Continuous-time OTA-C Filters Professor Yichuang Sun
This part is concerned with structures and design methods of OTA-C filters, which use operational transconductance amplifiers (OTA) and capacitors (C ) only. Further details of the topic can be found in the textbook:
T. Deliyannis, Y. Sun and J. K. Fidler, Continuous-time Active Filter Design, CRC Press, Florida, USA, January 1999, ISBN: 0-8493-7893-1.
Advantages: OTA-C filters have the advantages of high frequency or wide bandwidth, electronic tunability or programmability, monolithic integrability (in CMOS), and simple structure and reduced component count, over Active-RC filters.
1. Concepts and Models of OTA
An ideal operational transconductance amplifier (OTA) is a voltage controlled current source (VCCS), with infinite input and output impedances and constant transconductance. The symbol and ideal small-signal model of the OTA are shown below:
Vi+ Vi -
gm IABC
Io
Vi +
+
Io gm(Vi+-Vi-)
Vi -
-
The output current of the ideal OTA is given by
I o = g m (Vi + − Vi − ) where Vi + and Vi − are the voltages at the noninverting and inverting input terminals.
The transconductance is given by
g m = hI ABC where the proportional constant, h, is dependent upon temperature, input device geometries, and the process, and IABC is the external DC bias current. In practice, gm can be electronically adjusted by varying IABC. Thus, for example, when gm is a filter parameter, IABC can be used to control or program the characteristic of the filter. The bias current IABC is often dropped from the OTA symbol to simplify notation in filter circuits and m dropped off from gm for simplicity. The OTA is used as an open loop amplifier in OTA-C filter design in contrast to the opamp which is used in a closed feedback loop in active-RC filters, one of the reasons why OTAC filters can work at higher frequencies than active-RC filters.
2. Basic OTA-C Building Blocks
Voltage amplifier and gain:
K=
Vo g = 1 Vi1 − Vi 2 g 2
Ideal integrator and function:
H (s) =
Vo 1 = Vi1 − Vi 2 s (C1 / g1 )
Lossy integrator and function (also first order filter)
H ( s) =
Vo g1 = Vi1 − Vi 2 sC1 + g 2
OTA-resistor and resistance:
g Zin
R = Z in =
1 g
3. Design of OTA-C Filters
The most popular method for high-order filter design is the cascade method due to its modularity of structure and simplicity of design and tuning. Second-order filters are the basic sections in cascade filter structures.
3.1 Typical Second-order Filter Characteristics:
Lowpass Filter:
H LP − Desired ( s ) =
K LP ω 0 s2 + s
ω0 Q
2
+ ω0
2
KLP, gain at DC (zero frequency), or passband gain; fo, cut-off frequency, Bandwidth, BW =3dB frequency, f3dB =fo: BW = fo Q, quality factor: Q<=0.707, flat passband in magnitude frequency response; Q>0.707, there will be a peak at fo. As Q increases, the transition band between passband and stopband becomes narrower.
Bandpass Filter:
H BP − Desired ( s ) =
K BP s2 + s
ω0 Q
ω0 Q
s
+ ω0
2
fo, center frequency; KBP, gain at fo or passband gain, Relation of bandwidth BW, center frequency fo, quality factor Q: BW =fo/Q, As Q increases, the filter becomes more selective (the selectivity becomes higher).
Highpass Filter:
H HP − Desired ( s ) =
K HP s 2 s2 + s
ω0 Q
+ ω0
2
KHP, gain at infinity frequency or passband gain; Q, quality factor; fo, cut-off frequency
3.2 Second-order OTA-C Filters
Three most popular two integrator loop OTA-C filter structures are given below.
Two-OTA or canonical OTA-C filter The structure below is the canonical two integrator loop OTA-C filter which consists of two ideal integrators and minimum number of components.
Vi
g1
Vo1
g2
Vo C2
C1
Nodal voltage equations (using KCL): sC1Vo1 = g1 (Vi − Vo ) sC 2Vo = g 2 (Vo1 − Vo )
Solving the equations, the circuit has a second-order lowpass transfer function given by H LP ( s ) =
Vo g1 g 2 g1 g 2 C1C 2 = 2 = 2 Vi s C1C 2 + sg 2 C1 + g 1 g 2 s + s ( g 2 C 2 ) + g 1 g 2 C1C 2
Matching the coefficients between the circuit function the desired characteristic
H LP ( s ) =
K LP ω o2 g 1 g 2 C1C 2 = = H LP − Desired ( s ) s 2 + s ( g 2 C 2 ) + g1 g 2 C1C 2 s 2 + s (ω o Q ) + ω o2
and solving the resulting matched equations we have
K LP = 1, ω o =
g1 g 2 , C1C 2
Q=
g 1C 2 g 2 C1
Four-OTA or Tow-Thomas (TT) OTA-C Filters
The TT OTA-C filter consists of an ideal integrator and a lossy integrator. This filter is most popular in practice due to low parasitic effects.
g2
Vo2 gg11 C2
Vi
Vo1 C1
g3
g0
Writing and solving nodal equations, for the given input Vi, the transfer functions of the circuit at output Vo1 and Vo2 can be derived as
H LP ( s ) =
− g 0 g1 V o1 = 2 = Vi s C 1C 2 + sg 3 C1 + g 1 g 2
V sg 0 C 1 H BP ( s ) = o 2 = 2 = Vi s C 1C 2 + sg 3 C 1 + g 1 g 2
ω0 =
g1 g 2 C 1C 2
Q=
K LP =
g0 g2
K BP =
1 g3
− K LP ω 0 s +s 2
ω0
s +s 2
+ ω0
Q
K BP
ω0 Q
ω0 Q
2 2
= H LP − Desired ( s )
s
+ ω0
= H BP − Desired ( s ) 2
g1 g 2C 2 C1
g0 g3
The TT OTA-C filter has a lowpass function from Vo1 and a bandpass from Vo2. Equations also show that filter gain can be tuned (via go) independently of the quality factor and frequency and the quality factor can be tuned (via g3) independently of the frequency.
Six-OTA OTA-C Filters or Kerwin-Hueslman-Newcomb (KHN) OTA-C Filter
Figure below shows the six-OTA or KHN two integrator loop OTA-C filter. The circuit is versatile as it can simultaneously generate lowpass (LP), bandpass (BP), and highpass (HP) functions at output Vo2, Vo1, and Vo3, respectively. Vi
g4
g0 Vo3
g3 g5
g1
Vo1
g2 C1
Nodal equations can be written as: g 5Vo 3 = g 0Vi − g 3Vo1 − g 4Vo 2 sC1Vo1 = g1Vo 3 sC 2Vo 2 = g 2Vo1 Solving the equations leads to: g g − 0 1s V g 5 C1 H BP ( s ) = o1 = g g g g g Vi s2 + 3 1 s + 4 1 2 g 5 C1 g 5 C1C 2 g 0 g1 g 2 V g 5 C1C 2 H LP ( s ) = o 2 = g g g gg Vi s2 + 3 1 s + 4 1 2 g 5 C1C 2 g 5 C1 g0 2 s Vo 3 g5 H HP ( s ) = = g g g gg Vi s2 + 3 1 s + 4 1 2 g 5 C1C 2 g 5 C1 −
Vo2 C2
3.3 OTA-C Simulation of Inductors
The cascade method is simple in design and tuning. However it is very sensitive to component variations. Passive lossless LC filters have very low sensitivity. For high performance, OTA-C filters are thus normally designed by simulating passive LC filters.
One way to design OTA-C filters based on passive LC filters is to substitute inductors in LC filters by OTA-C counterparts.
A grounded inductor requires two OTAs and one capacitor to simulate: g1 g2
C
The equivalent inductance of the grounded OTA-C inductor can be derived as: L=C/g1g2. A floating inductor needs three OTAs and one capacitor.
The equivalent inductance of the floating OTA-C inductor is given by: L=C/g1g2. The inductor substitution technique leads to an OTA-C realization that has the same topology as the original passive LC network. The difference is that each inductor is replaced by a circuit using OTAs and the capacitor.
4. Effects and Reduction of OTA Nonidealities
Practical OTAs have finite input and output impedances. At very high frequencies, the OTA transconductance is frequency-dependent. These nonideal characteristics will degrade frequency performances of OTA-C filters.
A nonideal small-signal CMOS OTA macro-model is shown below. Io Vi+ Ci Vi-
+ Vi
-
G(s)Vi
Co
Go
Nonideal parameters of the CMOS OTA includes the input capacitance Ci, output capacitance Co, output conductance Go, and frequency dependent transconductance
G(s) =
g ≅ ge − jϕe 1 + s / ωb
where ωb is the bandwidth of the OTA, ϕe=ω/ωb is the excess phase and g is the DC transconductance.
The input conductance of the CMOS OTA is normally very small and can be ignored.
OTA input and output capacitances will increase the total capacitances or cause parasitic
poles, both reducing the operation frequency.
Parasitic capacitances can be absorbed into the grounded circuit capacitances. The absorption approach determines the real component values by subtracting the parasitic induced increments from the nominal values.
The finite output resistance will cause phase shift in integrators and reduce quality factor.
For TT OTA-C filter, we can derive
Q' =
Q 1 + 2(1 / gR0 − ϕ e )Q
The finite OTA output resistance Ro or finite OTA DC voltage gain gRo will reduce the filter quality factor and filter gain. In high-Q applications this effect may need to be eliminated by using negative resistance.
The OTA frequency dependent transconductance causes instability at higher
frequencies, limiting the operation frequency.
Transconductance frequency dependence or excess phase has the Q enhancement effect. We can derive the changed Q' of TT OTA-C filter as Q' =
Q 1 + 2(1 / gR0 − ϕ e )Q
Clearly, the excess phase increases the quality factor.
This effect may be compensated for by connecting a resistor in series with the capacitor. In IC design, this resistor is normally realized using a MOSFET in the triode or ohmic region. The equivalent resistance value can be adjusted by voltage VB. Vi
Vo
g C
VB R
We can derive
H ( s) =
g 1 + sRC sC 1 + s /ω b
It is clear that setting R=1/ωbC the circuit will become an ideal integrator. The effect of finite bandwidth is thus compensated for.
5. Balanced/Fully Differential Filter Structure
Balanced structures are widely utilized in continuous-time integrated filter design. This is because balanced structures can increase the common-mode rejection ratio, eliminate the even-order harmonic distortion components and reduce the effects of power supply noise. In mixed-signal or system on chip design, balanced structures are especially important for reducing interference/noise from digital circuits on the same chip.
Balanced configurations can be obtained from single ended structures. The single ended to balanced conversion can be generally achieved by first mirroring the whole single ended circuit at ground (duplicating all the components and changing the terminal polarities of all mirrored active elements) and then combining each original amplifier and its mirrored counterpart into a balanced differential input-differential output device with invertingnoninverting gains.
As an example, the balanced structure of TT bandpass OTA-C filter is shown below. Note that the OTA with a pair of fully differential inputs and outputs is used. Note also that inverting is achieved by a simple cross connection.
6. Automatic Filter Tuning
In practical fully integrated high frequency OTA-C filter design, automatic tuning circuitry is usually included on the same chip to overcome/reduce effects of temperature, parasitic, mismatch, process parameter variation, and environmental change.
The outline of a typical tuning system is shown below. A well defined reference signal is applied to the filter input. One or more parameters of the filter output signal are measured by the frequency tuning control circuit and compared to a reference. The resultant error signal is used by the control circuit to calculate a correction signal, which is then applied to the frequency tuning input of the filter. Thus the system forms a closed feedback loop in which the filter is forced to converge on the desired frequency response. In a similar way, if implemented, the Q control circuit generates a tuning signal that corrects the Q of the filter.
