12449


FACTA UNIVERSITATIS  
Series: Electronics and Energetics Vol. 37, No 2, June 2024, pp. 249 - 260 

https://doi.org/10.2298/FUEE2402249R 

 

© 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Invited paper  

ENERGETIC PROPERTIES OF THE (FAST) CHARGING AND 

DRIVING OF AN ELECTRIC VEHICLE – A RAGONE ANALYSIS 

Alfred Rufer  

EPFL Ecole Polytchnique Fédérale de Lausanne 

ORCID iD:  Alfred Rufer   https://orcid.org/0000-0002-3271-1996 

Abstract. The energy efficiency of EVs is analysed for the charging and for the driving 

process where real parameters of a product of the automotive industry are considered. 

The efficiency in the charging mode is analysed and represented in function of the 

charging time. The maximum charging power allowed by the manufacturer is defining 

the lowest efficiency. After the calculation in a situation with an ideal charging station 

the influence of the resistance of the charging cable is taken into consideration. For the 

driving mode, the efficiency is estimated with the help of a special tool called “The 

Theory of Ragone plots”. The driving efficiency is calculated for the maximum possible 

propulsion power of the car, and represented in a dimensionless Ragone representation. The 

rated power of the car is represented in reference to the maximum possible power that can 

deliver the car battery. 

Key words: EV battery, energy efficiency, charging time, driving efficiency, Ragone 

representation 

1. INTRODUCTION  

During the next decade, Electric Vehicles EVs are expected to grow rapidly in the 

scene of the individual transportation worldwide, in order to reduce significantly the 

greenhouse emissions which are responsible for the climate change. In comparison with 

internal combustion engines (ICEs) vehicles, EVs have not only zero emission propulsion 

system but present a significant higher energy efficiency. The zero-emission characteristic 

must however be considered in a context where the electric power is coming from renewable 

sources or from nuclear plants. One of the typical disadvantages of EVs concerns the 

relatively long charging time which is not able to compete with as well ICE vehicles as 

fuel-cell and Hydrogen propulsion systems. With the aim to overcoming this handicap 

the automotive industry has spent many efforts for allowing a fast charge, where the 

battery cooling has been specially designed. Other important efforts have been made in 

the field of the power distribution and of the charging infrastructure [1,2]. Fast charge of 

 
Received January 20, 2024; accepted January 20, 2024 

Corresponding author: Alfred Rufer 
EPFL Ecole Polytchnique Fédérale de Lausanne. 

E-mail: alfred.rufer@epfl.ch 

https://orcid.org/0000-0002-3271-1996


250 A. RUFER 

a battery means a high power or high current causing losses in the internal series 

resistance of the battery elements and in the charging cable. These losses will be analysed 

in the present paper in dependency of the charging power and charging time and a 

corresponding energy efficiency will be calculated. Fig. 1 shows the structure of the 

analysed system with the charging station, the car battery and the electric propulsion 

system. The efficiency of the charging process is indicated in the figure (ch). 

 

Charging 
station

72%ch

drive   

Fig. 1 Electric vehicle with the charging station 

For the driving mode of the EV the maximum power level is defined according to a 

performance criterium which makes the EV’s competitive with the actual standard 

performances of ICE vehicles. This means that the power limit is chosen as high as 

possible for a given battery capacity. The present paper tries to characterise the energetic 

performance of the driving mode of the EV and will use a specific method called the 

Theory of Ragone plots [3, 4]. The Ragone plot indicates additionally the theoretical 

performance limit of a given battery as the maximum extractable power in the discharge 

mode. The efficiency of the driving mode is indicated in Figure 1 (drive). 

The present study is realized on the base of available battery parameters and should 

induce further investigations, especially in the direction of always higher charging 

powers as indicated in Table 1. 

Table 1 Actual and future battery capacity and charging power 

Manufacturer Battery name Capacity Charging power 

Porsche Macan 95kWh 270kW 

Zeekr Golden Brick 75kWh 500kW 

Catl Quilin 100kWh  600kW 

2. ENERGY EFFICIENCY 

2.1. A general model for the efficiency 

The quality of a storage system is quantified through its energy efficiency, taking into 

the account the internal losses during charging and discharging, the self-discharge effect 

during the time the energy is maintained in the storage device, even if the exchanged power 



 Energetic Properties of the (Fast) Charging and Driving of an Electric Vehicle – A Ragone Analysis 251 

is set to zero. In some specific cases, the energy needed for the auxiliaries must be considered. 

These auxiliaries are for example the circulating pumps of a vanadium redox flow battery 

(VRB), the vacuum pumps of a flywheel for the reduction of the aerodynamic drag, or the 

cryogenic system of a superconductive magnetic energy storage device (SMES). Figure 2 

shows the energy flow of a storage system where all the listed effects are represented [5]. 

For an internally stored amount of energy, the primary needed amount can be significantly 

higher (Energy to be stored). The typical example of this mechanism is analysed in the 

following sections for electrical vehicles when so called Ultra-Fast Chargers are used. 

Similarly, at the output of the storage system, the recovered energy can be strongly reduced 

in comparison with the initially existing amount of accumulated energy. The penalty in the 

discharge process will further be illustrated using the “Theory of Ragone Plots”. 

 

Internally stored 
energy  E0

Internal losses  
Ech/disch

Self discharge Esd

Auxiliaries Eaux

Energy to be stored Recovered Energy

 

Fig. 2 Energy flow and losses in the transfer to and from an energy storage device 

2.2. Structure and parameters of an example of EV 

The aim of the present analysis is to set in evidence the real behaviour of the charging 

and driving of an EV vehicle. A quantitative estimation of the energetic performance 

must be based on a true example of the actual automotive industry. The example of the 

Volkswagen ID3 is chosen [6]. 

The structure of the ID3 battery is given in Fig. 3 with its serial connection of 9 

modules, each composed of a parallel connection of 2 times 12 elements in series. Figure 

4 is an equivalent circuit of one element where the local parameters are defined. The 

internal resistance of the elements Rielem is equal to 1.857 m and the internal voltage 

source U0elem is chosen at 3.59V. This value represents an average value of the variable 

voltage which in reality depends on the state of charge. The charge of one element is 

given as Q0elem = 74.76Ah. The energy stored in one element can be calculated as 

 
0 0 3.59 74.76 268.5elem elem elemE U Q V Ah Wh=  =  =  (1) 



252 A. RUFER 

With the series number of modules and their internal connection the total number of 

elements is 

 9 (12 / 2 ) 216eln s s p= =  (2) 

And the total energy capacity of the car becomes: 

 216 268.5 57.97batE Wh kWh=  =  (3) 

which value confirms the indicated nominal capacity of 58kWh on the nameplate. 

For the internal voltage of the car battery, there are 216/2 elements in series. The 

voltage becomes: 

 
0 216 / 2 3.59 387.7batU V V=  =  (4) 

The internal resistance of the car battery is 

 108 / 2 0.001857 0.1003ibatR  =  =  (5) 

 

Fig. 3 Structure of the ID3 battery 

Rielem

U0elem

Q0elem

 

Fig. 4 Equivalent scheme of one battery element 



 Energetic Properties of the (Fast) Charging and Driving of an Electric Vehicle – A Ragone Analysis 253 

3. THE CHARGING EFFICIENCY OF THE ID3 

3.1. Equivalent scheme 

For the evaluation of the energetic performance of the charging process, the equivalent 

scheme of Fig. 4 will first be considered, and the performance of the charging of the car 

will be calculated. Further the influence of the connecting cable will also be estimated. The 

corresponding equivalent scheme for the charging process is given in Fig. 5.  

Rcab Ribat

U0batCharging 
station

Ichbat

Ebat

 

Fig. 5 Equivalent scheme of the charging process 

3.2. Energetic performance 

The energetic performance of the battery is calculated in dependency of the charging 

time tch. 

First, the charging power is calculated internally of one element (Pi_elem). 

 _ ( ) elem

i elem ch

ch

E
P t

t
=  (6) 

Then, the internal losses on the series resistance are calculated with the help of the 

charging current: 

 
_

_

0

( )
( )

i elem ch

ch elem ch

elem

P t
I t

U
=  (7) 

The external power of one element corresponds to the sum of the internal power and 

the resistive power dissipation. 

 2

e_ _ _ _( ) ( ) ( )elem ch i elem ch i elem ch elem chP t P t R I t= +   (8) 

The two curves of the internal and the external power of one element in dependency 

of the charging time are represented in Fig. 6. The values of the powers are represented 

for a charging time range comprised between 4 hours and 6 minutes. After the value of 2 

hours of charging time, the curve is very flat. But under the value of half an hour, the 

power increase is very steep. 



254 A. RUFER 

 

Fig. 6 Internal and external charging power in function of the charging time 

Then, the efficiency of the charging process is calculated also in function of the charging 

time. The efficiency is calculated as the ratio of the internal power to the external one: 

 
_

_

( )
( )

( )

i elem ch

elem ch

e elem ch

P t
t

P t
 =  (9) 

Figure 7 shows the curve of the charging efficiency in an ideal case where the influence 

of the cable resistance is neglected. The curve shows that the efficiency is over 90% for a 

charging time above one third of an hour. For charging times under 15 min, the efficiency 

becomes problematic. 

The charging current delivered to the car from the charging station is equal to the 

double value of the current in the battery elements (2 parallel ways): 

 _bat _elem( ) 2 ( )ch ch ch chI t I t=   (10) 

 

Fig. 7 Charging efficiency in function of the charging time 



 Energetic Properties of the (Fast) Charging and Driving of an Electric Vehicle – A Ragone Analysis 255 

The curve of the current delivered to the car is represented in Fig. 8. For a charging 

time of half an hour, the current reaches already the value of more than 250 A. 

 

Fig. 8 Current delivered to the car in function of the charging time 

And the power delivered to the car is: 

 
e_216car elemP P=   (11) 

It is also possible to define a car internal power if the equivalent internal resistance of 

the car is considered. 

 2

_car e_car _car _car( ) ( ) ( )i ch ch i ch chP t P t R I t= −   (12) 

The two curves of the internal and external power of the car are represented in Fig. 9. 

The limited power level indicated by the manufacturer of 120kW is also indicated. 

 

0 0.5 1 1.5 2 2.5 3 3.5 4

Charging time [hours]

0

1

2

3

4

5

6

7

8

c
a
r 

p
o
w

e
r 

in
t 
(b

lu
e
) 

e
xt

 (
re

d
)[

W
]

105

Pch_max=120kW

 

Fig. 9 Internal and external power of the car battery 



256 A. RUFER 

Another criterium for the fast charge of the car is to have sufficient voltage from the 

charging infrastructure. Figure 10 indicates the minimum needed voltage to be able to 

provide the charging current. 

 

Fig. 10 Minimum charging voltage 

3.3. The influence of the charging cable 

Between the charging station and the EV a cable of 5m length with 70mm2 section is 

considered [6]. With a value of the copper resistivity of 0.017 mm2/m, the equivalent 

resistance of the cable becomes: 

 
2

2

2 0.017 / 2 5
0.0024

70
cab

l mm m m
R

s mm

 


   
= = =  (13) 

 2

i_car _car _car( ) ( ) ( ) ( )station ch ch i cab ch chP t P t R R I t= + +   (14) 

The power delivered to the car has only slightly different energy efficiency if the 

resistance of the cable is considered or not. The two efficiencies are represented in Fig. 11. 

 
Fig. 11 Efficiency of the charging process with and without the influence of the cable 

resistance 



 Energetic Properties of the (Fast) Charging and Driving of an Electric Vehicle – A Ragone Analysis 257 

4. THE EFFICIENCY OF THE TRACTION MODE  

4.1. The equivalent scheme 

For the evaluation of the efficiency of the traction mode the equivalent scheme of Fig. 

12 is chosen. The efficiency corresponds to the ratio of the amount of energy transmitted 

to the propulsion to the amount of energy stored initially in the battery. This ratio can be 

calculated using the Theory of the Ragone plots [3, 4]. 

Ribat

U0bat
Traction 
system

Itrac

Ebat

U
P

 

Fig. 12 Equivalent scheme for the traction mode 

4.2. The theory of Ragone plots 

This method applied to the discharge of an ideal battery under a constant power 

solicitation uses the basic relation between current and power for the equivalent circuit of 

Fig. 12. For the ideal battery the leakage resistor is neglected. 

 0( )P U I U RI I=  = −  (15) 

where U is the terminal voltage and I is the current. The solutions of the quadratic 

equation are: 

 
2

0 0

22 4

U U P
I

R RR
 =  −  (16) 

From the relations (15) and (16), and under the condition that the battery is empty at 

time tinf = Q0/I , the Ragone plot can be drawn. Q0 is the initial charge related to the initial 

energy E0 = Q0U0. The Ragone curve is the trajectory of the energy that can be transferred 

to the load when the discharge is driven by a constant power. The transmitted energy is 

depending on the power level of the exchange and can be calculated through rel. (17). 

 0

2

0 0

2
( )

4
b

RQ P
E P P t

U U RP
=  =

− −
 (17) 

There is a maximum power, 
2

max 0 / 4P U R=   associated with an energy E0/2. Relation 

17 represents the section of the curve where the second solution of the expression of the 

current is considered (-sign) in the equation and corresponds to a power varied from a 

vanishing low value up to the maximum Pmax. 

The expression of the Ragone plot can be given with dimensionless units using the 

reference values 

0 0/b be E Q U=     and  2

04 /p RP U=  as 

 1
( )

2 (1 1 )
b

p
e p

p
=

− −

  (18) 



258 A. RUFER 

This expression is represented in Fig. 13 and corresponds to the trajectory of eb(p) when 

the power is varied. This variation can be produced by a variable resistance Rload which can 

be changed from an infinite value (open circuit) down to zero (short-circuited battery). The 

zero value of the current (Rload=  ) corresponds to a zero power and is represented by the 

upper left end of the trajectory. Reducing the load resistance increases the current as well as 

the power but the energy extracted decreases. The power can be increased up to a maximum 

value of P=U0
2/4R corresponding to the right end of the figure (p=1). Then a further 

increase of the current up to the short circuit current corresponds to a decreasing power also 

to a strong reduction of the extracted energy. This second section of the curve (dotted line in 

Fig. 13) corresponds to the first solution of the expression of the current (+sign) in the 

equation (16). The double solution for the extracted energy for one single value of the 

power is an illustration of the double solution of the quadratic equation of the expression of 

the power as a function of the current (rel. 16). In Fig. 13 the point indicated (p=0.55, 

e=0.83) corresponds to the maximum power developed by the car with the example of the 

ID3 (150kW) as will be described in the next section. 

100

10-1

10-2

10-3

10010-110-210-310-4

p [4RP/U0
2]

e 
[E

/Q
0
U

0
]

RL=

p=0.55
e(p)=0.83

 

Fig. 13 The Ragone plot of the ideal battery in the dimensionless representation 

4.3. The representation of the highest power of the ID3 in the Ragone plot 

The name plate of the ID3 indicates a maximum traction power of 150kW. According 

to the definitions of the Ragone theory, the highest extractable power from the battery is 

given as: 

 
2

max 0 / 4P U R=  (19) 

At the level of the car, the parameters of an equivalent scheme according to Fig. 12 

become: 

 
0batt 0 108 3.59V 387.7Vs elemU n U=  =  =  (20) 

 
ibatt 108 / 2 54 0.001857 0.1ielemR R  =  =  =  (21) 



 Energetic Properties of the (Fast) Charging and Driving of an Electric Vehicle – A Ragone Analysis 259 

From rel. 19, the maximum value of the power that can be extracted from the battery 

is calculated: 

 2

max (387.7 ) / 4 0.1003 374.7P V kW=  =  (22) 

When the mechanical power developed by the car is of 150kW, the extracted power 

from the battery is increased (divided by) the efficiency of the propulsion system (Fig. 1). 

The value 0.72 results from the efficiencies of the motor an of the converter (90% and 

95%) and takes into the account the standby and idle losses (8%) as of the driveline and 

of the auxiliaries (6%). 

 
maxbatt 150 / 0.72 208.33P kW kW= =  (23) 

Then the dimensionless value of the power p becomes: 

 208.332 / 374.7 0.55p kW kW= =  (24) 

And the dimensionless energy  

 
1 0.55

0.83
2 1 1 0.55

e =  =
− −

 (25) 

These two values are reported in the dimensionless Ragone plot of Fig. 13. The 

parameter e can be interpreted as the efficiency of the battery discharge process. The value 

of 0.83 is only reached if the car is driven with a power of 150 kW what occurs very rarely. 

4.4 .The energy efficiency of the driving mode 

For the final value of the efficiency in the driving mode, the efficiency of the battery 

discharge is combined with the efficiency of the propulsion system: 

 _ 0.83 0.72 0.597drive disch batt prop  =  =  =  (26) 

This result is valid for the maximum propulsion power but shows the realistic energy 

properties of the typical EV. 

An extreme consideration would be to integrate into the energy balance the efficiency 

of the charging process. The charging efficiency (by maximum charging power of 

120kW) multiplied by the discharging one (Pprop=150kW) and by the efficiency of the 

propulsion system becomes: 

 _s _ 0.925 0.83 0.72 0.55drive to w ch disch batt prop   − − =   =   =  (27) 

This value would correspond to an equivalent efficiency “source-to-wheel” where the 

source corresponds to the electric distribution grid. 

CONCLUSIONS 

Electric vehicles tend to replace classical ICE vehicles with always larger battery 

capacities and shorter charging times. Fast charging has the consequence of internal 

losses and need an accurate analysis of the power to be provided by the charging station. 

Resultant energy efficiency has been calculated in function of the charging time. The 



260 A. RUFER 

influence of the resistance of the charging cable has only a small influence on the 

efficiency. For the driving mode, a dedicated method called the Theory of Ragone plots 

has been used for the evaluation of the efficiency in the driving mode of the vehicle. The 

design of the ID3 vehicle shows that the maximum power of the vehicle corresponds to 

0.55 of the maximum extractable power from the battery. A resultant efficiency for this 

power level is of 0.83 for the battery discharge. The final value of the driving efficiency 

is of 0.597 when the maximum driving power is used. 

REFERENCES 

 [1]  H. Hoimoja et al., Toward Ultrafast Charging Solutions of Electric Vehicles, CIGRE 2012, Paris. 

 [2]  A. Rufer, "Towards a Decarbonized Individual Mobility-Challenges and Solutions", J Phys Res Appl, 

vol. 7, no. 4, p. 1000047 (7 pages), 2023. 
 [3]  A. Rufer, Energy storage Systems and Components, CRC Press, Taylor and Francis Group, Boca Raton 

FL, 2018, ISBN 13: 978-1-138-08262-5 (Hardback). 

 [4]  T. Christen, M. W. Carlen, "Theory of Ragone plots", Journal of Power Sources, vol. 91, pp. 210–216, 2000. 
 [5]  K. Papailiou (ed), Springer Handbook of Power Systems, Chapter 16 Energy Storage by A. Rufer, 

Springer Nature Singapore Pte Ltd 2021, ISBN 978-981-32-9937-5 

 [6]  N. Wassiliadis, M. Steinstr€ater, M. Schreiber et al., "Quantifying the state of the art of electric 
powertrains in battery electric vehicles: Range, efficiency, and lifetime from component to system level 

of the Volkswagen ID.3", eTransportation vol. 12, p. 100167, 2022. 

 [7]  Amphenol Tuchel Industrial GmbH, CA HVCO 70 T 050 (Distrelec)