P. Bruschi: Notes on Mixed Signal Design
App. 3.4
Application of the cut-insertion theorem (decomposition theorem) to the design of
closed loop, op-amp based networks.
1.1 Ideal block diagrams of feedback systems.
Figure 1 shows a typical block diagram used in control theory to model feedback systems. All blocks
are ideal and unidirectional. By simple calculations:
α βA
VOUT
=−
β βA − 1
VS
(1)
When |βA| tends to infinity, Vout/Vs tends to -α/β. In order to obtain a precise overall transfer function,
transfer functions α and β must be precise, i.e. they should be designed to have as small as possible
temperature coefficients, reduced dependence on pressure variations and high time stability. On the
other hand, block A (amplifier) should exhibit high gain (to guarantee an high |βA| value, since
generally β ≤ 1 ), and provide the required power to drive the load.
Feedback
Network
β
VE
α
VS
A
VOUT
Fig.1: Typical block diagram used in control theory
Equation (1) can be re-written as:
[
VOUT
α
1 
α
−1
= − 1 +
 ≅ − 1 + (β A)
VS
β  βA − 1 
β
]
(2)
where the rightmost approximation holds when |βA|-1<<1, that is |βA|>>1. Eq.2 gives an estimate of the
relative error with respect to the ideal transfer function (-α/β):
ε AL ≡
∆AL
−1
≅ βA
AL
(3)
For example, if we need to design a closed loop transfer function with AL=1 (unity gain amplifier or
buffer), we simply can make α=1, β=−1. Than if we require an accuracy better than 1%, we simply
have to guarantee that A>100. Similarly, it can be shown that to get the same accuracy (1%) but with
an overall gain of 100 (e.g. α=1, β=−0.01), we need a block A with at least a gain of 104. The
1
P. Bruschi: Notes on Mixed Signal Design
App. 3.4
requirement on |βA| deriving from the relative error specification through (3) should clearly hold over
the whole frequency interval of interest.
1.2 Feedback systems in the electrical domain.
When dealing with real electrical networks, the analysis is more complicated, since signals are
embodied by voltage and/or currents that should satisfy Kirchhoff laws: blocks are often bidirectional
and loading effects occur when blocks are cascaded. Figure 2(a) shows the implementation of the
system of Fig.1 using electrical blocks.
Feedback Network
Feedback
Network
vs
I E
ve
O
AMP
βN
vs
vout
vo
αN
(a)
ve
vin
AMP
vout
(b)
Fig.2: (a) Implementation of the block diagram of Fig.1 in the electrical domain; (b) definition of the Feedback network
transfer functions.
In Fig.2(b) amplifier have been detached from the feedback network in order to measure the individual
transfer functions of the blocks. Let us indicate with αN and βN the transfer functions from the input
and output ports (I and O, respectively) to the error port (E) of the feedback network and with AOL the
amplifier gain. These transfer functions are defined by:
v 
α N ≡  e  ;
 v s  vo = 0
v 
β N ≡  e  ;
 vo vs = 0
v
AOL =  out
 vin



(4)
It would be desirable to be able to model the circuit of Fig.2(a) with the block diagram of Fig.1, where
α=αN, β=βN and A=AOL, so that Eqns. (1-3) would be applicable. Unfortunately, such a simple
schematization is not correct since:
•
•
•
The amplifier gain depends on the loading effect of the feedback network.
The feedback network transfer functions, calculated with the amplifier unconnected as in
Fig.2(b), are different from the actual transfer functions occurring in the closed loop circuit of
Fig.2(a).
Blocks are not unidirectional.
A possible approach is given by the cut-insertion method introduced by B. Pellegrini end described in
refs.[1,2], by which it is possible to obtain the network of Fig.3, where the following network transfer
functions can be defined:
2
P. Bruschi: Notes on Mixed Signal Design
v
α ≡  r
 vs

 v
 ; β ≡  r
v p =0
 vout
v
γ =  out
 vs

 ;
 vs = 0
 vp 


; Zi =  
i 
 v p =0
 p  vs = 0
v
A =  out
v
 p



 vs = 0
App. 3.4
(5)
 ip 
ρ =  
 vs  v p = 0
It can be shown [1] that the network of Fig.3 is equivalent to the original network in Fig.2(a) if the
following conditions hold true:
vp =
α
vs ;
1 − βA
1
1 ρ
= + (1 − β A)
Z p Zi α
(6)
Feedback
Network
Ip
vs
Z p vr
vp
AMP
vout
Fig.3: Application of the cut-insertion method to the network of Fig.2.
In these conditions, the overall closed loop transfer function of the network is given by:
vout
α βA
=−
+γ
vs
β βA − 1
(7)
Eqn, (7) is very similar to (2), except for the term γ, which is apparently a feed-forward path for the
signal vs . This approach represents a powerful method for the analysis of the network, since it allows
splitting the overall transfer function into simpler network functions. The study of the system stability
is also much facilitated. Complications might arise from the calculation of Zp, but, since the amplifier
can be approximated as a unidirectional block, ρ=0 and Zp is simply the input impedance of the
amplifier itself.
In order to fully exploit the significant simplification offered by the cut-insertion theorem also for
design purposes, the following conditions would be required:
•
•
the feed-forward term γ should be negligible, since it introduces an error with respect to the
nominal –α/β transfer function and its magnitude is not affected by increase of the βA term.
the network functions α and β should coincide with αN and βN, respectively, calculated as in
Fig.2(b), in order to greatly simplify the design of the feedback network and make it as
independent as possible of the amplifier characteristics.
3
P. Bruschi: Notes on Mixed Signal Design
App. 3.4
Both requirements are met if the following conditions hold:
Z i >> Z e
(8)
Z out << Z o ;
(9)
where Zout and Zi are the output and input impedances of the amplifier, respectively, while Zo and Ze are
the impedances seen across port O and port E, respectively, when the other ports are short-circuited.
While the eq.(8) can be assumed to be verified, at least as a first approximations, (9) is difficult to
fulfill with modern, low voltage operational amplifier, where, in order to maximize the output swing,
common drain output stages are used, with output resistances in the range of several tens of kilo-Ohms.
In this case, γ is often non negligible and α strongly depends on the amplifier output resistance, which
is a parameter that is difficult to be controlled and even to be reliably simulated.
Nevertheless, it is interesting to observe that, if |βA| tends to infinity, the closed loop transfer function
tends to:
v
lim  out
β A →∞ v
 s

α*
 = −
β

(10)
where α* is vr/vs calculated with vp=0 and the output port short-circuited. Note that α* does not depend
on the amplifier output impedance and also coincides with αN in the case that the input impedance of
the amplifier is much larger than port E output impedance, i.e. condition (8) holds.
With this new definition of α, the closed loop transfer function tends to a simple expression just as in
the case of the ideal block diagram of Fig.1. In particular, Eqn.(10) is not affected by the feed forward
term γ.
In order to demonstrate Eqn.(10) let us start by defining the two quantities α* and β*, using the
network of Fig.4, obtained from the network of Fig.3. We keep vp turned off and place an ideal voltage
source vo across the output terminations. This is clearly possible only if the amplifier output impedance
is not zero, which is true in all real cases. Then:
v 
v 
α* ≡  r 
; β* ≡  r 
 v s  v p ,vo = 0
 vo  v p ,vs = 0
(11)
Feedback
Network
vp=0
vs
AMP
vr’=α* vs+β *vo
Fig.4: Network used for calculation of the definition of α* and β*.
4
vo
P. Bruschi: Notes on Mixed Signal Design
App. 3.4
Now, let us consider Fig.5 that shows the network used for determining parameters α and γ, according
to definitions (5). If we place an ideal voltage source of value γvs across the output termination (leaving
vs turned on), the network shown in Fig. 5 is not altered, therefore the value of vr does not change. We
can then consider the network of Fig.5 as a particular case of Fig.4, with vo=γvs. Then:
α vs = vr
v p =0
= vr'
v p = 0 , v 0 = γv s
= α*vs + β* γvs
(12)
Since this expression must hold true for whatever value of vs, then the following relationship can be
found:
α = α * + β* γ
(13)
Feedback
Network
vp=0
vs
vout=γ vs
AMP
vr=α vs
Fig.5: Network used for calculation of α and γ.
Substituting this expression of α into (7) we get:
(
)
(
)
vout
α * + γβ* β A
α* βA
γ
β* − β A
=−
+γ=−
+
+γ
vs
β
βA − 1
β β A − 1 1 − βA
1 − βA
(14)
In order to find a further simplification, it is useful to find the relationship between β and β*.
First, we refer to Fig. 6, where the voltage vr (indicated here with vr’’) is expressed as a function of vp
and vo (with vs=0). Clearly, when only vo is on, we are in the same conditions of Fig.4 with vs=0, then
the transfer function from vo to vr is β*. We introduce a new transfer function between vp and vr with
vs=0 and vo=0 (i.e. output termination shorted):
v 
η≡ r 
v 
 p  v s , vo = 0
(15)
5
P. Bruschi: Notes on Mixed Signal Design
App. 3.4
Feedback
Network
vp
vs=0
vo
AMP
vr’’=β*vo+η vp
Fig.6 Network used for calculation of the dependence of vr on vp and vo for vs=0
Then let us consider Fig.7, showing the configuration used to calculate parameter β. Clearly, vr does
not change if a voltage source of value Avp is placed across the output termination. This corresponds to
setting vo=Avp in Fig. 6, therefore:
β Av p = vr
vs = 0
= vr''
v p = 0 , v0 = Av p
= β* Av p + ηv p
(16)
From which, we find:
η
β* = β −
A
(17)
Feedback
Network
vp
vs=0
AMP
vout=Avp
vr=βvout
Fig.7: Network used for definition of A and β.
Note that, with the particular configuration of the network resulting from the application of the cutinsertion theorem to the original circuit of Fig.2(a), voltage vp may affect vr only trough the output
termination. Since the latter is shorted in (15), η must be zero.
As a result:
β* = β
(18)
With (18), Eqn. 14 becomes:
6
P. Bruschi: Notes on Mixed Signal Design
vout
α* βA
γ
=−
+
vs
β βA − 1 1 − β A
App. 3.4
(19)
It can be easily shown that the block diagram corresponding to (19) is that of Fig.8:
Feedback
Network
vS
β
ve
α∗
A
vout
γ
Fig.8: Block diagram equivalent to Eqn.(19).
If we calculate the limit of (19) for |βA| that tends to infinity, we just obtain Eqn. (10). For |βA|>>1, the
relative error with respect to the ideal transfer function (-α*/β) is given by:
∆AL
γβ
γβ 
−1
−1 
≅ β A 1 + * ≤ β A 1 + *  (20)
AL
α
α 

Using Eq. (20) it is possible to estimate the minimum value of the gain loop |βA| to make the error
smaller than the maximum allowed value specified by the design constraints.
Finally we consider that, if we neglect the input impedance of the amplifier (i.e. we consider Zi->∞),
we can make the following approximations:
ε AL ≡
 α* ≅ α N

*
β = β ≅ β N
(21).
A possible design flow based on the cut-insertion theorem can be the following:
1. design the feedback network in such a way that the target design function is given by –αN/βN;
2. design (or choose) the amplifier in such that, once loaded by the feedback network, its gain is
still large enough to maintain the relative error, given by 20, below the maximum allowed
value;
3. Calculate (or simulate) the actual feedback transfer functions (α*, β*) with the amplifier
impedance connected and check if (21) can be considered acceptable.
References.
[1] B. Pellegrini, “Considerations on the feedback theory,” Alta Frequenza, vol. 41, no. 11, pp. 825–
820, Nov. 1972. http:// brahms.iet.unipi.it/elan/scompos.pdf.
[2] B. Pellegrini, “Improved Feedback Theory”, IEEE Trans. on Circuit and Systems —I: Regular
Papers, vol. 56, no. 9, September 2009.
7