Revisiting the Power-Efficiency Trade-Off on a DC Voltage...

Revisiting the Power-Efficiency Trade-Offon a DC Voltage Source

Arturo Fajardo Jaimes 1,2, Fernando Rangel de Sousa 1

1- Radiofrequency Laboratory| Department of Electrical and Electronics Engineering | UFSC | Florianpolis, Brazil

2- Department of Electronics Engineering | Pontifical Xavierian University (PUJ) | Bogota, Colombia

Table of Content RFLaboratory

IntroductionPower-efficiency trade-offThis work

Efficiency in a voltage source

Power-efficiency trade-offPETO for lossless voltage source (Rss = 0,Rsp =∞PETO for voltage source w/o loss of self consumption(Rss 6= 0,Rsp =∞)PETO for general voltage source (Rss 6= 0,Rsp 6=∞)

Experimental verification

Conclusions

References

2UNIVERSIDADE FEDERALDE SANTA CATARINA

Power-efficiency trade-off I RFLaboratory

Energy conversion processes are characterizedby a trade-off between output power and effi-ciency.

Earliest work about electric-mechanic conversion (1978)Edison and his chief assistants re-searched the design techniques for DCelectric generators, and determinedthat a generator with smaller internalresistance than its load is more effi-cient than a generator with internal re-sistance equal to its load [1].


Power-efficiency trade-off II RFLaboratory

Nanoscale energy conversion Molecular motors (2016)At the nanoscale new possibilities and phe-nomena arise. For example, chemical energycan be converted directly into useful work(molecular motors). This diagram shows thecalculated output power versus efficiency ofa molecular motor (dashed) near equilibriumand (solid) far from equilibrium for (blue) per-fect coupling and (red) weak loss processes[2].


Power-efficiency trade-off III RFLaboratory

PETO and Portable Electronic devices

• Most of the electronic devices must beportable and must be connected to theInternet [3]

• Typically, the device portability is associ-ated to the energy technologies that hasbeen adopted [4].

• The designer of PEDs must face a trade-off between functionality (i.e. dissipatedpower) and portability (i.e. battery run-ning time, weight, size) [5].


About this work RFLaboratory

The problemThe designer of PEDs need to take into account the PETO in orderto achieve the required specifications and, as a result, an optimumoperating point (i.e. optimum load) could be found and then could beused as a constraint in the electronic design of the system.

The Proposed SolutionIn this work an analytic expression is developed for quantifying thePETO on a DC Voltage Source with and without limited energy.

ηsc =1− pL

ξ

1 + ξk·pL


Efficiency in a voltagesource I

RFLaboratory

Modeling the DC sourceA common technique for modeling a real voltage source is theuse of the Norton or Thevenin equivalent circuit as its model.

VthRth

RL

RthRL

Vth/Rth

ModelsRS

RL

Real

DC

Voltage

Source

Rth<<RL

Rth>>RL

Vth

RL

RL

Vth/Rth


Efficiency in a voltagesource II

RFLaboratory

ObservationThe equivalent circuit predicts only the current and voltage per-formance of the source, from the load point of view.

Efficiency values predicted by both models

Without knowledge about thephysical laws that govern the in-ternal source behaviors, we can-not predict accurately its effi-ciency.


PETO for voltage source RFLaboratory

The PETO is discussed for any energy source that can be modeledusing the circuits shown:

With infinite energy

V RssRsp RLvL

iL

vs

is

DC source

With finite energy

V

QnRss

Rsp RLvL

iL

vs

isDC source


Ideal battery model RFLaboratory

In [6] was proposed a battery model based on the circuit shown:

Symbol

V

Qnsv

is

Circuit model

C=Qn

SoE

-isVz=1

Vf=0

0

0 0

V SoE

SoEsv>ì

= í=î

is


PETO Rss = 0,Rsp =∞ RFLaboratory

Efficiency and power expressions

For this circuit the efficiency is aload independent variable, there-fore PETO does not occur.

PLa =V 2

RL; (1)

ηsa = 1. (2)

(a) Load power and efficiency vs load (b) Efficiency vs power


PETO Rss 6= 0,Rsp =∞ I RFLaboratory


PLb= 4Pavs

RLRss(

1 + RLRss

)2; (3)

ηsb =ηsa

1 + RssRL

; (4)

Pavs =V 2

4 · Rss; (5)

pL =PLb

Pavs

=4 · RL

Rss(1 + RL

Rss

)2. (6)



PETO Rss 6= 0,Rsp =∞ II RFLaboratory

Quantifying the PETO

• The better compromise be-tween efficiency and poweroccurs in the load rangewhere the PETO exists.

• The limits are: 100% of ηand 100% of pL.

• We cannot achieve theselimits simultaneously. (pL =0, η = 1) or (pL = 1, η = 0.5)

• A good compromise of thisPETO is pL = 0.75, η =0.75.

In all the operating points of thisPETO, the relationship betweenthe pL and the η is described by:

ηsb = 1− pLξ

;

where, ξ=2(1 +√

1− pL), and

RL ≥ Rss.


PETO Rss 6= 0,Rsp 6=∞ I RFLaboratory


PLc = PLb= 4Pavs

RLRss(

1 + RLRss

)2; (7)

ηsc =ηsb(

RssRsp

(RLRss + 1

)+ 1

) ; (8)

Pavs =V 2

4 · Rss; (9)

pL =PLb

Pavs

=4 · RL

Rss(1 + RL

Rss

)2. (10)



PETO Rss 6= 0,Rsp 6=∞ II RFLaboratory

Quantifying the PETO

• The PETO range is limited bythe maximum η and minimumpL point and the minimum ηand maximum pL point.

• Defining the source quality fac-tor as k = Rsp

Rss :

maxηsc

=

k√k + 1(

1 + k +√k + 1

) (k +√

k + 1)

max pL = 1

• The extreme Load val-ues are:

RL = RLp = Rss

RL = RLη = Rss√k + 1

• RLp is the load valuethat maximizes pL. RLηis the load value thatmaximizes η


PETO Rss 6= 0,Rsp 6=∞ III RFLaboratory

Quantifying the PETOWhen RL is in the PETO range (RLp ≤ RL ≤ RLη), the load powercan increase only if the efficiency decrease and vice versa. In all theoperating points of this PETO, the relationship between the pL and theη is described by:

ηsc =1− pL

ξ

1 + ξk ·pL

where, Rss√k + 1 ≥ RL ≥ Rss and ξ=2

(1 +√

1− pL). Having a high

k value means that the power supply is good, i.e. both the self con-sumption and the internal conduction losses are low.


Experimental verification RFLaboratory

Experimental Setup

RL

Real DC

voltage

source

vL

iL

• The implemented system is a demon-strator of the validity of the analytic func-tion rather than targeting a specific ap-plication.

• The real DC source (k = 90) was emu-lated using a power supply (1 V < V <10 V) and two discrete resistors (Rsp =100 kΩ, Rss = 1.1 kΩ), the electric loadwas an adjustable resistor (100mΩ <RL < 10 MΩ).

• Using this setup, both the resistive loadand the input voltage were swept.


Experimental andtheoretic results I

RFLaboratory

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(a) Efficiency vs load power


Experimental andtheoretic results II

RFLaboratory

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(b) Efficiency vs normalized load power


Conclusions RFLaboratory

• In this paper an analytic expression was developed and used forquantifying the power-efficiency trade-off involved in DC voltagesource that supplies energy to an electric load.

• The expression was validated by an experimental demonstrator,with good agreement between the results and the predicted val-ues.

• The analytical work developed would be a useful tool for design-ers and students in order to understand and solve a lot of confu-sion and myths about power supply efficiency and life time of thebattery.


References I RFLaboratory

[1] R. E. Stross, The wizard of menlo park: how thomas alva edison invented the modern world. New York: ThreeRivers Press (CA), 2008.

[2] [Online]. Available: http://www.physics.otago.ac.nz/research/epg/research

[3] Z. Pang, “Technologies and architectures of the internet-of-things (iot) for health and well-being,” Dept. Elect.Eng, Ph.D. thesis, Royal Institute of Technology Univ., Stockholm, Sweden. 2013.

[4] R. W. Erickson and D. Maksimovic, Fundamentals of power electronics. New York: Springer, 2001.

[5] S. D’Ambrosio, S. De Pasquale, G. Iannone, D. Malandrino, A. Negro, G. Patimo, A. Petta, V. Scarano, L. Serra,and R. Spinelli, “Mobile phone batteries draining: Is green web browsing the solution?” in Proc. 2014 GreenComputing Conf., Nov 2014, pp. 1–10.

[6] A. Fajardo and F. Rangel de Sousa, “Ideal energy power source model and its implications on batterymodeling,” presented at the XXII IBERCHIP Workshop, Florianopolis, Brasil. 2016.


http://www.physics.otago.ac.nz/research/epg/research

ObrigadoE-mail: [email protected]

Site: http://lrf.ufsc.br/

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