frequency domain analysis of linear circuits using ... and fft.pdf · frequency domain analysis of...

Frequency domain analysis of linear

circuits using synchronous detection

Physics 401, Fall 2013.

Eugene V. Colla

http://www.uiuc.edu/


1. Fourier transform,. Discrete Fourier transform. Some properties.

2. Time domain and Frequency domain representation of the data.

3. Frequency domain spectroscopy (FDS)

4. Lock-in amplifiers

5. Practical application of lock-in’s in FDS

6. Taking data and simple data analysis using OriginPro.

Frequency domain analysis of linear circuits using synchronous detection

Outline

9/9/2013



in 1822, Jean Baptiste Fourier developed the theory that shows that

any real waveform can be represented by the sum of sinusoidal

waves.

Jean Baptiste Joseph

Fourier

(1768 – 1830)

Let we try to create the square wave as a sum of

sine waves of different frequencies

Square wave.F=40Hz, A=1.5V

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1A sin(2πωt) 1 3 3sin(2 ) sin(2 3 )A t A t

1 3 3

5 5

sin(2 ) sin(2 3 )

sin(2 5 )

A t A t

A t

1 3 3

5 5 7 7

sin(2 ) sin(2 3 )

sin(2 5 ) sin(2 7 )

A t A t

A t A t

9/9/2013



Fourier Transform

+

2πjft

-

H(f)= h(t)e dt; j= -1

+

-2πjft

-

h(t)= H(f)e df

The continues Fourier transformation of the signal h(t) can be

written as:

H(f) represents in frequency domain mode the time domain signal h(t)

Equation for inverse Fourier transform gives the correspondence of the

infinite continues frequency spectra to the corresponding time domain

signal.

In real life we working with discrete representation of the

time domain signal recorded during a finite time.



Discrete Fourier Transform

It comes out that in practice more useful is the representation the

frequency domain pattern of the time domain signal hk as sum of

the frequency harmonic calculated as:

12 /

0

1( )

Nkn N

n n k

k

H H f h eN

D is the sampling interval, N – number of collected points

DFT

Time domain Frequency domain



Discrete Fourier Transform

For periodic signals with period T0:

0

1 10 0

2 2( ) cos sin

2n n

n n

a nt ntF t a b

T T

0 0

0

0 0 0 00 0

0

0 0

2 2 2 2( )cos ; ( )cos ;

2( ) ;

T T

n n

T

nt nta F t dt b F t dt

T T T T

a F t dtT



Discrete Fourier TransformNow how I found the amplitudes of the harmonics

to compose the square wave signal from sine waves of different frequencies.

DFT

Time domain signal

Decomposition the signal into the sine wave harmonics. The only

modulus's of the harmonics amplitudes are presented in this picture.



Frequency Domain Spectroscopy(linear system)

Studied objectAsin(t) B1sin(t)+B2cos(wt)

Applied test signal Response of the studied system

We applying the sine wave signal to tested object and measuring the

response. Varying the frequency we can study the frequency

properties of the system

x

y

reference

V0sin(t+) x

0 100 200 300 400 500 600 700

-0.6

0.0

0.6

-0.6

0.0

0.6

-0.6

0.0

0.6

time (msec)

Vin=sin(t+/4)



Lock-in amplifier

Now about the most powerful tool which can be used in frequency

domain technique.

PSD*

Signal

amplifier

VCO**

Signal

in

Reference

in

Signal

monitor

Reference

out

Low-pass

filter

DC

amplifier

output

*PSD - phase sensitive detector;**VCO - voltage controlled oscillator

John H. Scofield

American Journal of Physics 62 (2)

129-133 (Feb. 1994).



The DC output signal is a magnitude of the product of the input and reference

signals. AC components of output signal are filtered out by the low-pass filter

with time constant t (her t=RC)



Lock-in amplifier. Phase shift.

x

y

reference

V0sin(t+)

=/4, Vout=0.72Vin

0 100 200 300 400 500 600 700

-0.6

0.0

0.6

-0.6

0.0

0.6

-0.6

0.0

0.6

time (msec)

Vin=sin(t+/4)

reference

output



Lock-in amplifier. Two channels demodulation.

In many scientific applications it is a great advantage to measure both

components (Ex, Ey) of the input signal. We can use two lock-ins to do

this or we can measure these value in two steps providing the phase

shift of reference signal 0 and /4. Much better solution is to use the

lock-in amplifier equipped by two demodulators.

Ein=Eosin(t+)

sin(t)

cos(t)

to Ex channel

to Ey channel

x

y

Ey

Ex



Digital Lock-in amplifier

ADCInput

amplifier

einDSP

External reference signal

Internal

Function

generator

Asin(t+f)

DAC

Analog outputs

Digital interface



Digital Lock-in amplifier

Digital lock-in SR830

Lock-in demo



lock-in_demo.vee

Experiments. Main idea.

Investigating the frequency response of circuit.

H()

Response function →

Frequency domain representation of the system

Linear engineering systems are those that can be modeled by

linear differential equations.



9/9/2013

Typical application of the lock-in amplifier

He4

AC drive signal

Transmitter (heater)

Receiver



Experiments. Main idea.

Calculation of the response function in

frequency domain mode.

Example 1. High-pass filter.

C

R

2( )( ) ( )* ( ) ( )

1( ) 2( )out in in

ZV H V V

Z Z

Applying the Kirchhoff

Law to this simple network



Experiments. Calculation of the response function

in frequency domain mode. High-pass filter

C

R

2( )( ) ( )* ( ) ( )

1( ) 2( )out in in

ZV H V V

Z Z

R

L

C

Z = R

Z = j L

1 jZ = = -

j C C

R

L L

C -1

C

Z =R+...

Z =jωL+R

1 1Z = =

jωC jωC+R

Ideal case More realistic





C

R

2 2

1

2 2

2

( ) ;1 1 1 1

where ;

1( ) ; ( )

( )

( )

( )

( )arctan arctan

1

c

I

I

R

R

R

I

R j RCH

HH

jH j

j RC jR

j C

j

C

H

HH

H

R

t t

t t

t

t

t

t

t

t – time constant of the filter

C - cutoff frequency



C

R





High-pass filter. Fitting

Fitting parameters: V0, t, Voff

0

2( ) ;

1out in

V V H V RCt

tt

Fitting function



C

R





C

R


in frequency domain mode. Low-pass filter

2 2

1

2 2

2

1

1 1( ) ;

1 1 1 1

whe

1( )

re ;

1( ) ; ( ) arct

( )

an( )

( )arctan

1

R

R

I

R

c

I

I

j CH j

j RC jR

j

H

HH

C

RC

H

HHH

j

t t

t

t

t

t



Application of the lock-in amplifier for

study of the transfer function of the RLC circuit

R

C

Lock-in SR830

input FG output

Setup for measuring of

the response function

of the Hi-pass filter

Input (A) Sine wave output

• Use internal reference mode

• Do measurements on harmonic no1

• Take care about time constant – should

be at least ~10 times larger than period of

measuring frequency

• Avoid overloading of the lock-in





R

LC

R0

0

1 1 12 ;

o

Lf Q

RC R CLC

2

0

20

2

0

1

1

1

1

R j LH

R LC j RC

R j L

Rj RC

0 RLCR Z





0

1 1 12 ;

o

Lf Q

RC R CLC

0

2 20 0

2 2

0 0 0

2

2

0 0

2 22

0

2 2

0 0

11

11 1

11 1

11

j QR j L R

HR R

j RC jQ

j QQR

R

Q



Another style of the transfer function expressions

2 2 220 00

22 2 2 2

00

2 2

0 0

22 2 2 2

00

2 2200

22 2 2 2

00

0

0

;

;

;

1; ; Q=

out

out

out

jLV

R

LX

R

jLY

R

R

LLC

2

0

0

L

R

0 Fitting parameters: - scaling coefficient and



The resonance curves obtained on RLC circuits with two different

damping resistors





The example of fitting of the RLC circuit date to

the analytical expression could be found in: \\engr-file-03\PHYINST\APL Courses\PHYCS401\Common\Simple Examples\Lab 3 Frequency Domain

Analysis_example.opjj\

Fitting function

Fitting parameters



From time domain to frequency domain.

Experiment.

Lock-in SR830input Reference in

Wavetek

Out Sync

Time domain pattern

Frequency

domain ?

0

0

0

00 0

00 0

00

0

2 2( )cos ;

2 2( )sin ;

2( )

T

n

T

n

T

nta F t dt

T T

ntb F t dt

T T

a F t dtT

F(t) – periodic function F(t)=F(t+T0):

0

0

0

0 0

0 ; 2

2

TV V t

TV t T

V0

T0




Experiment with SR830. Results.

Time domain pattern

V0

T0

Time domain

Frequency domain

Spectrum measured by

SR 830 lock-in amplifier




FFT using Origin. Results.

V0

T0

Time domain taken

by Tektronix scope

Data file can be used to convert

time domain to frequency domain




FFT using Origin. Results.

V0

T0

Time domain taken

by Tektronix scope

Data file can be used to convert

time domain to frequency domain

Spectrum calculated by

Origin.

Accuracy is limited because

of the limited resolution of

the scope




Using of the Math option of the scope.

V0

T0

Time domain taken

by Tektronix scope

Spectrum calculated by

Tektronix scope.

Accuracy is limited because

of the limited resolution of

the scope




Using of the Math option of the scope.

Time domain taken

by Tektronix scope

Spectrum of the square wave

signalSpectrum of the pulse signal




Different waveforms. Lock-in.

ramp

pulse



Appendix #1

Reminder: please submit the reports by e-mail in MsWord or pdf format.

Strongly recommend the file name structure as:

L1_lab2_student1

Lab section Lab number Your name



Appendix #2

Origin templates for the Lab are available in:

\\engr-file-03\PHYINST\APL Courses\PHYCS401\Common\Origin

templates\frequency domain analysis

…



References:

1. John H. Scofield, “A Frequency-Domain Description of a Lock-in

Amplifier” American Journal of Physics 62 (2) 129-133 (Feb. 1994).

2. Steve Smith “The Scientist and Engineer's Guide to Digital Signal

Processing” copyright ©1997-1998 by Steven W. Smith. For more

information visit the book's website at: www.DSPguide.com" *

• You can find a soft copy of this book in:

• \\engr-file-03\phyinst\APL Courses\PHYCS401\Experiments



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