16.8 Miscellaneous Signal Modification Circuitry

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In addition to the signal modification devices discussed so far in this chapter, there are many other types

of circuitry that are used for signal modification and related tasks. Examples are phase shifters, voltageto-

frequency converters, frequency-to-voltage converters, voltage-to-current converts, and peak-hold

circuits. The objective of the present section is to discuss briefly several of such miscellaneous circuits and

components that are useful in the instrumentation of dynamic systems.

16.8.1 Phase Shifters

A sinusoidal signal given by

v ¼ va sinðvt þ fÞ ð16:101Þ

has the following three representative parameters:

va ¼ amplitude

v ¼ frequency

f ¼ phase angle

Note that the phase angle represents the time

reference (starting point) of the signal. The phase

angle is an important consideration only when two

or more signal components are compared. The

Fourier spectrum of a signal is presented as its amplitude (magnitude) and the phase angle with respect

to the frequency.

Phase-shifting circuits have many applications. When a signal passes through a system, its phase

angle changes due to dynamic characteristics of the system. Consequently, the phase change provides

very useful information about the dynamic characteristics of the system. Specifically, for a linear

constant-coefficient system, this phase shift is equal to the phase angle of the frequency-response

function ( frequency-transfer function) of the system at that particular frequency. This phase-shifting

behavior is, of course, not limited to electrical systems and is equally exhibited by other types of

systems including mechanical vibrating systems. The phase shift between two signals can be

determined by converting the signals into the electrical form (using suitable transducers) and shifting

the phase angle of one signal through known amounts, using a phase-shifting circuit, until the two

signals are in phase.

Another application of phase shifters is in signal demodulation. For example, one method of

amplitude demodulation involves processing the modulated signal together with the carrier signal. This,

however, requires the modulated signal and the carrier signal to be in phase. Usually, however, since the

modulated signal has already transmitted through electrical circuitry having impedance characteristics,

its phase angle has changed. Then, it is necessary to shift the phase angle of the carrier until the two

signals are in phase, so that demodulation can be performed accurately. Hence, phase shifters are used in

demodulating, for example, when demodulating LVDT displacement-sensor outputs.

A phase-shifter circuit, ideally, should not change the signal amplitude while changing the phase angle

by a required amount. Practical phase shifters can introduce some degree of amplitude distortion (with

respect to frequency) as well. A simple phase-shifter circuit can be constructed using resistance (R) and

capacitance (C) elements. A resistance or a capacitor of such an RC circuit is made fine-adjustable so as to

obtain a variable phase shifter.

An opamp-based phase shifter circuit is shown in Figure 16.24. We can show that this circuit provides

a phase shift without distorting the signal amplitude. The circuit equation is obtained by writing the

current balance equations at nodes A and B, noting, as usual, that the current through the opamp leads

Input

vi

Output

vo

+

B − R

R

R A c

C

FIGURE 16.24 A phase-shifter circuit.

16-56 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

can be neglected due to high input impedance; thus

vi 2 vA

RC ¼ C

dvA

dt

vi 2 vB

R þ

vo 2 vB

R ¼ 0

On simplifying and introducing the Laplace variable, we obtain

vi ¼ ðts þ 1ÞvA ðiÞ

and

vB ¼

1

2 ðvi þ voÞ ðiiÞ

in which, the circuit time constant, t; is given by

t ¼ RcC

Since vA ¼ vB as a result of very high gain in the opamp, by substituting Equation ii into Equation i, we

obtain

vi ¼

1

2 ðts þ 1Þðvi þ voÞ

It follows that the transfer function GðsÞ of the circuit is given by

vo

vi ¼ GðsÞ ¼ ð1 2 tsÞ

ð1 þ tsÞ ð16:102Þ

It is seen that the magnitude of the frequency-response function GðjvÞ is

lGðjvÞl ¼

ffiffiffiffiffiffiffiffiffiffiffiffi

p1 þt 2v2

ffiffiffiffiffiffiffiffiffiffiffiffi

p1 þt 2v2

or

lGðjvÞl ¼ 1 ð16:103Þ

and the phase angle of GðjvÞ is

/GðjvÞ ¼ 2tan21 tv 2 tan21 tv

or

/GðjvÞ ¼ 22 tan21 tv ¼ 22 tan21 RcCv ð16:104Þ

As needed, the transfer function magnitude is unity, indicating that the circuit does not distort the signal

amplitude over the entire bandwidth. Equation 16.104 gives the phase lead of the output, vo; with respect

to the input, vi: Note that this angle is negative, indicating that actually a phase lag is introduced. The

phase shift can be adjusted by varying the resistance, Rc:

16.8.2 Voltage-to-Frequency Converter

A voltage-to-frequency converter (VFC) generates a periodic output signal whose frequency is

proportional to the level of an input voltage. Since such an oscillator generates a periodic output

according to the voltage excitation, it is also called a voltage-controlled oscillator (VCO).

A common type of VFC uses a capacitor. The time needed for the capacitor to be charged to a fixed

voltage level will depend on the charging voltage (it is inversely proportional). Suppose that this voltage is

governed by the input voltage. Then, if the capacitor is made to periodically charge and discharge, we

have an output whose frequency (inverse of the charge – discharge period) is proportional to the charging

voltage. The output amplitude will be given by the fixed voltage level to which the capacitor is charged in

Signal Conditioning and Modification 16-57

© 2005 by Taylor & Francis Group, LLC

each cycle. Consequently, we have a signal with a fixed amplitude and a frequency that depends on the

charging voltage (input).

A VFC (or VCO) circuit is shown in Figure 16.25(a). The voltage-sensitive switch closes when the

voltage across it exceeds a reference level, vs; and it opens again when the voltage across it falls below a

lower limit, voð0Þ: The programmable unijunction transistor (PUT) is such a switching device.

Note that the polarity of the input voltage, vi; is reversed. Suppose that the switch is open. Then,

current balance at node A of the opamp circuit gives

vi

R ¼ C

dvo

dt

As usual, vA ¼ voltage at positive lead ¼ 0 because the opamp has a very high gain, and current through

the opamp leads ¼ 0 because the opamp has a very high input impedance. The capacitor charging

equation can be integrated for a given value of vi: This gives

voðtÞ ¼

1

RC

vit þ voð0Þ

The switch is closed when the voltage across the capacitor voðtÞ equals the reference level vs: Then,

the capacitor will be immediately discharged through the closed switch. Hence, the capacitor charging

time, T, is given by

vs ¼

1

RC

viT þ voð0Þ

Input

−vi

Oscillator

Output

v + o

A − R

vs

Reference Level

(a)

Voltage-

Sensitive

Switch

C

(b) Time t

Output

vo(t)

0 T 2T 3T 4T

vs

vo(0)

(vs vo(0))

vi

RC

T = −

FIGURE 16.25 A voltage-to-frequency converter (voltage-controlled oscillator): (a) circuit; (b) output signal.

16-58 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Accordingly,

T ¼

RC

vi ðvs 2 voð0ÞÞ ð16:105Þ

The switch opens again when the voltage across the capacitor drops to voð0Þ; and the capacitor again

begins to charge from voð0Þ up to vs: This charging and instantaneous discharge cycle repeats periodically.

The corresponding output signal is as shown in Figure 16.25(b). This is a periodic (sawtooth) wave with

period T: The frequency of oscillation of the output ð1=TÞ is given by

f ¼

vi

RCðvs 2 voð0ÞÞ ð16:106Þ

It is seen that the oscillator frequency is proportional to the input voltage vi: The oscillator amplitude

is vs; which is fixed.

VCOs have many applications. One application is in analog-to-digital conversion. In the VCO type

analog-to-digital converters, the analog signal is converted into an oscillating signal using a VCO. Then,

the oscillator frequency is measured using a digital counter. This count, which is available in the digital

form, is representative of the input analog signal level. Another application is in digital voltmeters.

Here, the same method as for ADC is used. Specifically, the voltage is converted into an oscillator signal

and its frequency is measured using a digital counter. The count can be scaled and displayed to provide

the voltage measurement. A direct application of the VCO is apparent from the fact it is actually a

frequency modulator, providing a signal whose frequency is proportional to the input (modulating)

signal. Hence, the VCO is useful in applications that require frequency modulation. Also, a VCO can be

used as a signal (wave) generator for variable-frequency applications; for example, it can be used for

excitation inputs for shakers in vibration testing, excitations for frequency-controlled DC motors, and

pulse signals for translator circuits of stepping motors.

16.8.3 Frequency-to-Voltage Converter

A frequency-to-voltage converter (FVC) generates an output voltage whose level is proportional to

the frequency of its input signal. One way to obtain a FVC is to use a digital counter to count the

signal frequency and then use a DAC to obtain a voltage proportional to the frequency. A schematic

representation of this type of FVC is shown in Figure 16.26(a).

Digital

Counter

Voltage

Output

Frequency

Signal DAC

−

Comparator

(a)

(b)

Switching

Circuit

Capacitor

Circuit

Switching

Circuit

Voltage

Output

Frequency

Signal

Threshold

Signal

Charging

Voltage

vs

FIGURE 16.26 Frequency-to-voltage converters: (a) digital counter method; (b) capacitor charging method.

Signal Conditioning and Modification 16-59

© 2005 by Taylor & Francis Group, LLC

An alternative FVC circuit is schematically shown in Figure 16.26(b). In this method, the frequency

signal is supplied to a comparator along with a threshold voltage level. The sign of the comparator output

will depend on whether the input signal level is larger or smaller than the threshold level. The first sign

change (negative to positive) in the comparator output is used to trigger a switching circuit that will

respond by connecting a capacitor to a fixed charging voltage. This will charge the capacitor. The next

sign change (positive to negative) of the comparator output will cause the switching circuit to short the

capacitor, thereby instantaneously discharging it. This charging – discharging process will be repeated in

response to the oscillator input. Note that the voltage level to which the capacitor is charged each time

will depend on the switching period (charging voltage is fixed), which is in turn governed by the

frequency of the input signal. Hence, the output voltage of the capacitor circuit will be representative of

the frequency of the input signal. Since the output is not steady due to the ramp-like charging curve and

instantaneous discharge, a smoothing circuit is provided at the output to remove the noisy ripples.

Applications of FVC include demodulation of frequency-modulated signals, frequency measurement

in mechanical vibration applications, and conversion of pulse outputs in some types of sensors and

transducers into analog voltage signals.

16.8.4 Voltage-to-Current Converters

Measurement and feedback signals are usually

transmitted as current levels in the range of 4 to

20 mA rather than as voltage levels. This is

particularly useful when the measurement site is

not close to the monitoring room. Since the

measurement itself is usually available as a voltage,

it has to be converted into current by using a

voltage-to-current converter (VCC). For example,

pressure transmitters and temperature transmitters

in operability testing systems provide current

outputs that are proportional to the measured

values of pressure and temperature.

There are many advantages to transmitting current rather than voltage. In particular, the voltage level

will drop due to resistance in the transmitting path, but the current through a conductor will remain

uncharged unless the conductor is branched. Hence, current signals are less likely to acquire errors due to

signal weakening. Another advantage of using current instead of voltage as the measurement signal is that

the same signal can be used to operate several devices in series (for example, a display, a plotter, and a signal

processor simultaneously), without causing errors through signal weakening due to the power lost at each

device, because the same current is applied to all devices. AVCC should provide a current proportional to

an input voltage without being affected by the load resistance to which the current is supplied.

An operational-amplifier-based voltage-to-current convert circuit is shown in Figure 16.27. Using the

fact that the currents through the input leads of an unsaturated opamp can be neglected (due to very high

input impedance), we write the current summation equations for the two nodes, A and B, thus:

vA

R ¼

vp 2 vA

R

and

vi 2 vB

R þ

vP 2 vB

R ¼ io

Accordingly, we have

2vA ¼ vP ðiÞ

Input

Voltage

vi

Output

Current io

−

+

B

R R

R A

R

Load

RL

P

FIGURE 16.27 A voltage-to-current converter.

16-60 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

and

vi 2 2vB þ vP ¼ Rio ðiiÞ

Now, using the fact that vA ¼ vB for the opamp (due to very high gain), we substitute Equation i into

Equation ii. This gives

io ¼

vi

R ð16:107Þ

in which

io ¼ output current

vi ¼ input voltage

It follows that the output current is proportional to the input voltage, irrespective of the value of the load

resistance, RL; as required for a VCC.

16.8.5 Peak-Hold Circuits

Unlike a S/H circuit that holds every sampled value

of the signal, a peak-hold circuit holds only the

largest value reached by the signal during the

monitored period. Peak holding is useful in a

variety of applications. In signal processing for

shock and vibration studies, what are known as

response spectra (e.g., a shock response spectrum) are

determined by using a response spectrum analyzer

that exploits a peak holding scheme. Suppose that

a signal is applied to a simple oscillator (a singledegree-

of-freedom second-order system with no

zeros) and the peak value of the response (output) is determined. A plot of the peak output as a function

of the natural frequency of the oscillator, for a specified damping ratio, is known as the response

spectrum of the signal for that damping ratio. Peak detection is also useful in machine monitoring and

alarm systems. In short, when just one representative value of a signal is needed in a particular

application, the peak value is a leading contender.

Peak detection of a signal can be conveniently done using digital processing. For example, the signal

may be sampled and the previous sample value replaced by the present sample value if and only if the

latter is larger than the former. By sampling and then holding one value in this manner, the peak value of

the signal is retained. Note that, usually, the time instant at which the peak occurs is not retained.

Peak detection can be done using analog circuitry as well. This is, in fact, the basis of analog spectrum

analyzers. A peak-holding circuit is shown in Figure 16.28. The circuit consists of two voltage followers. The

first voltage follower has a diode at its output that is forward biased by the positive output of the voltage

follower and reverse-biased by a low-leakage capacitor, as shown. The second voltage follower presents the

peak voltage that is held by the capacitor to the circuit output at a low output impedance, without loading

the previous circuit stage (capacitor and first voltage follower). To understand the operation of the circuit,

suppose that the input voltage, vi; is larger than the voltage to which capacitor is charged (v). Since the

voltage at the positive lead of the opamp is vi and the voltage at the negative lead is v; the first opamp will be

saturated. Since the differential input to the opamp is positive under these conditions, the opamp output

will be positive. The output will charge the capacitor until the capacitor voltage, v; equals the input voltage,

vi: This voltage (call it vo) is in turn supplied to the second voltage follower which presents the same value to

its output (gain ¼ 1 for a voltage follower), but at a very low impedance level. Note that the opamp output

remains at the saturated value only for a very short time (the time taken by the capacitor to charge). Now,

suppose that vi is smaller than v: Then, the differential input of the opamp will be negative, and the opamp

output will be saturated at the negative saturation level. This will reverse bias the diode. Hence, the output

of the first opamp will be in open circuit, and as a result the voltage supplied to the output voltage follower

+

v

−

Peak Value

(Output) vo

Input

Signal

vi

+

−

Diode

Reset

Switch

Output

Voltage Follower

FIGURE 16.28 A peak-holding circuit.

Signal Conditioning and Modification 16-61

© 2005 by Taylor & Francis Group, LLC

would still be the capacitor voltage and not the output of the first opamp. It follows that the voltage level of

the capacitor (and hence the output of the second voltage follower) would always be the peak value of the

input signal. The circuit can be reset by discharging the capacitor through a solid-state switch that is

activated by an external pulse.

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