A Study on Linear Single-Loop Feedback SystemsUsing Geometric Vectors

Qinfeng Zhang, Fengyi HuangSchool of Information Science and Computer Engineering, Southeast University

Nanjing, ChinaEmail: {zqffdr,fyhuang}@seu.edu.cn

Abstract—An analysis method based on vector triangle andvector locus is introduced in this paper. By using vector triangles,a set of quantities generally used in feedback theory can be intu-itively visualized, which is impossible by using other theories. Thevector locus records the evolvement of the vector triangle versusfrequency in polar coordinates, and combines the magnitude andphase plot together in one track. Examples are given to show thatthe proposed method gives intuitive insights while still retains ageneral method to characterize feedback systems.

Index Terms—Feedback control theory, linear network theory,vector, positive feedback, circuit analysis

I. INTRODUCTION

A solid understanding of feedback theory is crucial forsuccessful circuit design. Generally there are two main ap-proaches to feedback theory: elementary [1] and general [2].The elementary theory divides circuits into two parts: theopen-loop transfer function Ho(s) and the feedback coefficientβ(s). Equations comprised of these quantities are used tocharacterize the underlying feedback system. In the case whenthis kind of decomposition is obscure or impossible, thegeneral theory, based on signal flow graphs (SFG) [2] andits variations, has to be used. Compared with the elementarytheory, the general one, which analyzes the network as awhole, has found wide application at the cost of losing intuitiverepresentations.

There have been lots of publications [2]–[4] devoted to thistopic, yet all of them use complex symbolic calculations, suchas transfer functions, to characterize feedback systems. Theseanalytic techniques inevitably turn the physical phenomenainto abstract mathematics, ignoring the fact that an alternativemethod coupled with one’s sensory experience will be easier tobe understood. Davison [5] indeed gives a graphic procedureto analyze stability issues using describing function technique,but no information regarding other important quantities aregiven. Inspired by [6], we introduce geometric vectors to char-acterize feedback systems. As expected, this visual analysismethod provides geometric intuitions to a set of quantities offeedback theory, such as closed-loop gain, open-loop phaseshift and closed-loop phase shift, and gives insight into whatphysically happens in feedback systems.

The paper is organized as follows. Section II introduces theconcept of vector triangle and illustrates some features that canbe obtained from it. Then vector locus, which gives completegain and phase shift plot of feedback systems, is derived in

(a) (b)

Fig. 1. Block diagram of (a) negative feedback systems, and (b) positivefeedback systems.

Section III. Examples for the vector locus are also given inthis section to further illustrate it. Section IV concludes thepaper.

II. VECTOR TRIANGLE

The vector triangle is based on the theory of phasor diagram.Variables in the frequency domain at a specific frequency canbe represented by vectors: the length of a vector is a measureof the magnitude of the corresponding variable, while theangle between a vector and a reference vector represents theamount of phase shift a corresponding variable leads or lagswith respect to a reference variable. Based on these notations,the concept of vector triangle can be derived for both negativeand positive feedback systems.

A. Negative Feedback

Negative feedback, invented by H. S. Black [1], has ex-traordinary linearity within closed-loop bandwidth and isinsensitive to open-loop gain variations. The price paid forthis, however, is its reduced gain. Fig. 1(a) shows the blockdiagram of negative feedback systems. It has an open-looptransfer function Ho(s) and a feedback coefficient β(s). Itsinput, feedback and output quantities are represented as xin(s), xfb(s) and yout(s) respectively.

For negative feedback systems shown in Fig. 1(a), it isstraightforward to obtain these two equations:{

yout(s) = (xin(s)− xfb(s)) ·Ho(s),

xfb(s) = β(s) · yout(s).(1)

Hence xfb(s) can be expressed as

xfb(s) = LG(s) · (xin(s)− xfb(s)) , (2)

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where LG(s) = β(s) ·Ho(s) is the “loop gain” of feedbacksystems. As a result, the following equation, by replacingxfb(s) with (2), is derived:

xin(s)− xfb(s) = xin(s)− LG(s) · (xin(s)− xfb(s)) . (3)

The above frequency domain equation indicates a vectortriangle formed by three vectors: xin − xfb, xin and xfb =LG · (xin − xfb), as shown in Fig. 2(a). The edge lengths ofthe vectors in Fig. 2(a) represent the magnitude of the cor-responding quantities while the directed angles between twovectors reflect their relative phase shift. For example, we canhave |xin| = |xin(s)| and ∠(xfb/xin) = ∠(xfb(s)/xin(s))for any valid feedback systems.

Despite its simplicity, a quite amount of information can beobtained, as elaborated below.

1) Closed-loop gain: The closed-loop gain can be repre-sented as

|Hc| =∣∣∣∣youtxin

∣∣∣∣ = 1

|β|·∣∣∣∣xfb

xin

∣∣∣∣ , (4)

where Hc is the vector representing closed-loop transferfunction Hc(s).

If |LG| ≫ 1, the length of xfb, which is equal to |LG·(xin−xfb)|, would be much longer than that of xin − xfb, givingrise to the vector triangle shown in Fig. 2(b). The triangleshows that |xfb| ≈ |xin| under this condition. Therefore, theclosed-loop gain |Hc| given by (4) approximates to 1

|β| . Onthe other hand, if |LG| ≪ 1, the length of xfb would bevery short compared with the length of xin − xfb. The vectortriangle now becomes what is drawn in Fig. 2(c) and showsthat |xin − xfb| ≈ |xin|. Consequently, by combining (2) and(4), we have |Hc| = |β ·Ho · (xin − xfb)/(β · xin)| ≈ |Ho|.

2) Open-loop and closed-loop phase shifts: Two directedangles in the vector triangle can be related to open-loop andclosed-loop phase shifts respectively as

∠ xfb

xin − xfb= ∠β ·Ho · (xin − xfb)

xin − xfb

= ∠ϕo + ∠β = ∠LG, (5)

∠xfb

xin= ∠β · yout

xin= ϕc + ∠β, (6)

where ϕo = ∠Ho is the open-loop phase shift, ϕc = ∠Hc =∠yout

xinis the closed-loop phase shift, ∠β is the phase shift

introduced by β, and ∠LG = ∠ϕo + ∠β is the loop phaseshift.

The relationships indicated by the above two equations arealso depicted in Fig. 2(a). If β does not introduce any phaseshift, that is, ∠β = 0 at all frequencies (e.g., by using resistivefeedback or capacitive feedback network only), (5) and (6) canbe simplified to

∠ xfb

xin − xfb= ϕo, (7)

∠xfb

xin= ϕc. (8)

(a)

(b) (c)

(d) (e)

Fig. 2. (a) Illustration of the vector triangle of negative feedback systems.(b)-(e) Vector triangles of negative feedback systems at different stages.

3) Closed-loop bandwidth: In general, the closed-loopbandwidth is defined as the frequency range [0, ωcb], whereωcb is the frequency corresponding to unity loop gain, thatis, |LGcb| = |LG(jωcb)| = 1. Recognizing that |xfbcb | =|xincb

− xfbcb | when |LGcb| = 1, we find that the vectortriangles at this particular frequency become isosceles, asshown in Fig. 2(d).

4) Phase margin: The phase margin (PM) can be definedas

PM = 180◦ + ∠LGcb. (9)

The above equation indicates that the internal angle formed byxfbcb and xincb

− xfbcb just equals PM when ∠LGcb < 0, asillustrated in Fig. 2(d). In addition, for the isosceles trianglesin Fig. 2(d), closed-loop gain and closed-loop phase shiftexpressed by (4) and (6) become

|Hccb | =1

2 |βcb| · sin(PM2 )

, (10)

ϕccb =PM

2− 90◦ − ∠βcb. (11)

For example, with PM = 90◦ and ∠βcb = 0, the vectortriangle becomes a right angled isosceles triangle, and theclosed-loop gain and closed-loop phase shift at ωcb given by(10) and (11) are calculated to be 1√

2·|βcb|(-3dB lower than

1|βcb| ) and −45◦ respectively . In addition, with PM = 60◦

and ∠βcb = 0, the triangle becomes equilateral and the twoquantities become 1

|βcb| and −60◦ respectively.Both Fig. 2(d) and (10) show that |Hccb | increases with

decreasing PM and that when PM reaches zero, |Hccb | ap-proaches infinity.

5) Stability: As is known to all, negative feedback systemswould oscillate when |LG(jωstb)| = 1 and ∠LG(jωstb) =−180◦. The vector triangle with ∠LGstb = −180◦ is shown

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in Fig. 2(e). Accordingly, following equation can be derived:

|xinstb− xfbstb | = LGstb · |xinstb

− xfbstb |+ |xinstb| . (12)

If |LGstb| = 1, the above equation gives |xinstb| = 0, which

means the negative feedback system would oscillate at thisfrequency without the need for input.

B. Positive Feedback

Though the concept and application of positive feedbackamplifiers appear earlier than that of negative feedback ampli-fiers [7], their disadvantages of reduced closed-loop bandwidthand high sensitivity of closed-loop gain to open-loop gainhave constrained their use. However, in some cases positivefeedback is still one of the choices to enhance gain (e.g., indeep-submicron process, fT is large but intrinsic gain is small).

The block diagram of positive feedback systems is shown inFig. 1(b). Contrary to negative feedback systems, the feedbackquantity xfb in positive feedback systems is added to xin.Hence, by replacing −xfb in (2) with +xfb, we get the vectortriangle for positive feedback systems as

xin + xfb = xin + LG · (xin + xfb) , (13)

which is shown in Fig. 3(a).An inquiry into Fig. 3(a) gives the following information:1) Closed-loop gain: The closed-loop gain equation for

positive feedback systems is identical with (4).When ϕo0 = ϕc0 = ∠βo = 0 (e.g., at DC or resonant

frequency), we have the vector triangle shown in Fig. 3(b). Itfollows that |xin0 + xfb0 | = |xin0 |+ |xfb0 |. Therefore |xfb0 |can be expressed as

|xfb0 | = |LG0| · (|xin0 |+ |xfb0 |) . (14)

Combining (4) and (14), the closed-loop gain can be derivedas

|Hc0 | =|Ho0 |

1− |LG0|. (15)

For positive feedback systems to be stable, |LG0| should besmaller than 1, otherwise (14) and (15) have no meaningfulsolutions and vector triangles can not be formed. But it doesnot mean that positive feedback systems with |LG0| ≥ 1are useless. For example, they can be found in latches andoscillators for the purpose of fast-settling and oscillation,respectively.

2) Open-loop and closed-loop phase shifts : Two directedangles in Fig. 3(a) can be expressed as

∠ xfb

xin + xfb= ∠β ·Ho · (xin + xfb)

xin + xfb= ∠LG, (16)

∠xfb

xin= ∠β · yout

xin= ϕc + ∠β. (17)

The relationships indicated by the above two equations areshown in Fig. 3(a).

(a)

(b)

Fig. 3. (a) Illustration of the vector triangle of positive feedback systems. (b)Vector triangle of positive feedback systems when ϕo0 = ϕc0 = ∠βo = 0.

III. VECTOR LOCUS

If we put the vector triangle in polar coordinates and let xin

be the base of the triangle with coordinate (1, 0), the endpointof vector xfb will move in the polar coordinates as frequencyvaries from zero to infinity. The locus of the point is the socalled vector locus.

To plot the locus, expressions for xfb (ρ, ϕ) in polar co-ordinates need to be derived. We will show in the followingdiscussion that the expressions for ρ and ϕ can be representedby |LG| and ∠LG. In other words, once Ho and β are known,the vector locus will be determined and can be plotted.

Note that after setting xin to be (1, 0), the expression forclosed-loop gain given by (4) can be modified to

|Hc| =1

|β|· |xfb| . (18)

A. Negative Feedback

For negative feedback systems, Fig. 2(a) indicates ρ = |xfb|and ϕ = ϕc + ∠β, therefore, |xin − xfb| can be representedas

|xin − xfb| =|xfb||LG|

=ρ

|LG|. (19)

By applying the law of cosines to ∠LG and ϕ respectively,we derive the following two equations:{

1 = ρ2

|LG|2 + ρ2 + 2ρ2

|LG| · cos (∠LG) ,

1 = ρ2

|LG|2 − ρ2 + 2ρ cos (ϕ) .(20)

After solving the above two equations, we get the expressionsfor ρ and ϕ as

ρ =|LG|√

1 + |LG|2 + 2 |LG| cos (∠LG), (21)

ϕ = − arccos

|LG|+ cos (∠LG)√1 + |LG|2 + 2 |LG| cos (LG)

− 2πn, n ∈ integers. (22)

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Fig. 4. Vector loci of a negative feedback system.

2πn in (22) is needed since arccos() is only in the range of[−π, π].

As an example, several vector loci are plotted in Fig. 4. Theyhave the same open-loop transfer function, but with various βvalues. For simplicity, all of them are selected to be constantreal numbers. By (21), the start point of the locus can becalculated as

ρ0 =|LG0|

1 + |LG0|. (23)

So, these loci start from ( |LG0|1+|LG0| , 0) at DC, rotate counter-

clockwise as ω increases and end at (0, 0) when ω approachesinfinity.

The blue concentric semicircles drawn in Fig. 4 are servedas a measure of deviation from 1

|β| . To understand this, notethat points on a semicircle of radius r have |xfb| = r, hencethe closed-loop gain given by (18) becomes

|Hc| =1

|β|· r. (24)

Generally, in negative feedback systems, it is desirable thatthe closed-loop gain retains the value of 1

|β| within the closed-loop bandwidth. From Fig. 4, it can be obtained that in thisexample a PM around 60◦ best fulfills the requirement, sincewith a PM of 60◦ the locus is very close to the semicircle ofradius 1 over the entire closed-loop bandwidth.

B. Positive Feedback

Use the same method for negative feedback systems, theequations of ρ and ϕ for positive feedback systems can becalculated as

ρ = 1 +|LG|√

1 + |LG|2 − 2 |LG| cos (∠LG), (25)

ϕ = − arccos

cos (∠LG)− |LG|√1 + |LG|2 − 2 |LG| cos (∠LG)

+ 2πn, n ∈ integers. (26)

As an example, vector loci for positive feedback systemsare depicted in Fig. 5. Still, for simplicity, the values of β are

Fig. 5. Vector loci of a positive feedback system.

selected to be constant real numbers. By (25), the start pointof the locus can be calculated as

ρ0 = |xin0 |+ |xfb0 | = 1 +|LG0|

1− |LG0|=

1

1− |LG0|. (27)

So, these loci start from ( 11−|LG0| , 0), rotate counterclockwise

as ω increases and end at (1, 0) when ω approaches infinity.The closed-loop bandwidth for positive feedback systems

can be defined as the frequency range [0, ωcb], where ωcb isa specific frequency at which ϕccb = −45◦. As illustrated inFig. 5, the triangles consisting of point (1, 0), end point ofxfbcb and point ( 1

1−|LG0| , 0) can be well approximated by aright angled isosceles triangle. Thus, the closed-loop gain atωcb can be represented as

|Hccb | =1

β· |xfbcb | =

1

β· |xfb0 |√

2=

1√2· |Hc0 | . (28)

That is, the closed-loop gain at ωcb is about -3dB lower thanthe value at DC.

IV. CONCLUSION

An analysis method based on geometric vectors for bothpositive and negative single-loop feedback systems is pro-posed. The vector triangle is derived and defined, which givesa geometric view of a set of quantities generally encounteredin feedback theory. Based on the vector triangle, vector locusis also derived and illustrated. We have shown that such amethod gives geometric intuition while still offers a generalway of characterizing feedback systems.

REFERENCES

[1] H. S. Black, “Stabilized feed-back amplifiers,” American Institute ofElectrical Engineers, Transactions of the, vol. 53, no. 1, pp. 114–120,1934.

[2] S. J. Mason, “Feedback theory-some properties of signal flow graphs,”Proceedings of the IRE, vol. 41, no. 9, pp. 1144–1156, 1953.

[3] H. W. Bode, Network analysis and feedback amplifier design, ser. BellTelephone Laboratories series. New York,: D. Van Nostrand company,inc., 1945.

[4] W.-K. Chen, Circuit analysis and feedback amplifier theory. BocanRaton, FL: Taylor & Francis, 2006.

[5] E. J. Davison, “Application of the describing function technique in asingle-loop feedback system with two nonlinearities,” Automatic Control,IEEE Transactions on, vol. 13, no. 2, pp. 168–170, 1968.

[6] T. Needham, Visual complex analysis. Oxford New York: ClarendonPress ; Oxford University Press, 1997.

[7] E. H. Armstrong, “Some recent developments in the audion receiver,”Radio Engineers, Proceedings of the Institute of, vol. 3, no. 3, pp. 215–238, 1915.

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