component; it is usually incorporated as part of a feedback loop to obtain some desired dynamic characteristic for the closed-loop system. The reason for not using it alone is that it integrates its own dc bias current and offset voltage, causing its output, in the absence of an input signal, to drift into saturation slowly and linearly. In the circuit of Figure 6.16, set Vs = 0 and consider the op amp ideal, so vi′ = 0. Thus, the node voltage V1 must equal VOS. Using superposition, and considering IB and VOS to be steps applied at t = 0, the integrator’s time-domain output can be written:

Vo(t) = −t IB/CF + VOS (1 + t/RCF)

(6.60)

			CF
R	V1	VOS
		+
		vi’	Vo
Vs	IB		IOA

FIGURE 6.16

A dc model of a voltage op amp integrator. The op amp is assumed ideal and its offset voltage and bias current are externalized to its inverting (vi′ ) node. The integrator’s behavior is analyzed in the text.

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264	Analysis and Application of Analog Electronic Circuits
			CF
			R
C	vi’	ena
		ina	Vo
Vs			OA

FIGURE 6.17

Circuit for an op amp differentiator. In this problem, the op amp’s frequency response is considered, as well as its short-circuit input voltage noise root power spectrum, and input current noise root spectrum.

IB and VOS can have either sign, depending on the op amp design and its temperature. VOS magnitude is typically in the range of hundreds of microvolts and IB can range from tens of faradamperes to microamperes, again depending on OA design and temperature.

If stand-alone analog integration must be done, the integrator drift must be considered. Op amps with low VOS and IB are used and a means of shorting out CF is used to ensure zero initial conditions at the start of integration. Digital integration is no panacea; it will integrate small analog offset voltages present in the anti-aliasing filter, the sample and hold, and the ADC.

Often, when an integrator is used in a closed-loop system, it is desirable to give its transfer function a real zero, as well as the pole at the origin. The zero is used to give the system better closed-loop dynamic performance and is easily obtained by placing a resistance, RF, in series with CF. Assuming an ideal op amp, the transfer function is:

Vo		RF + 1 sCF		(	s R C	F	+ 1		Ki (sτf	+ 1)
	(s) = −		= −		F		)	= −			(6.61)
V					s RC				s
		R					F
s

which is what is desired.

Figure 6.17 illustrates a conventional op amp analog differentiator. If the op amp is treated as ideal and CF is neglected, then the transfer function is:

	Vo	(s) = −s RC	(6.62)
	V
	s

which is the transfer function of an ideal time-domain differentiator with gain −RC.

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Now consider what happens to the differentiator’s transfer function when a real frequency-compensated op amp is modeled by the open-loop gain:

	Vo		−Kvo
		(s) =					(6.63)
	V′		τ	A	s + 1
	i
Now the node equation on the Vi′ node is:
Vi′ (G + s C) − Vo G = Vs s C							(6.64)

Using Equation 6.63, Vi′ in Equation 6.64 can be eliminated and the overall transfer function written:

Vo	(s) =	−s RC Kvo (1+ Kvo )
		s2 τA RC (1+ Kvo )+ s(τA + RC)	(1+ Kvo )+ 1
Vs
			(6.65)

−s RC

s2 τARCKvo + s(τA + RC) Kvo + 1

Note that Equation 6.65 has the form of the transfer function of a quadratic band-pass filter shown in the following equation:

V	(s) =			−s K			(6.66)
o
V		s2	ω 2	+ s 2ξ ω	n	+ 1
s			n

From Equation 6.66 and Equation 6.65, the resonant frequency is:

ωn Kvo (τARC) r s					(6.67)
and the damping factor is:
ξ =	(τA + RC)		Kvo		(6.68)
	2 Kvo		τARC

When typical numerical values are substituted for the op amp and circuit

parameters (Kvo = 106; τA = 0.001 sec; R = 106 Ω; and C = 10−6 F), ωn = 3.16 ∞ 104 r/s and ξ = 1.597 ∞ 10−2. This damping is equivalent to a quadratic BPF

Q = 1/(2ξ) = 31.3, which is a sharply tuned filter. Thus, any fast transients in Vs will excite a poorly damped sinusoidal transient at the differentiator output.

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266	Analysis and Application of Analog Electronic Circuits

Another problem with the simple R–C differentiator is the differentiation of the op amp’s short-circuit equivalent voltage noise, ena. The transfer function for ena is easily shown to be:

Vo = Ena (1 + s RC)

(6.69)

Because ena has a broadband spectrum, the differentiator’s output noise has a spectrum that increases with frequency, making the output very noisy. (Note that the ina component in Vo is not differentiated.) When a feedback capacitor, CF, is added in parallel with R, the differentiator’s frequency response is found from the new node equation:

Vi′[G + s(C + CF )]− Vo (G + sCF ) = Vs sC

(6.70)

Using Equation 6.63, Vi′ in the preceding equation can be eliminated and the transfer function written:

Vo	(s) =	−s RC
		s2 τAR(C + CF ) Kvo + s[(τA + RC) Kvo + RCF ]+ 1	(6.71)
Vs

The differentiator transfer function still has the format of a BPF. Examine its ωn and damping: all parameters are the same as the preceding ones and CF = 6.7 ∞ 10−11 F (67 pF).

	ω 2 =	Kvo	=	106		= 9.99933 ∞ 108
		τAR(C + CF )		10−3 ∞ 106 ∞ (10−6	+ 6.7 ∞ 10−11)
	n

¬

ωn = 3.1622 ∞ 104 r/s

The damping factor is now found from:

2ξ ω

n

=

τA + RC

+ RC

Kvo

F

ξ = 0.5 ∞ 3.1622 ∞ 104

10−3 + 1

+ 6.7 ∞ 10−11 ∞ 106

ˆ

= 1.075

˜

10

6

↓

(6.72)

(6.73)

(6.74)

(6.75)

The ωn has changed little, but now the system is slightly overdamped, i.e., both poles are close on the real axis in the s-plane. The differentiator’s frequency response is still band pass with its peak gain at the same frequency,

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Operational Amplifiers			267
			GF
			CF
			vi’
•			Vo
ix = F d	CT	GT	EOA

FIGURE 6.18

An electrometer op amp is used to make a charge amplifier to condition the output of a piezoelectric force transducer. See text for analysis.

but is no longer severely underdamped. Making CF larger will make the system more overdamped; it will have two real poles, one below ωn and the other above it. Thus, for practical reasons, an op amp, analog differentiator should have a small feedback capacitor in parallel with the feedback R to limit its bandwidth and give it reasonable damping so that input noise spikes will not excite “ringing” in its output.

6.6.3Charge Amplifiers

A charge amplifier is used to condition the output of a piezoelectric transducer used in such applications as accelerometers, dynamic pressure sensors, microphones, and ultrasonic receivers. Figure 6.18 illustrates the schematic of an electrometer op amp used as a charge amplifier. The shunt capacitance, CT, includes the capacitance of the piezotransducer, the connecting coax (if any), and the input capacitance of the op amp. Similarly, the conductance GT includes the leakage conductance of the transducer, the connecting cable, and the op amp’s input conductance. The op amp is assumed to have a finite gain and frequency response, i.e., it is nonideal in this respect. Thus, the summing junction voltage, vi′, is finite. The op amp’s gain is modeled by:

	V = −V′		Kvo		(6.76)
			τ s + 1
	o	i
	The node equation for the summing junction is:
	Vi′[s(CF + CT )+ GT + GF ]− Vo[sCF + GF ] = F(s)d = sF(s)d				(6.77)

from which:

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268				Analysis and Application of Analog Electronic Circuits
				Vi′=		sF(s)d + Vo[sCF + GF		]
						[s(CF + CT )+ CT + GF		]		(6.78)
							¬
			V	τs +	1 = −K		s F(s)d + Vo[sCF		+ GF ]	(6.79)
							vo [s(CF + CT )+ GT		+ GF ]
			o[		]
							¬
	Vo	(s) =					−s Kvo d
			s2τ(CF + CT )+ s[τ(GF + GT )+ CF + CT + KvoCF ]+ KvoGF							(6.80)
	F

To put Equation 6.80 into time-constant format, divide top and bottom by Kvo GF and note that certain terms in the denominator are negligibly small. Let s = jω in order to write the charge amplifier’s frequency response:

Vo	(jω)	− jω d RF	(6.81)
F		(jω)2 τ(CF + CT )RF Kvo + jω CFRF + 1

The frequency response function of Equation 6.81 is band pass with two real poles. The low-frequency one is set by the feedback R and C and is s1

−1/(RFCF) r/s; the high-frequency pole can be shown to be approximately s2 −(Kvo/τ)[CF/(CF + CT)] = ωT CF/(CF + CT) r/s. ωT is the op amp’s radian gain-bandwidth product. For example, set Kvo = 106; τ = 10−3 sec; CF = 10−9 F;

CT = 10−9 F; GF = 10−10 S; and GT = 10−12 S. Now the low break frequency is at flo = 1.59 ∞ 10−2 Hz and the high break frequency is at fhi = 80 MHz, which is adequate for conditioning 5-MHz ultrasound.

6.6.4A Two-Op Amp ECG Amplifier

Op amps can be used to make high-gain differential amplifiers with high common-mode rejection ratios, as described in Section 3.7.2 in Chapter 3. The two-op amp DA was found to have a DM gain:

Vo = (Vs − Vs′) 2(1 + R/RF)

(6.82)

R and RF should be chosen so that the amplifier’s DM gain is 103. To use this amplifier as an ECG system front end, it is necessary to define the ECG band pass with a highand low-pass filter. An easy design is to cascade the Sallen and Key high-pass and low-pass filters, as shown in Figure 6.19. The HPF is given complex-conjugate poles with an fn = 0.1 Hz and a damping

factor of ξ = 0.707. Using the design formulas given in Section 7.2.2 in Chapter 7, C1 = C2 = C and, for ξ = 0.707, R1 = 2 R2. fn = 1/(2πC R1R2) = 0.1,

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Operational Amplifiers			269
		C1
R1	HPF	LPF
C		V2 R
V1		O R C2	V3
C	OA		A
	R2

FIGURE 6.19

A band-pass filter for electrocardiography made from cascaded Sallen and Key high-pass filter and low-pass filters.

so one can solve for R1 and R2 after choosing C = 3.3 μF. A little algebra gives R2 = 3.31 ∞ 105 and R1 = 6.821 ∞ 105 Ω.

The parameters for the S and K LPF are found from ξ = C2/C1 = 0.707 and fn = 1/(2πR C1C2) = 100 Hz. Clearly, C2 = 2C1 for the desired damping. Choose C1 = 0.1 μF, so C2 = 0.2 μF. Using the fn relation, R = 1.125 ∞104 Ω. The op amps need not be special low-noise or high fT designs, the ECG signal has limited bandwidth, and the noise performance is set by the differential front end amplifier having the gain of 103.

Now that the ECG signal has been amplified and its bandwidth defined, it is necessary to consider patient galvanic isolation. One easy way to achieve isolation is to run the DA and filter op amps on isolated battery power with no ground connection common with the power line mains. A pair of 12-V batteries can be used with IC voltage regulators to give ±8 V to run the four signal conditioning op amps. One cannot simply connect the output of the filter to an oscilloscope or computer to view the ECG. The common ground from the oscilloscope or computer destroys patient galvanic isolation, so some means of coupling the signal without the necessity of a common ground must be used. One means is to sample the isolated ECG and convert it with a serial output isolated ADC and then couple the digitized ECG to the outside using a photo-optic coupler (POC). (A transformer can also be used.) The digitized ECG must then be converted to analog form with a serial input DAC in order to be viewed on an oscilloscope or be an input to a strip-chart recorder.

A simpler form of galvanic isolation that uses POCs and gives an analog output is shown in Figure 6.20. OA1 is run from the isolated battery power supply of the signal conditioner. The top LED in the dual POC IC is biased on with the 3-mA dc current source, causing an average light intensity, I1o, to be emitted. The isolated conditioned ECG output, V3, causes the light intensity to vary proportionally around the average level, I1o. The light from LED1 is transmitted through a plastic light guide to photodiode, PD2. An increase in the light hitting the photojunction of PD1 causes a proportional increase in its (reverse) photocurrent. PD1’s increased photocurrent is the base current of the top npn BJT, so its collector current increases, causing the inverting input of op amp OA2 to go negative and making OA2’s output go positive. The positive increment on OA2’s output increases the current in

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