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Basic linear feedback equations

Feedback is everywhere where electronic circuits are. A good basic unterstanding of feedback helps to solve electronic feedback problems, having special requirements as:

  • AC and DC voltage stabilization
  • Operational Amplifiers
  • Active filters
  • DC-stabilisations of RF-amplifiers
  • Automatic gain control, AGC
  • Intermodulation products cancelation , IPC.
  • Phase Look Loops, PLL.
  • Industrial control electronics

1.0 The linear regulator block diagram

Each linear regulator can be simplified to look like the block diagram  of Fig.1. The control block Ks and the feedback block Kr. Ks may be a machine; a power device or a frequency mixer. Kr is the feedback amplifier including phase and offset networks. Fig.1 This is the basic schematic of each regulator.The input Z is the unstable value which has to be regulated. The output X is the stabilized value of Z. Then there is  the feedback path which controls Z via the control value Y. The input value Z can be a voltage, a frequency , the speed of a machine, a.s.o.The control value must not be the same type a the Z value. But can be a voltage, a frequency, a current or a digital byte.

Fig.1 Basic regulator schematic.

Fig. 2 Simple voltage regulator.

Now, we look to the circuit of the simple DC voltage regulator of Fig.2. Transistor 1 is the control block, transistor 2 the feedback amplifier. The Input voltage is Z, the output voltage X. The base current of T1 is the control value Y. If the input voltage does not change we have a stable balanced condition, which is inticated by this big letters, X,Y, and Z. But this values may change if the input Z changes a small amount dz. The other values then will have the deviations dx and dy. This deviations are considered very small. compared to the stable condition values.

Now we divide Ks into two blocks, one for the input and an other for the controlling and get a new but more detailed schematic.Fig3

 Fig.3 Basic regulator schematic to compute feedback formulas.

We get a new block Kz which is indeed the block Ks having at the input the deviation dz of Z. But has an fixed control value Y , the outcome of a certain stable condition. At Ks it is vice versa. We led the input Z be the stable value and the control value have the deviation dy. The feedback regulator gets a new external control input W to control  X via the adjustment dw. To get am more detailed equaition, we look again to the simple regulator of Fig.2 and see in the feedback the resistotors R1 and R2. This are the measurement resistors for the feedback whereas the zenerdiode adjustes the output and is W.  In regulators the measurement often is frequency depentend we get a additional block Km. Km  is part of Vo and must be consitered computing Vo. Yet we can find the main regulator formulas :

2.0 Behavior to input deviations dx/dz

  • dx/dz = Kz / ( 1-KrKsKm).
  • KrKsKm is total loop gain ; Vo =   KrKsKm .             
  • It is obvious, that V must be a negative value to get regulation, we get:
  • dx/ dz = Kz / (1+Vo) ;                          

3.0 Behavior to external control adjustment dx/dw

  • dx/dw = -KrKsKm/(1-KrKsKm)
  • dx/dw = Vo / (1+ |Vo|)
  • If Vo >> 1; dx = dw.
  • Regulation quality depends on Vo .
  • As no regulator is perfect the output X will not exactly follow the external control W:
  • The deviation of X due of dw is dxw, dxw/dw = -1/ (1+|Vo|)

The deviations dx, dz, and dy. have been considered to be very small . Yet we must distinguish between two kinds of deviations. Either small jumps in a certain short time , having the value dz = z ore small sinusoidal voltages having a certain frequency. dz = dz . The time jump will produce another jump x and the frequency will produce the same frequency but with an other amplitude dx

Using s = the feed back formulas becomes :

4.0 Behavior to Frequency input Deviations dz(s)/dx(s)

 

  • dx(s)/dz(s) = Kz(s) / ( 1-Kr(s)Ks(s)Km(s)).    [1]
  • If Kr(s)Ks(s) = Vo(s) >> 1 ; dx(s)/dz(s) = Kz(s) / Vo(s)

Fig.4 Measurement Block Mr

5.0 Behavior to external Control Deviations dx(s)/dw(s)

The Formula then becomes

  • dx(s)/dw(s) = -Vo(s) / (1+ Vo(s))
  • dx(s)/dw(s) = 1 / (1-(1/Vo(s))           [2]
  • If Kr(s)Ks(s)Km = Vow(s) >> 1 , dx(s)/dw(s) = 1;

 This makes it clear, to the loop gain has to be as high as possible.

6.0 Transmission Definition of Regulator Blocks

 The Gain or Ks(s) or Kz(s ) is expressed as transmission gain :

This general Formula is valid for Kz or Ks including filters or V or any electronic circuit which is connected as the definition of Fig.4 shows.. The input of the driving source is zero and the load resistance is infinite.This is true for  low frequency OP-AMP circuitry.

         Fig.4 Definition of F(s) [Gain] for feedback design

As the main regulator equations are expressed as function of the frequency it is obvious, that to work with and design a feedback circuit, means to analyze and compute  equations like  F(s). To ease this work we express F(s) in terms of multiplicands to ease the writing of a computer program and to work graphically in a bode plot.

The multiplicands M1 to M5 are standard expressions and can exist in n numbers.

7.0 The Logarithmic Frequency diagram (Bode plot)

In the Bode plod or an equivalent computer program, the multiplication at  F(s ) changes to addition of gain in dB and addition of phases in degree:

Gain :

 

Phase:    

 

 

              

 The Multiplicands are:

  • V = Gain
  • M1 = First order :       

 Bode plot values of the denominator:  Gain = -20 dB/Dekade . Phase = 0 to -90 degree

 Bode plot values of the numerator :   Gain = 20 dB/Dekade . Phase = 0 to 90 degree

  • M2 = Second order    .

                   Damping :     < 0.8 = Gain overswing  

Bode plot values of the denominator:   Gain = -40 dB/Dekade . Phase = 0 to -180 degree.

Bode plot values of the numerator :      Gain = 40 dB/Dekade . Phase = 0 to 180 degree.

  • M3 = Integrator or Differentiator: .

Bode plot values of the denominator:   Gain = 20 dB/Dekade .Phase = -90 degree, Integrator

Bode plot values of the numerator :    Gain = 20 dB/Dekade .Phase = 90 degree; Differentiator:

  • M4 = Dead time term:

Bode plot values of the denominator:   Gain = 0 ; Phase = 0 to -90 degree. -57,3 degree at omega = 1/Td

Bode plot values of the numerator :      Gain = 0 ; Phase = 0 to 90 degree.  57,3 degree at omega = 1/Td          

  •     M5 =

The  bode diagram of M1 to M3 low pass behavior, shows Fig.5,6:

 

 

 

 

 

Fig.5

 

Fig.5 Bode plod of factors M1, M2, M3 Low pass behavior

Fig.6 Phase of factors M1, M2, M3  Low pass behavior

8.0 The time behaviors of regulator Blocks

The  jump (time  behavior) dz = z can be computed using complex integration of F(s) :

 

Another solution is the use of related standard time tables or a software, referred to a certain F(s)

See: F. Frauenberger , Regelungstechnik Teubner Verlag 1967.

Better is to use a linear circuit analysis like LISA to compute the time behavior of circuits.

In case of known time behavior the frequency behavior can be computed using Laplace transformation: or use Laplace transformation tables or software.

The above equations are the basic equations to design a feedback. Besides the two design goals, regulation quality , and control quality,  feedback stability is the primary goal. As we have seen, in equation [1] and [2] the gain Vo should be very high.(20 to 50 dB is normal). and  must have a negative Phase of -180 degree. As he phase of the gain versus frequency will change an other negative angel of -180 degree may exist at a certain frequency. At 360 degree ,the result will be an unwanted oscillation or only an enhancement of noise and jitter, if the loop still has gain. The following limits are valid to get feedback stability in a linear feedback :

9.0 The Stability rules for simple linear feedback      

The computed frequency behavior of overall open loop gain V must be in the following limits to prevent oscillation:

  • Phase of Vo(s) at the frequency of crossing 0 dB gain :    Ph(0db) = 20 to 60 degree
  • Gain of Vo(s) at the frequency where the angel is -180 degree (total phase than would be 360 degree):     G(-180) = -3 to -20 dB.
  • Gain versus frequency of Vo(s) must cross the 0 dB line very flat.
  • Gain of Vo(s) must be very high at the maximum regulation frequency or speed.

10.0 A mixed CAD method to optimize  feedback loops and regulators

We have seen the feedback equations 1, 2 and F(s) , are mathematical formulas, containing the data’s of the regulator itself.  But parts of the feedback, for example  mixer gain transmission , feedback amplifiers, filters and sensors , must be designed as circuits. Therefore , to design a critical feedback , a mix of a mathematical program and a  LCA (linear circuit analysis) program is necessary. One can use the following ProgCad (Combination of mathematical programs and linear circuit analysis ) to optimize regulators:

  • Write a mathematical program for the regulator formulas using C+ or FORTRAN
  • Let the results be written into a touchstone S21 file, fitting into a linear circuit analysis)
  • Make a circuit file for a S parameter Block containing the computed data’s , which is matched on the input and output to match the S-Block, as F(s) like Fig. 4 . Fig.7

Fig.7 Matched S-Block

 

 

 

  • Add this S- parameter circuit to the regulators analyzing LCA (linear circuit analysis, for instance “ Analyser Pro.... Number 1 Systens” )  circuit file.
  • Change Parameters in the FORTRAN program and see how the results fit to total circuit running LCA and vice versa.

This ProgCad methode allows too, to desing the feedback of regulators in the time domain using time CAD programs for circuits.

11.0 Examples of feedback. Links to it:

>>>Goto to the example of  DC current regulator

>>>Go to the example of  power stage transmission F(s)

>>> Go to an example of regulator open loop

>>>Go to an example of a PLL feedback filter

RETURN

 

 

Computing the transfer of regulator blocks

Beside lumbed element active filters, which are prescribed in in many filter catalogue’s,( use the internet keywords: “active RC-Filters ” ), a lot of R-C-L filters and circuits are needed at analog electronics.The transfer of such circuits is easy to calculate,  using voltage divider technique with resistors. Example 1 shows how to do it:

The transfer of the circuit of Fig.1 will be computed.We use resistors R and complex resistors              L(s) and 1/ C(s), and divide the voltages:

 Vout / Vin = F (s) =

Now we must simplify this equation into a bode plot formula to get a rough idea what the roll off will be :

Fig.1 Low pass network

. Yet we see, we have 2 corner frequencies and two slopes, -20dB and +20dB per decade and +90 and -90 degree of phase. The precise values can be found using a CAD-program like No.1 Systems “Analyser”.

An similar easy way of computing circuits connected to OP-Amps is used in example 2: If we have an input and a feedback network around an OP-Amp, only the the shorted circuit of the input and feedback resistors must be calculated to find the voltage transfer.  Fig.2 shows the way of looking into the networks using the red dotted lines.

.   The Voltage transfer becomes:   

This is valid if the corner frequncy of the OP-Amp is higher then about 10 times the working frequency.

  

 Fig.2 Resistive Networks around an OP-Amp

 

RETURN

 

Regulator Feedback Filters

As electronic regulators mostly are instable, Phase correction networks must be used to reduce negative loop phase to gain stability. Often a very small circuit may do th job, but some regulators as noise free Phase Look Loops, need a complicated elctronic filter network to be stable and reduce noise. Here are several simple and complicated loop fiters, I have used for Yeahrs in many feedback circuits to optimize stability and time behavior. Some of this circiuts are well known, others are not. The first line is the circuittype number, the second the circiut and the thirt the transfer formula F(s) = Vout(s)/ Vin(s) to be used in a bode plot to optimize stability.The definition of a regulator block transfer is explained as . The Formulas itself are defined using only first order or second order multiplicants:

 

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#1 Regulator Block : Lowpass-Circuit with load.

This Circuit may be used, to stabilze a regulator and decrease regulaters speed.

 

Fig.1 Integrating Circuit with load

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#2 Regulator Block : Low Pass-Circuit and OP-AMP.

This Circuit may be used, to stabilze a regulator and decrease regulaters speed.

 

 

 Fig.2 Integrating Circuit and OP-AMP.

 

#3 Regulator Block : First Order High Pass-Circuit

This this differential circuit may be used, to optimize regulator speed and time behavior

Fig.1 Diffrentiating Circuit Equations

 

Fig.3 Diffrentiating Circuit

 

#4 Regulator Block : Second Order High pass-Circuit

This this differential circuit may be used, to optimize a regulators speed and time behavior. It also can be used as an RF- linearizing circuit.

 Fig.1 Second Order High pass equation

Fig.2 Second Order High pass2 circuit

 

#5 Regulator Block : Regulator Optimizing-Circuit

This is the classical circuit to optimize regulators time behavior for industrial power regulators. As high pass and low pass behavior of a loop filter can not optimize a regulators time behavior and regulation qualtity, the filter must have a integration behavior, what means one pole at zero F( s ) = 1 / (s) .The circuit of Fig.1 has :

  • Low pass behavior
  • High pass behavior
  • 1 pole at zero ( integrator)

The transfer formula is:

 

 

 

 

#11 Filter Block : Rolloff circuit for digital Modulators

Filters in digital modulators must have an rolloff and a gain overswing.This circuit can realise roll off and overswing.As both cornerfrequencies are equal, it also can be used to produce positive phase in a regulator feedback.

The transfer formula is:

Fig.1 Formula for digital modulator filter

 

 

 

Fig.2 circuit for digital modulator filter

 

 

 

 

 

#12 Phase correction circuit for linerar voltage regulators

As todays power transistors are very fast., the loop gain of linear transistor voltage regulators decreases very fast, more than -40 dB/ Dek. The result is high frequency oscillation. A phase correction Network must be used therefore.To get stabilty, the feedback gain must be decreased, by means of an -20dB/Dek, slope, until stability is reached. Fig 1 shows the classical circuit. It is important, that the supply voltage, blocked for higher frequencies [MHz range]. The voltagetransfer is :

 

 

 Fig.1 Phase correction circuit for linerar voltage regulators