Experiment 17

Two Differentiators Circuit

Analog Computing

• Analog computers– First were mechanical systems. Electrical analog

computers were developed in the early 1940’s and used extensively.• Maximum speed of response was limited to less than

10 MHz.

• Analog controllers– PID (proportional-integral-derivative) controllers• Thermostats

Capacitors

)()(1

)(

)(

1

oC

t

t

CC

CC

tvdttiC

tv

dt

dvCti

o

iC

i = 0

iR

iR + iC + i =0 where i = 0mA

iC =C1 dVin/dt

iR = [0V – Vo]/R1

Vo = -R1C1 dVin/dt

If Vin = 0.5V sin(wt), then Vo =[ -R1C1 w cos(wt)] x 0.5V

where -R1C1 w is the maximum gain of the differentiator.

Conversions for Sinusoids

A sin(wt +f) A cos(wt + f - 90o)- A sin(wt +f) A sin(wt + f + 180o )

Or A sin(wt + f - 180o )

- A cos(wt +f) A cos(wt + f + 180o )Or

A cos(wt + f - 180o )

A sin(wt +f) A sin (wt + f - 360o)Or

A sin (wt + f + 360o)

A cos(wt +f) A cos (wt + f - 360o)Or

A cos (wt + f + 360o)

Sine – Cosine Conversion

• Vin = sin(wt)

• Vo = -R1C1 w cos(wt)

• Vo = R1C1 w cos(wt – 180o)

• Vo = R1C1 w sin(wt – 180o + 90o)

• Vo = R1C1 w sin(wt – 90o)

When the input voltage is sinusoidal, the output voltage has a phase shift of 90o with respect to the input voltage.

Limitation of Ideal Differentiator

• If the input contains electronic noise with high frequency components, the magnitude of the high frequency components will be amplified significantly over the signal of interest and the system could become unstable. – It is necessary to modify the circuit to reduce or

eliminate such effects (see Practical Differentiator circuit in Experiment 16 of the lab manual).

Circuit to be constructed

Velleman Function Generator

Circuit Construction

• Position of the two switches changes the maximum gain of the differentiator.– R1 is determined by setting w R1 C1 = 1 when f is 7.23 kHz. Use

0.1 µF for C1.

– R2 is determined by setting w R2 C1 = 1 when f is 723 Hz.

• Use two of the three slide switches in the parts kit– Three pin black rectangle with knob

• Middle pin should be connected to the circuit• Either one of the outer two pins should be connected to the other

portion of the circuit– Sliding the knob from right to left changes which outer pin is shorted to the

middle pin

Phase Shift

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2

f

1T is wavesine theof period the wheredegrees360

angle. phase theis where)sin( )90sin()cos()sin(

T

t

tttdt

td o

Phase Shift as a Function of Frequency

• The phase shift between the input voltage and the output voltage of the op amp will change from 90o to 180o.

Caution: PSpice Transient Analysis Issue

Information in first half cycle is incorrect because the initial charge on the capacitor is zero.

Measurement of Phase Angle

• There are two sets of instructions in the Week 11 Module.– Phase Delay.pdf, which explains how to make a phase

angle calculation using the information displayed when the Oscilloscope function of the Velleman oscilloscope is used.• You should become familiar with this technique.

– Magnitude and Phase.pdf, which explains how to use the automated measurement tools on the Velleman scope to obtain the magnitude and phase of a signal at a single frequency and over a range of frequencies.

dB

• dB is an abbreviation for decibels

V

V log 20 dB

P

Plog 10 dB

in

out

in

out

0.707 2

2

V

V when occurs 3dB-

2

1

P

P when occurs 3dB-

in

out

in

out