Phasors, Capacitors, Impedance

Spin My World Right Round

LPF Application: Gastric Electrical Activity

60-70 million people suffer from GI disorder

Electrically active organ

http://www.virtualmedicalcentre.com/anatomy/gastrointestinal-system/7

Image from P. Du et al. Biophys J. (2010) 99:2784-2792.

Electrogastrogram and Slow Wave mapping

Recording the EGG (6 electrodes)Flexible Electrode Array (256 electrodes)

Typical gastric slow wave recording (freq ~ 0.05 Hz)

Mapping SW activity with high density electrodes

Real life example

Signal of interest

Noise (oscillations so fast you can’t see the individual waves)

What the electrode records (ruh-roh!)

One possible solution: Low Pass Filter

• High frequency components: Kill off (strongly attenuate) • Low frequency components: Pass through essentially untouched

Vout(t)

LPF in Action

Strongly attenuated 60Hz

Barely attenuated 0.05Hz

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-5

-4

-3

-2

-1

0

1

2

3

4

5

Time (s)

Sig

nal (

V)

60Hz power line noise

0 10 20 30 40 50 60-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Time (s)

Sig

nal (

V)

Gastric Slow Wave

Raw vs. Filtered Signal

0 10 20 30 40 50 60-6

-4

-2

0

2

4

6

Raw S

ignal

(V)

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Filter

ed Si

gnal

(V)

Time (s)

0 10 20 30 40 50 60-6

-4

-2

0

2

4

6

Raw

Signa

l (V)

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Filter

ed Si

gnal

(V)

Time (s)

INPUT

(raw gastric + raw powerline)

OUTPUT

(filtered gastric + raw powerline)

Capacitors come in all shapes and sizes

Typical ones in the lab Some big, some small

These are capacitors too!

Cell membrane Icky cells (bacteria)

And so is this!

PC boardRed = top layerGreen = bottom layer

Phasors: Magnitude and Phase

Projection onto real axis traces out Acos(wt + f)

Magnitude and Phase

http://resonanceswavesandfields.blogspot.com/2007/08/phasors.html#phasor-beating

Phasors!

coswt-coswt

sinwt

-sinwt

Rotates in timeBut we just read off the magnitude and phase information at one instant (usually, t = 0)

Raw vs. Filtered Signal

0 10 20 30 40 50 60-6

-4

-2

0

2

4

6

Raw S

ignal

(V)

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Filter

ed Si

gnal

(V)

Time (s)

0 10 20 30 40 50 60-6

-4

-2

0

2

4

6

Raw

Signa

l (V)

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Filter

ed Si

gnal

(V)

Time (s)

(50 k )W

(10 mF)

We now interrupt this broadcast to do some phasor math…

Implementing a low pass filter (LPF)

Strongly attenuated 60Hz

Barely attenuated 0.05Hz

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-5

-4

-3

-2

-1

0

1

2

3

4

5

Time (s)

Sig

nal (

V)

60Hz power line noise

0 10 20 30 40 50 60-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Time (s)

Sig

nal (

V)

Gastric Slow Wave

60 Hz signal zoomed in

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

-1

-0.5

0

0.5

1

1.5

2

Time (s)

Sig

nal (

V)

60Hz power line noise

f

Bode Plot

-5 -4 -3 -2 -1 0 1 2 3 4 5-120

-100

-80

-60

-40

-20

0

log10

()

G(

) (d

B)

-5 -4 -3 -2 -1 0 1 2 3 4 5-120

-100

-80

-60

-40

-20

0

log10

()

G(

) (d

B)

-5 -4 -3 -2 -1 0 1 2 3 4 5

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

log10

()

(d

eg)

gastric power linecutoff

What else is it good for?

GET RID OF BIEBER

Beauty and the Beat

- Sweet bass line (f <= 150 Hz)

- Screechy Bieber vocals (f >= 450 Hz)

- Raw song SNR = Bass/Bieber = ½

- Design a circuit that:- increases SNR >= 1- keeps phase shift for bass

frequencies |f|<= 40 deg.