lektronik
abor
Linear Control Loop Theory
Prof. Dr. Martin J. W. Schubert
Electronics Laboratory
Regensburg University of Applied Sciences
Regensburg
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
Abstract. A general control loop model consisting of a forward
network (A) and a feedback network (B) is studied theoretically and
approximated with electronic circuitry such, that first and second
order system models result. The particular circuit models are
compared to the generalized first and second order models, so that
conclusions can be drawn for the circuits amplification, bandwidth
and damping factor.
1 Introduction
Feedback loops dominate our life. This document considers fundamental aspects of linear
control loops from a general aspect, i.e. regardless whether they are time-continuous
(modeled in s) or time-discrete (modeled in z).
The organization of this document is as follows:
Section 2 presents the definition of what is linear in a signal processing sense.
Section 3 evaluates the widely accepted control loop model for a single loop.
Section 4 investigates function inversion and error attenuation obtainable with control loops.
Section 5 extends this derivation for nested loop topologies typical for higher-order systems.
Section 6 is concerned with stability,
section 7 brings the theory to application as required in the laboratories for this course.
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M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
2 Definition of Linear and Time-Invariant (LTI) Systems
2.1 Linearity
2.1.1
The Linearity Axiom
y[ c1⋅x1(t) + c2⋅x2(t) ] = c1⋅y[ x1(t) ] + c2⋅y[x2(t)].
(a)
x1(t)
(2.1)
(b)
c1
x1(t)
H
x2(t)
H
c1
y(t)
y(t)
H
x2(t)
c2
c2
Fig. 2.1.1: (a) linear superposition of two signals, (b) equivalent system.
Linearity for signal processing systems is defined according equation (2.1), illustrated by
Fig. 2.1.
2.1.2
The Proportionality Implication.
Setting c2=0 in equation (2.1) shows: Linearity implies proportionality:
y[ c⋅x(t)] = c⋅y[x(t)]
(2.2)
as illustrated in Fig. 2.1.1-2. Proportionality allows to shift constants over LTI systems and
therefore to combine several constants within the circuit mathematically to a single constant.
(a)
x(t)
(b)
c
H
y(t)
x(t)
H
c
y(t)
Fig. 2.1.2: Proportionality: Systems (a) and (b) are equivalent for linear circuits.
2.1.3
The Zero-Offset Implication.
Setting c=0 in equation (2.2) shows: Proportionality implies zero offset:
y[0] = 0⋅y[x(t)] = 0
(2.3)
Conclusion :
The resistive divider in Fig. 2.1.1-3(a) is linear, because U2 = constant ⋅ U1.
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M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
The circuit with OpAmp in Fig. system in Fig. 2.1.1-3(b) is non-linear, as U2≠0 when U1=0.
(a)
U1
(b)
R1
R2
R1
U1
R2
U2
Uoff
U2
Fig. 2.1.3: (a) resistive divider, (b) circuit using OpAmp with offset voltage Uoff≠0.
Remark: Linearity according to Eq. (2.1) is a signal processing definition. From a
mathematical point of view any system Uout = a⋅Uin + b with any constants a, b is be linear.
2.2 Time-Invariance
A system is time-invariant when its impulse response h(t) is not a function of time:
h(t) = h(t-τ)
Fig. 2.1.4:
Time-variant system
when Uctrl varies with
time
(2.4)
Uin
L
Ck1
Cv
Ck2
Uctrl
Uout
Most systems we use are time-invariant. An example for a time-variant system is shown in
Fig. 2.1.2, where response of Uout to impulses at Uin depends on the control voltage Uctrl,
which varies with time.
2.3 Causality : y(t) = f[x(τ)] with τ≤t
The present state of a system, y(t), is a function of the past and present state of its input, but
not of future inputs.
2.4 Stability : Bounded Input Bounded Output (BIBO)
There exist constant values for M and K, so that from |x(t)|≤M follows |y[x(t)]| ≤ K⋅M.
Question: Is an ideal integrator BIBO stable? (Hint: consider f→0!)
-4-
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
3 General Considerations for LTI Systems with Feedback
(b)
(a)
A
X
X
A
Y
Y
B
W
Y
A
(c)
A1
X
(d)
A2
loop
Y
B
k
Y
X
A1
A2
(f)
E
A1
Y
A2
k
(e)
X
B
A2
Y
X
A1
Y
A2
W
k
k
Fig. 3: Evaluation of the loop equations with network A2 being common to A and B.
In Fig. part (a) the transfer function of the system is
Y = A⋅X + B⋅W
In Fig. part (b) closing the loop delivers W=Y and therefore
Y=
A
X.
1− B
The so-called signal-transfer function STF=Y/X is then
STF =
A
A
B →∞
⎯⎯
⎯→ − .
1− B
B
(3.1)
In Fig. parts (c) and (d) we see that forward and backward network have a common part A2.
According to the linearity axiom (2.1) figure parts (c) and part (d) above are identical. For
high loop amplifications B the STF becomes
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M. Schubert
STF =
Linear Conrol Loop Theory
A
A
AA
A
B →∞
⎯⎯
⎯→ − = − 1 2 = − 1 .
1− B
B
kA2
k
Regensburg Univ. of Appl. Sciences
(3.2)
In Fig. part (e) an error E is introduced into the loop. Its error-suppression capability is
termed noise transfer function (NTF):
NTF =
Y
E
=
X =0
1
B →∞
⎯⎯⎯→ 0
1− B
(3.3)
To derive the NTF from the STF we simply set A=1, as there is no forward network between
the introduced error E and the output Y, as E is added directly before the feedback branch.
The main goal of any control circuit is to obtain a STF ≈ k-1 together with a high noise
suppression NTF → 0. Thereby we use an important observation:
Loop behavior STF ≈ k-1 is not a function of A2 when |k·A2| → ∞. The loop quality
depends on the qulity of its feedback-network k and its input (e.g. offset).
Fig. part (f) deals with the question how forward network A and open-loop network B can be
measured. Open the loop at any point of the feedback network k. Set E=0 (if possible). The
forward network is
A=
Y
X
(3.4)
W =0
and the open-loop network B in Fig. part (f) is obtained from
B=
Y
W
(3.5)
X =0
Remarks:
• B can be measured at any two points in the feedback loop that are connected when the loop
is closed.
• Mostly it is easiest to measure B from B=Y/W as shown in Fig. part (f).
• E=0 is sometimes difficult to realize, e.g. when E is the built-in error of a device.
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M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
4 Objectives Obtainable with Control Loops
4.1 Function Inversion
It is seen from (3.2) that the loop itself has the behavior ~k-1, i.e. the reverse behavior of its
feedback network k. We take advantage from this fact in many situations, where we can
construct k directly but not its reverse function k-1.
Examples:
• If k is the resistive divider k=R1/(R1+R2), then
the loop behaves as 1/k for kA>>1.
• If k is a D/A-converter (DAC), then k-1 is an
A/D-converter (ADC).
• If k is a voltage controlled oscillator (VCO),
then k-1 is a phase-locked loop (PLL).
Av
Uip
Uin
Uout
Uim
R2
R1
Fig. 4.1
4.2 Error Attenuation by NTF
(a)
(b)
xerr
x
A
xk
2
xerr,ges =
aerr
y
x
2
2
2
xerr + kerr + (aerr/A)
A
y
xk
kerr
k
k
Fig. 4.2: Error aerr is attenuated by the NTF while xerr , kerr cannot be attenuated by the loop.
(Time-averaged signals add in ampliude (~x) when they are correlated (i.e. interdependent) and in power (~x2) when they
are uncorrelated (independent from each other). So error sources were assumed to be uncorrelated in Fig. 4.2(b).)
Fig. 4.2(a) illustrates a control loop with three error sources:
(i) xerr at the loop’s input,
(ii) kerr located at the feedback networks output and
(iii) aerr at the output of the feedback network, e.g. nonlinearities of the amplifier.
Input error xerr, e.g. an OpAmp’s offset + other input noise, becomes part of the input signal.
The output error of the feedback loop (kerr), e.g. tolerances in R1, R2 in Fig. 4.1, adds directly
to the input node and cannot be suppressed by the loop, we will get STF ~ 1/(k+kerr).
The output error of the forward network (aerr) can be translated to aerr,in,equiv=aerr/A and then
processed like an input signal: yerr = STF⋅aerr,in,equiv = STF⋅aerr/A = NTF⋅ aerr.
-7-
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
5 Network Topologies
Higher order systems come often in one of two topologies:
• Topology 1: distributed feed-in common network, concentrated feed-out of it,
• Topology 2: distributed feed-out of common network, concentrated feed-in to it.
5.1 Topology 1: Distributed Feed-In to the Common Network
5.1.1
Transfer Functions for Topology 1
X
Fig. 5.1:
(a) Closed loop
with the
common feedforward
network of
type distributed feed-in,
concentrated
X
feed-out.
(b) Breaking the
loop at its
concentrated
point to measure networks
A and B.
ak
a2
F
a1
F
b2
b1
ak
a2
a1
F
bk
b2
a0
F
b1
E
Y
F
bk
F
a0
Y
Y
W
Open the loop of this network at its concentrated point as shown in Fig. 5.1(b). Then
determine the feed forward network A for W=0. Search any path on which a signal can come
from X to Y. For linear systems all the solutions superpose linear, i.e. they add:
A=
Y
X
W =0
= a0 + a1 F + a2 F 2 + ... + a k F k .
(5.1)
Then measure the open-loop network B as
B=
Y
W
X =0
= b1 F + b2 F 2 + ... + bk F k .
(5.2)
Signal and noise transfer functions can be computed from
STF =
a + a F + a2 F 2 + ... + ak F k
Y
A
,
=
= 0 1
X 1− B
1 − b1F − b2 F 2 − ... − bk F k
(5.3)
NTF =
1
1
Y
.
=
=
E 1 − B 1 − b1F − b2 F 2 − ... − bk F k
(5.4)
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M. Schubert
5.1.2
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
Stage Amplifications for Topology 1
A2: Common Forward Network
Fig. 5.1.2:
Stage-amplifi- Xin
cation study.
F
Xk-1
X2
bk
F
b2
X1
Y
F
b1
Particularly for ΔΣ modulators, where function F is an integrator, we have to estimate the
voltage amplitudes or required bit-widths of numbers for the internal quantities Xi of
Fig. 5.1.2, where Xin is the input signal.
In Fig. 5.1.2 we set the feed-in coefficients to ak=1, ai=0 for i=0...k-1. Then consider the
coefficient bk as feedback network and all the rest of the network in the dashed box as
common feed-forward network – corresponding to A2 in Fig. 3. For |bkA2| → ∞ we obtain
according to (3.2) the closed-loop amplification 1/bk, or
Y=
X in
bk
when |bkA2| → ∞ .
Note the precondition that the open-loop amplification >> 1! It is typically fulfilled for
low frequencies when F is an integrator, but not in digital filters where |F|=|z-1|=1.
As we can remove the outer loop and apply the same procedure to the k-1 loop, we can say
Y=
X in
X
X
= ... = 2 = 1
bk
b2
b1
for high open-loop amplification in any loop
(5.5)
Amplification with respect to input signal Xin is then
Xi =
bi
X in
bk
for high open-loop amplification in any loop
Often we find bk=1.
-9-
(5.7)
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
5.2 Topology 2: Distributed Feed-Out of the Common Network
(a)
E
(b)
V
Y
F
W
F
b1
F
Y
b2
bk
(c)
a0
E
a1
a2
F
F
F
F
F
F
ak
Y
U
X
W
b1
V
b2
bk
(d)
a0
X
U
a1
F
a2
F
b1
Fig. 5.2:
ak
F
b2
Y
bk
(a) Topology (AX+E) ⋅ [1/(1-B)]. (b) Topology [1/(1-B)] broken to measure openloop gain B, (c) Topology [1/(1-B)] ⋅ A(X+E) and (d) the typical structure.
- 10 -
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
Fig. 5.2(a) illustrates the System Y=(AX+E) ⋅ [1/(1-B)]. In this form we need the twice
number of blocks F than in (d), but we can distinguish between STF and NTF:
A=
V
proportionality
= a0 + Fa1 + F 2 a2 + ... + F k ak = ⎯⎯
⎯ ⎯⎯→ = a0 + a1 F + a2 F 2 + ... + ak F k .
X
(5.7)
Fig. 5.2 (b) shows how to break the feedback loop to measure open-loop gain B:
B=
Y
W
V = E =0
proportionality
= Fb1 + F 2b2 + ... + F k bk ⎯⎯
⎯⎯⎯
⎯→ = b1F + b2 F 2 + ... + bk F k .
(5.8)
As (5.7) = (5.1) and (5.8) = (5.2), the STF in Fig. part (a) is the same as given by (5.3). To
obtain the NTF we set X=0 → V=0 and compute Y=B⋅W + E. Substituting W=B yields the
same NTF as given by (5.4) for Fig. part (a).
Fig. 5.2(c) illustrates the System [1/(1-B)] ⋅ AX (i.e. coefficients ai after the feedback loop)
with common forward network. In this case the STF is the same as given by (5.3). It is
pointless to introduce an error function E as shown in Fig. part (b) because STF and NTF are
both [1/(1-B)] ⋅ A⋅(X+E). The loop is broken between points W and V to show a possibility
how we can measure loop-gain as B = [V / W ]X = E = 0 delivering the same result as (5.8).
Fig. 5.2(d): The branches realizing blocks F of feed-forward and feed-back network were
combined. No error is introduced, This was pointless as the NTF is equal to the STF.
Consequently, the topology in Fig. 5.2(d) can be used to obtain the same STF like Fig. 5.1 and
is frequently seen in digital filter structures.
- 11 -
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
6 Stability
6.1 The Phase-Margin Criterion
(a)
X
(b)
A
X
A
Y
W
Y
B
W
B'
loop
X
A1
loop
Y
A2
X
A1
k
Fig. 6.1:
A2
Y
k
A=A1A2, B=kA2, (a) with summation point, (b) with difference point: B = -B' .
In many control loops we see a difference point as shown in Fig. 6.1(b) instead of the
summation point in (a). Substituting for the difference point B = -B' delivers
STF =
Y
A
A
=
=
X 1 − B 1 + B'
NTF =
(6.1)
1
1
Y
=
=
E 1 − B 1 + B'
(6.2)
It is obvious from (6.1) and (6.2) that STF→∞ and NTF→∞ when B → 1 Ù B' → -1. The
significance of TF→∞ is that there may be |Y|>0 while |X|=0 (e.g. an oscillator). We define
B = 1 corresponds B = e j ( 2π −ϕ M ) with phase margin φM=0.
(6.3)
j (π −ϕ M )
(6.4)
B' = -1 corresponds B' = e
with phase margin φM=0.
For a summation point we measure the phase margin against -2π or 0 Ù -370° or 0°.
For a difference point we measure the phase margin against -π Ù -180° .
- 12 -
(6.5)
(6.7)
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
6.2 Considering Poles and Zeros
Computing poles (i.e. zeros of the denominator) and zeros (of the numerator) of STF and NTF
is often complicated and requires computer-aided tools. If we have these poles and zeros, we
have deep system insight. For X=0 both STF and NTF become
Y⋅(1-B) = Y⋅(1+B') = 0
(6.7)
Consequently, the output signal must follow the poles that fulfill (6.7):
Time-continuous:
y (t ) = ∑ Ci e
s pi t
(6.8)
i
Time-discrete:
y (n) = ∑ Ci e
s p nT
i
= ∑ Ci z npi
(6.9)
i
With Ci being constants and index pi indicating pole No. i. Fig. 6.2 illustrates the relation
between the location of poles sp=σp+jωp in the Laplace domain s=σ+jω and the respective
transient behavior. For a polynomial with real coefficients complex poles will always come as
σ t
σ t
complex pairs sp=σp±jωp that can be combined to e p cos(ω pt ) or e p sin(ω pt ) , because
ejx+e-jx=2cos(x) and ejx-e-jx=2j⋅sin(x). It is obvious that these functions increase with σp>0 and
decay with negative σp<0.
s T
The conclusion from a time-continuous to time-discrete is obvious from z=esT and e p with
T=1/fs being the sampling interval. Increasing time corresponds to increasing n for z np .
Consequently, stability requires σp<0 or |zp|<1 for all poles sp,i or zp,1, respectively.
(a) Location of poles
sp1
Butterworth
jω0 = j 1/LC
aperiodic
(dead-beat)
limit case
Fig. 6.2:
A(1-espat)
σ
sp2
Butterworth
4.29%
Butterworth
s'p1
45°
A(1-espt)
A
β
spa
s'p2
(b) step response
jω = j ωo2-σ2
A(1-es'p1t)
0
-jω0
(a) locus of poles,
Zeit
(b) transient response using these poles.
- 13 -
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
7 Applications
7.1 Time-Continuous Applications
7.1.1
Linearization of a Time-Continuous Integrator
(a)
(a)
Ua
OAx
UB
Ra
Ux
(c)
U'a
0V
U'a
OAx
Ra
Cx
Rx
Uy
U'x
Uvg
Rb
ax
Cx
Rx
U'y
U'x
U'y
-ωx
s
0V
Rb
bx
Ub
U'b
U'b
Fig. 7.1.1: (a) Real Situation, (b) circuit linearized, (c) equivalent signal-flow model.
First of all we have to linearize the circuit. Assume a 3-input integrator as shown in part (a) of
the figure above. Its behavior is described by
U out − U vg = −
U a − U vg
sRa C x
−
U x − U vg
sR x C x
−
U b − U vg
(7.1)
sRb C x
where the virtual ground voltage Uvg is given by
U vg = U B − U off − U out / AV
(7.2)
with AV being the OpAmp’s amplification. To linearize (7.1) we have to remove the constant
term, i.e. UB-Uoff. The amplifier’s offset voltage, Uoff, is either compensated for or neglected
assuming Uoff≈0V. To remove UB we remember that any voltage in the circuit can be defined
to be reference potential, i.e. 0V, and define
(7.3)
U' = U – UB
Ù
U = U' + UB
(7.4)
This yields the circuit in Fig. part (b) which is linear in the sense of (2.1). Things are
facilitated assuming also AV→∞ so that Uvg=UB so that U'vg=0V. Then (7.1) facilitates to
'
U out
=−
U a'
U x'
U b'
−
−
sRa C x sR x C x sRb C x
(7.5)
It is always possible to factor out coefficients such, that the weight of one input is one, e.g.
R
R
⎡−ω ⎤
1
'
(7.7), a = x (7.8), b = x (7.9)
U out
= aU a' + U x' + bU b' ⋅ ⎢ x ⎥ (7.7) with ω x =
Rx Ck
Ra
Rb
⎣ s ⎦
(
)
- 14 -
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
Application to a Time-Continuous 1st-Order System
7.1.2
(a)
(b)
OA1
0V
X
a1
a0
Uerr1
U'in
C1
R1
E1
Y
F1
U'out
U'out1
b1
Rb1
(c)
U'in
E1
1
−ω1
s
U'out
b1
Fig. 7.1.2: (a) Real Circuit, (b) general model, (c) particular model for circuit in (a).
Fig. 7.1.2 (a) shows the analog circuit under consideration. Fig. part (b) illustrates the very
general form of the topology. To get forward network A set b1=E1=0. Identify all paths a
signal can take from X to Y. Assuming a linear system the sum of all those partial transfer
function delivers the STF. Open-loop network B is found by summing all paths from Y to Y:
A=
Y
X
= a0 + a1F1 ,
B=
(7.10)
E1 =b1 =0
Y
Y
= b1 F1
(7.11)
X =0
Adapting the general topology in part (b) to the particular model in Fig. part (c) we can select
either ak or bk free. Let’s set ak=1. Here, with order k=1, we get
a1 = 1 , a0 = 0 , b1 =
R1
1
ω
, ω1 =
, F1 = − 1
Rb1
R1C1
s
(7.12)
Signal and noise transfer functions can then be computed from
STF =
ω1
A
a +aF
F
(−ω1 / s )
= 0 1 1= 1 =
.
=−
1 − B 1 − b1F1 1 − F1 1 − b1 (−ω1 / s )
s + b1ω1
Hence :
NTF =
STF = −
ω1
s + b1ω1
1
1
1
=>
=
=
1 − B 1 − b1F1 1 − b1 (−ω1 / s )
=−
Rb1
1
1
R
→0
⎯s⎯
⎯→ − = − b1
R1 1 + sRb1C1
b1
R1
(7.13)
s
sRb1C1
→0
=
⎯s⎯
⎯→ 0
s + b1ω1 1 + sRb1C1
(7.14)
NTF =
- 15 -
M. Schubert
7.1.3
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
Application to a Time-Continuous 2nd-Order System
(a)
OP2
0V
OP1
0V
Uerr2
C2
U'out2
R2
U'in
Uerr1
C1
U'out1
R1
Rb2
U'out
Rb1
-1
(b)
(c)
X
X
a2
a1
F2
-b2
a0
F1
1
E1
−ω1
s
−ω2
s
Y
b1
E1
-b2
Y
b1
Fig. 7.1.3: (a) Real Circuit, (b) general model, (c) particular model for circuit in (a).
Fig. 7.1.3 (a) shows the analog circuit under consideration. Fig. part (b) illustrates the very
general form of the topology. To get forward network A set b1=b2=E1=0. Identify all paths a
signal can take from X to Y. Assuming a linear system the sum of all those partial transfer
function delivers the STF. Open-loop network B is found by summing all paths from Y to Y:
A=
Y
X
= a0 + a1F1 + a2 F1F2 ,
(7.15)
E1 =b1 =b2 =0
B=
Y
Y
= b1F1 − b2 F1 F2
(7.17)
X =0
Adapting the general topology in part (b) to the particular model in Fig. part (c) we can select
either ak or bk free. Let’s set ak=1. Here, with order k=2, we get
a2 = 1 , a1 = a0 = 0 , b2 =
1
1
ω
ω
R2
R
, b1 = 1 , ω2 =
, ω1 =
, F2 = − 2 , F1 = − 1 (7.17)
R2C2
Rb 2
Rb1
s
R1C1
s
The signal transfer function can then be computed from
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M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
− ω1 − ω2
A
a0 + a1F1 + a2 F2 F1
F2 F1
s
s
=>
=
=
=
STF =
1 − B 1 − b1 F1 − (−b2 F2 F1 ) 1 − b1 F1 + b2 F2 F1 1 − b − ω1 + b − ω1 − ω2
1
2
s
s
s
STF =
ω1ω2
s + b1ω1s + b2ω1ω2
(7.18)
2
We compare this result to the general 2nd-order model
H 2 ndOrder =
A0ω02
s 2 + 2 Dω0 s + ω02
(7.19)
with DC amplification A0, cutoff frequency ω0 and damping constant D. (7.18)=(7.19)
delivers
STF =
A0ω02
ω1ω2
=
= H 2 ndOrder
s 2 + b1ω1s + b2ω1ω2 s 2 + 2 Dω0 s + ω02
Comparing coefficients of s delivers for DC amplification, cutoff frequency, damping factor
A0 =
1 Rb 2
=
b2 R 2
(30)
ω0 = b2ω1ω2 =
1
R1C1Rb 2C2
(31) D =
R1Rb 2
b1ω1
=
2ω0
2 Rb1
C2
C1
(7.20)
The noise transfer function is computed from
NTF =
s2
1
1
1
1
=
=
=
= 2
1 − B 1 − b1 F1 − (−b2 F2 F1 ) 1 − b1F1 + b2 F2 F1 1 − b − ω1 + b − ω1 − ω2 s + b1ω1s + b2ω1ω2
1
2
s
s
s
Hence :
NTF =
s2
s2
→0
=
⎯s⎯
⎯→ 0 2
s 2 + b1ω1s + b2ω1ω2 s 2 + 2 Dω0 s + ω02
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(7.21)
M. Schubert
Linear Conrol Loop Theory
Regensburg Univ. of Appl. Sciences
7.2 Time-Discrete Applications
7.2.1
Time-Discrete Finite Impulse Response (FIR) Filter
xn , X(z)
ak
ak-1
a2
z-1
a1
a0
z-1
z-1
yn
Y(z)
a0
a1
a2
xn
ak
z-1
z-1
z-1
yn
Y(z)
X(z)
Fig. 7.2.1: FIR filters in 1st (top) and second (bottom) canonical direct structure.
Fig. 7.2.1 shows some time-discrete FIR-filters. FIR filters have no recursion, i.e. B=0.
Therefore, there impulse response is finite and always stable. The symbol z-1 stands for a
signal delay by one sampling-clock cycle. Can you identify the different topologies
introduced in chapter 5?
In a direct structure the impulse response taps can be directly adjusted by the coefficients. The
topologies are canonic because the number of delay elements is equal to the filter order.
7.2.2
Time-Discrete Infinite Impulse Response (FIR) Filter
Fig. 7.2.2 shows some time-discrete IIR-filters. IIR filters have recursion, i.e. B≠0. Therefore,
there impulse response is theoretically infinite. IIR filter design requires uppermost care to
obtain stability. Can you identify the different topologies introduced in chapter 5?
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M. Schubert
Linear Conrol Loop Theory
xn , X(z)
dk
Regensburg Univ. of Appl. Sciences
d2
d1
z-1
d0
z-1
-bk
z-1
-b2
-b1
yn,Y(z)
yn
a0
a1
Y(z)
a2
xn,
X(z)
z-1
z-1
z-1
ak
z-1
z-1
-b1
-b2
-bk
a0
a1
a2
xn
z-1
z-1
ak
X(z)
yn
Y(z)
-b1
-b2
-bk
Fig. 7.2.2: IIR filters in 1st (top) and second (bottom) canonical direct structure. The structure
in the middle is not canonic because the number of delay elements is larger than the filter
order.
8 References
[1]
[2]
[3]
M. Schubert, Laboratory "Analog Systems of 1. and 2. Order", available: http://homepages.fhregensburg.de/~scm39115/homepage/education/courses/psk/psk.htm.
M.
Schubert,
Laboratory
"Delta-Sigma
Modulation",
available:
http://homepages.fhregensburg.de/~scm39115/homepage/education/courses/psk/psk.htm.
Regensburg University of Applied Sciences, available: http://www.hs-regensburg.de/.
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