Filters

6.002

CIRCUITS AND ELECTRONICS

Filters

http://electrical.globalautomation.info

Review

R

vI + –

C

+ vC –

ZR

Vi + –

ZC

+ Vc –

ZC Vc = ⋅ Vi ZC + Z R 1 Vc 1 jωC = = 1 Vi + R 1 + jωRC jω C Reading: Section 14.5, 14.6, 15.3 from A & L.

http://electrical.globalautomation.info

A Filter ZR

Vi + –

ZC

+ Vc –

ZC 1 ⋅ Vi = Vc = ZC + Z R 1 + jωRC Vc H (ω ) = Vi

1

Demo with audio

“Low Pass Filter”

ω

http://electrical.globalautomation.info

Quick Review of ImpedancesJust as I ab

A

R1

+ Vab

RAB

Vab = = R1 + R2 I ab

Z AB

Vab = = R1 + jωL I ab

R2

B

I ab

A

R1

–

+ Vab

j ωL B

–

http://electrical.globalautomation.info

Quick Review of Impedances Similarly

A

Z AB = R1 + Z C || R2 + Z L

R1 R2

C L B

= R1 +

Z C R2 + ZL Z C + R2

= R1 +

R2 + jω L 1 + jωCR2

http://electrical.globalautomation.info

We can build other filters by combining impedances Z (ω )

L Z

R C

ω

http://electrical.globalautomation.info

We can build other filters by combining impedances Z (ω)

L Z

R C

ω

H (ω )

HPF High Pass Filter

+ –

ω H (ω )

LPF Low Pass Filter ω

+ –

H (ω )

+ –

HPF ω

http://electrical.globalautomation.info

Check out:

C

L

+ R Vr –

+ –

Vi

Intuitively: Vr 1 Vi C

k bloc

eq r f w s lo

L bloc

ωo =

1

(1 − ω

ω RC

2

LC ) + (ω RC ) 2

freq

ω

LC

R Vr = 1 Vi jω L + +R jω C j ω RC = 1 − ω 2 LC + j ω RC Vr = Vi

ks hig h

2

At resonance, ω = ωo and ZL + ZC = 0, so Vi sees only R! More later…

http://electrical.globalautomation.info

What about:

Vlc

+ Vi + –

Vlc Vi 1

C open

L

– C

R

Band Stop Filter L open

ω

Check out Vl and Vc in the lab.

http://electrical.globalautomation.info

Another example: R + L

Vi + –

C

Vo –

Vo Vi ort h s L

BPF Cs

ωo

ho rt

ω

Application: see AM radio coming up shortly

http://electrical.globalautomation.info

AM Radio Receiver

antenna

R Vi + –

L

C

demodulator amplifier

Thévenin antenna model crystal radio demo

http://electrical.globalautomation.info

AM Receiver R Vi + –

L

C

demodulator amplifier

filter

signal strength 10 KHz

WBZ News Radio

f 540 …1000 1010 1020 1030 … 1600 KHz “Selectivity” important — relates to a parameter “Q” for the filter. Next…

http://electrical.globalautomation.info

Selectivity: Look at series RLC in more detail C

L

Vi + – Recall, Vr Vi

+ Vr –

R Vr R = Vi R + jω L +

1 1 2

1 jω C

higher Q

Δω

bandwidth

ωo ωo Define Q = Δω

ω

quality factor

high Q ⇒ more selective http://electrical.globalautomation.info

Quality Factor Q Q=

ωo Δω

ωο: R Vr = Vi R + jω L +

1 = 1 L 1 ⎞ ⎛ 1 + j⎜ ω − ⎟ jω C R CR ω ⎝ ⎠

at ωο =0 1 ωo = LC Δω ?

http://electrical.globalautomation.info

Quality Factor Q ωo Q= Δω Δω : Note that abs magnitude is when

Vr = Vi

i.e. when

1 2

1 1 = ⎛ L 1 ⎞ 1 ± j1 1 + j⎜ ω − ⎟ ω R CR ⎠ ⎝ 1 ωL − = ±1 R ω CR

ω2 ∓

ωR L

−

1 =0 LC

Looking at the roots of both equations, R 1 ω1 = + 2L 2

R2 4 + L2 LC

R 1 ω2 = − + 2L 2

R2 4 + L2 LC

R Δω = ω1 − ω2 = L

http://electrical.globalautomation.info

Quality Factor Q

Q= Q=

ωo Δω ωo R L

=

ωo L R

1 ωo = LC

The lower the R (for series R), the sharper the peak

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Quality Factor Q Another way of looking at Q : energy stored Q = 2π energy lost per cycle = 2π

1 L Ir 2

2

1 2 2π Ir R 2 ω0 ωo L Q= R

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