RLC Filters - CAT-SOOP

11/4/19

1

RLC Filters

TF 1

!"#$%"&'

=)*+

1 − *.+/0 + )*+

1 − *.+/= )*+0 − *.0+/ + )*+ =

)*0/

1+/ +

)*0/ − *.

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2

TF 2

!"#$%"&'

= )) + +,-

1 − ,0-1= ) − ,0)-1) − ,0)-1 + +,- =

1-1 − ,

0

1-1 +

+,)1 − ,0

TF 3

!"#$%"&'

= )) + +,- + 1

+,/= +,)/+,)/ − ,1-/ + 1 =

+,)-

1-/ +

+,)- − ,1

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TF 4

!"#$%"&'

=)*+ + 1

)*./ + )*+ + 1

)*.= −*1+. + 1)*/. − *1+. + 1 =

1+. − *

1

1+. +

)*/+ − *1

Larger Patterns?

!"#$%"&'

= )(+)+#- + /+21 − +-

+# Natural Resonant Frequency of SystemDamping Coefficient1

The relationship between 1 and +# dictates how the circuit will respond in time (But also frequency)!

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RLC Circuits (Examples of Second Order)

• Undriven (series) RLC circuit

Differential eqn

Solution has form:

Characteristic equation:

Damping factor:

Undamped resonant frequency

Roots:

Homogeneousresponse

!" =$% =

&',)*+,,)*

Characteristic impedance

For series RLCDepends on topology!!

Second Order Response• Three cases dictated by !" and #

# < !%Underdamped

&'( = *+,- ./*012- + .4*+012-&'( = *+,- .5cos(!:;) + .=sin(!:;)&'( = .@*+,-cos(!:; + .A)

Qualityfactor

# > !%Overdamped

Real,negative

&C( = ./*DE- + .4*DF-

# = !%Critically damped

&C( = ./*+,- + .4;*+,-

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!"#$%"&'

=)*+,

1., +

)*+, − *1

!"#$%"&'

=1., − *

1

1., +

)*+, − *1

!"#$%"&'

=)*+.

1., +

)*+. − *1

!"#$%"&'

=1., − *

1

1., +

)*+. − *1

2 = 3 2 = 24 2 = ∞

3 67 8 3 3 − 67

8 3 3 ∓ 67 8 3

3 67 8 3 3 − 67

8 3 3 ∓ 67 8 3

RLC circuit

!"#$%"&'+

-

!"#$%"&' =

1*+, ∥ *+.1*+, ∥ *+. +0

=*+.0

1−+2., + *+.0

3456

3457

08 > :;

02 < :;

• Can tune sharpness of resonance by changing R• Small R è blunt resonance• Large R è sharp resonance

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Band-pass filters• Combine R and parallel LC or series LC

BandwidthBW = !"

# = 2%&' ≫ &)

&' ≪ &)

1/√2

BW

&'

&)

2%

./

Band-stop or Notch filters• Combine R and parallel LC or series LC

!"

!#

2%

&'

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Sallen Key Topology

https://en.wikipedia.org/wiki/Sallen%E2%80%93Key_topology

What you drop in for each impedance block will affect the output. Certain patterns result in certain types of filters!

Changing op amp feedback path from a regular old buffer (gain of 1) to something with some resistors in the feedback path gives us freedom to increase the overall GAIN of the system R3/R4 with the op amp form a non-inverting amplifier, and if you were to drop the implication of that into our original derivation, you'd find that the original transfer function just gets scaled by this gain. In terms of frequency response if you have a high-pass filter

And the Sallen Key is super flexible! Change what parts you put where and you'll get a Low Pass Filter!