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Basics of Transconductance – Capacitance Filters Dr. Paul Hasler

Hasler GmCFilter01

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Page 1: Hasler GmCFilter01

Basics of Transconductance –Capacitance Filters

Dr. Paul Hasler

Page 2: Hasler GmCFilter01

Operational Amplifiers

OP Amp: Vout = Av (V+ - V- )

Balanced Op Amp: Vout

+ = Av (V+ - V- )Vout

- = Av (V- - V+)

V+

V- V-

V+

VoutVout

+

Vout-

Av is frequency dependent

Page 3: Hasler GmCFilter01

Transconductance Amplifiers

• Most op-amps can be used as OTAs, most OTAs can be used as op-amps. (depends which application is being optimized)

OTA: I = Gm (V+ - V- )

Balanced OTA: I+ = Gm (V+ - V- )I- = Gm (V- - V+)

V+

V- V-

V+

I I+

I -

Page 4: Hasler GmCFilter01

Major Issues for OTAs

Major Issues for OTAs: • Linearity• Offsets• Output Resistance (depends upon application)

Page 5: Hasler GmCFilter01

How to Build OTAs

Basic transistor differential amplifer, Wide-output-range differential amplifier …

Build with cascodes or folded cascode or differential approaches.

Page 6: Hasler GmCFilter01

Analysis of Diff-Pair

Page 7: Hasler GmCFilter01

Differential Pair Currents

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

0.5

1

1.5

2

2.5

3

Differential input voltage (V)

Out

put c

urre

nt (

nA)

Iout+Iout

-

Page 8: Hasler GmCFilter01

Basic Differential Amplifier

Page 9: Hasler GmCFilter01

Transfer Functions

Page 10: Hasler GmCFilter01

Wide-Output Range Differential Amplifier

Wide-output range amplifier

Simple differential amplifer

"Vmin" effect

Page 11: Hasler GmCFilter01

Simplest Gm-C Filter

IVoutVin

C1

GND

dVout(t) dtBy KCL: Gm(Vin - Vout ) = C1

Defining τ = C1 / Gm

dVout(t) dtVin = Vout + τ

Also, should mention the high-pass version as well.

We can set Gm and build C sufficiently big enough (slow down the amplifier), or set by C (smallest size to get enough SNR), and change Gm.

If an ideal op-amp, then Vin = Vout

Page 12: Hasler GmCFilter01

Tuning Frequency using Bias Current

6Hz1/25GΩ1pA

6kHz1/25MΩ1nA

6MHz1/25kΩ1µA

600MHz1/250Ω1mA

Cutoff FrequencyTransconductanceBias Current

This approach is the most power-efficient approach for any filter.

All electronically tunable: Advantage: we can electronically change the cornerDisadvantage: we need a method to set this frequency (tuning)

C = 1pF, W/L of input transistors = 30, Ith ~ 10µA.

Page 13: Hasler GmCFilter01

A Low-Pass FilterC (dVout/dt) = I

= Ibias tanh(κ(Vin - Vout)/(2 UT) )= ((κ Ibias )/(2 UT) ) (Vin - Vout)

Time-constant changeswith bias

τ = C UT / κ Ibias

τ (dVout/dt) + Vout = Vin

Page 14: Hasler GmCFilter01

Linear Range

C (dVout/dt) = Ibias tanh(κ(Vin - Vout)/(2 UT) )

This is not linear….“Small-signal analysis: How small is “small”?

Page 15: Hasler GmCFilter01

Another First-Order Low-Pass Filter

GND

CGND

10nS 1nSVin GND

Vout

Page 16: Hasler GmCFilter01

Second-Order Behavior

1 + (τ/Q) s + τ2 s2

1Vout

Vin

=

Gain = Q at ω = 1/τ,

Page 17: Hasler GmCFilter01

Second-Order Behavior

1 + (τ/Q) s + τ2 s2

1Vout

Vin

=

Gain = Q at ω = 1/τ,

High Q = very narrow band(width ∝ 1/Q)

Page 18: Hasler GmCFilter01

Basic Second-Order Sections

Vout

1kΩ

Vin

Gm1

Vin

Gm1

I

C1

GND

I

C1

GND

Gm2

Page 19: Hasler GmCFilter01

Diff2 Based Second-Order Filter

1 + (τ/Q) s + τ2 s2

1Vout

Vin

=

Vin

-+ Vout

V1 +-

Vτ2Vτ1

τ = (τ1 τ2)1/2

Q = (τ2/τ1)1/2

Page 20: Hasler GmCFilter01

Diff2 Responses

Page 21: Hasler GmCFilter01

Diff2 Second-Order Section

Vout

Vin

Gm1Gm2

I

C1

GND

I

C1

GND

We will see this circuit again for sample and hold elements

Issue of feedback with an outputbuffer for an op-amp.

Page 22: Hasler GmCFilter01

Tow-ThomasSecond-Order Section

Vout

Vin

Gm1Gm2

I

C1

GND

I

C1

GND

GND

GND

Page 23: Hasler GmCFilter01

Bandpass Second-Order Section

GND

Vout

Vin

Gm1

C

GND

CV Gm2

Page 24: Hasler GmCFilter01

Another Second-Order Section

GND

Vout

Vin

Gm1

CL

GNDGNDC1

GND

Cw

Gm2

Gm3

V1

Page 25: Hasler GmCFilter01

Even Bigger Gm-C Filter

GND

Vin

Gm1

C

GND

C Gm2

GND

Vout

Gm1

C

GND

C Gm2

1

Page 26: Hasler GmCFilter01

Filter Design Problem

Design example(s): Design a Chebyshev filter with the following specs:

150kHzfsb

100kHzfpb

-25dBTsb

-1dB ( 0.5088)Tpb

We get the following filter:

From our MATLAB functions, we get the following τ’s and Q’s:

Page 27: Hasler GmCFilter01

Filter Design Example

101 102 103 104 10510

-2

10-1

100

Frequency (Hz)

Mag

nitu

de

From MATLAB code: N = 5

101

102

103

104

105

106

10-3

10-2

10-1

100

101

Q = 5.5561e+000, tau = 1.6009e-006Q = 1.3987e+000, tau = 2.4290e-006

tau = 5.4974e-006