1
MANUSCRIPT ID NUMBER TIM-24-06142
Inc easing Whea s one B idge sensi i i y
Josep M. To en s, Julio Ga cia-Cal e e
Abs ac —
Many physical a iables o in e es (mo emen ,
o ce, o que, powe , p essu e, low o empe a u e) a e usually
measu ed wi h senso s whose ou pu (o an elec ical pa ame e )
a ies li le wi h espec o hose physical a iables o in e es .
Thus, in o de o ha e adequa e accu acies and esolu ions, i is
usually con igu ed in a di e en ial way h ough an elec ical
ci cui known as a Whea s one B idge. This pape p esen s a new
Whea s one B idge opology ha wi h squa e al e na ing signal
exci a ion inc eases he adi ional sensi i i y in ol age di e ence
in he wo b anches o he B idge wi h a senso elemen (ε/4) o a
sensi i i y o (ε/e), whe e (ε) is he di e ence in he senso (as pe
one) caused by he physical a iable o in e es . I is analyzed
analy ically, hen simula ed wi h calcula ion and inally measu ed
in an ad-hoc se -up.
Index Te ms—
AC B idge ci cui , capaci i e senso , di ec
in e ace. esis i e senso , sensi i i y analysis, squa e al e na ing
signal, s ep unc ion, u( ) signal, Whea s one B idge.
I.
INTRODUCTION
ANY
elec ical senso s a e based on he ac ha hei
esis ance, induc ance o elec ical capaci ance (R, L
o C) a ies depending on a physical a iable o
in e es such as empe a u e, o ce, p essu e, e c. In gene al, his
a ia ion o R, L o C is small wi h espec o i s absolu e alue.
To inc ease he esolu ion in he measu emen o he physical
a iable o in e es , a Whea s one B idge [1] is adi ionally
used, such as he one in Fig. 1. Whe e V
dc
is he exci a ion o
he B idge and R
1
, R
2
, R
3
and R
4
a e he esis ances o he B idge
(i will also be e e ed as RR Whea s one B idge when he
b anches ha e only esis o s). A leas one o he esis ances is
a unc ion o he physical a iable o in e es .
Fig. 1. Resis i e Whea s one B idge (o RR Whea s one
B idge).
When all ou esis o s o he B idge ha e he same alue
This pape was submi ed on June, 24
h
, 2024. This wo k was suppo ed in
pa by he UPC – Ba celona TECH.
Josep M. To en s is wi h he Elec onic Enginee ing Depa men , UPC –
Ba celona TECH, C4 Building, Campus No d, c. Jo di Gi ona, 1, 08034
Ba celona, SPAIN (e-mail: jos[email p o ec ed]).
(in equilib ium si ua ion and e e ence alue o he physical
a iable) occu s: 𝑉𝑉
When he e e ence physical a iable a ies, ( o example
R
3
is a unc ion o he physical a iable while R
1
, R
2
and R
4
emain cons an ) imbalance appea s in he B idge and:
𝑉𝑉
I is easily demons a ed [2] ha he maximum sensi i i y
(maximum change o he di e ence be ween V
a
and V
b
o a
gi en change o he measu and) o a a ia ion o he physical
a iable is eached when he ollowing equilib ium si ua ion is
gi en: 𝑅𝑅𝑅𝑅
Fig. 1 ep esen s an example in DC. The esul is main ained
when he B idge is exci ed in AC. In he case o AC exci a ion,
he esis o s can be eplaced by induc ances o capaci ances (L
o C) [3].
E en in a si ua ion o maximum sensi i i y (in equilib ium
R
1
=R
2
=R
3
=R
4
), he di e ence:
|𝑉𝑉|
is small compa ed o he exci a ion (V
dc
in Fig. 1). The usual
solu ion is o add a high-gain Ins umen a ion Ampli ie (IA) o
ampli y his di e ence. In o de o simpli y he elec onics and
sa e he IA, some au ho s p opose wo king in squa e
al e na ing signal exci a ion (u( ) signal) wi h di ec in e ace
[4] o e en amp-less in e ace [5].
Conside ha he a iable o in e es changes and causes a
small change (ε) in one o he R
3
esis ances. Fo example, o
(ε=0.01) conside an exci a ion o V
dc
=1 V. Gi en a si ua ion
o maximum sensi i i y [2] ( o a single sensing esis o ):
𝑅𝑅𝑅𝑅
and 𝑅𝑅1𝜀
Analyzing by Ki chho and applying McLau in se ies
expansion whe e app op ia e, he ollowing solu ion (o dashed
ace in Fig. 3) is eached:
|𝑉𝑉|
1 𝑉 𝑅
𝑅𝑅𝑅1𝜀
𝑅𝑅1𝜀1
21𝜀
2𝜀
1
21𝜀
21𝜀21
21𝜀1𝜀2
2
𝜀4 1
Julio Ga cía-Cal e e. was wi h UPC – Ba celona TECH. He is now an
independen esea che (e-mail: [email protected]).
Colo e sions o one o mo e o he Figu es in his a icle a e a ailable
online a h p://ieeexplo e.ieee.o g
M
This a icle has been accep ed o publica ion in IEEE T ansac ions on Ins umen a ion and Measu emen . This is he au ho 's e sion which has no been ully edi ed and
con en may change p io o inal publica ion. Ci a ion in o ma ion: DOI 10.1109/TIM.2025.3542674
This wo k is licensed unde a C ea i e Commons A ibu ion 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by/4.0/
2
MANUSCRIPT ID NUMBER TIM-24-06142
Then, ope a ing wi h a nume ical example is ob ained:
|𝑉𝑉|
1 𝑉0.00249
T adi ionally, he Whea s one B idge was adap ed o
di e en measu emen needs. Fo example, by eso ing o he
h ee-wi e o Siemens me hod when he senso is loca ed a
om he B idge ci cui [2], o when eac ances o di e en
na u e (made wi h L and C componen s in se ies and pa allels)
a e combined in he Whea s one B idge, aking he names o he
enginee s who p oposed hem: Owen, Sche ing, Maxwell,
Wein o Hay [6].
Gi en he impo ance o his ci cui as a measu emen
me hod, e y closely ela ed o all ypes o senso s, a ious
au ho s ha e wo ked and de e mined aspec s o he Whea s one
B idge o imp o e i s sensi i i y and accu acy and hus imp o e
he senso iza ion o physical a iables and he ins umen a ion
ha emba ks hem. In [7] o example, aspec s o he wo king
supply and de ec ion o he Whea s one B idge a e imp o ed.
In [8, 9], he sensi i i y o he Whea s one B idge is inc eased
by conside ing he powe dissipa ed in he b anches o he same.
In [10], he accu acy o measu emen s is imp o ed when ec o
signals a e conside ed in he exci a ion o he Whea s one
B idge.
Following hese changes o he o iginal Whea s one B idge,
his pape p oposes a new measu emen sys em ha inc eases
he sensi i i y o he Whea s one B idge in a gi en
con igu a ion. The goal o his a icle is o explo e a opology,
wi hou changing he senso (o i s sensi i i y), ha inc eases
he sensi i i y o he B idge. Tha is, o inc ease he alue o
│V
a
– V
b
│ wi hou inc easing he exci a ion.
II.
D
EVELOPMENT
Nowadays, all senso sys ems inco po a e some ype o
digi al elec onics, such as mic op ocesso s and
mic ocon olle s; al hough he condi ioning o he sensing
elemen o en con inues o be analog (based on ampli ie s).
So, i is easy o exci e a Whea s one B idge wi h a squa e o
s ep signal o he u( ) ype. In his case, he Whea s one B idge
AC opology illus a ed in Fig. 2 is conside ed (i will also be
e e ed as RC Whea s one B idge when he b anches ha e
capaci o s and esis o s).
Fig. 2. RC Whea s one B idge shows an inc eased sensi i i y
compa ed o he esis i e one wi h a single senso elemen ,
ei he R (Fig. 2a) o C (Fig. 2b) in e changeably.
Rega ding he DC opology shown in Fig. 1, he DC
exci a ion has been changed o a s ep signal and wo esis o s,
one in each b anch, o wo capaci o s. The inc eased sensi i i y
is he same whe he i is a esis i e o capaci i e sensing
elemen i he a ia ion in bo h o he sensing elemen emains
(ε).
The analysis p oceeds as in he esis i e o RR Whea s one
B idge om he in oduc ion sec ion, and wi h he same
equilib ium s a ing si ua ion (ε=0). Tha is o say:
𝑉𝑉
When a a ia ion appea s in he physical a iable i causes an
imbalance in he B idge (ε≠0) and consequen ly:
𝑉𝑉
Analyzing by Ki chho , sol ing he di e en ial equa ion and
applying McLau in se ies de elopmen whe e app op ia e, he
ollowing solu ion is eached:
|𝑉𝑉|
1 𝑉𝑒/𝑒/𝑒𝑒
𝑒𝑒𝑒
𝑒1𝑒
𝑒𝑡𝜀
𝜏 2
wi h τ being he RC ime cons an and >0 he ime a iable a
he beginning o he s ep.
The con inuous line plo s (2) in Fig. 3. In addi ion, (2) has a
maximum o ( = τ) and o ha momen :
|𝑉𝑉|
1 𝑉𝑒𝜏𝜀
𝜏𝜀/𝑒 2𝑏
Highe alue han ha ound wi h he RR Whea s one B idge
de eloped in he in oduc ion sec ion:
𝜀𝑒𝜀4
Then, ope a ing wi h he nume ical example om he
in oduc ion sec ion is ob ained:
|𝑉𝑉|
1 𝑉0.00368
Fig. 3 illus a es his esul wi h he nume ical alue used in
he in oduc ion sec ion (ε=0.01). The dashed line shows he
classic esul p esen ed in he in oduc ion sec ion (1) and he
con inuous line shows he esul o his opology exci ing wi h
a s ep signal (2). I is obse ed ha o measu emen s a ound
ime ( = τ) he sensi i i y o he RC Whea s one B idge is
highe han he classical opology RR Whea s one B idge in DC
o also in AC exci a ion.
Fig. 3. Resul wi h he nume ical alue used in he in oduc ion
This a icle has been accep ed o publica ion in IEEE T ansac ions on Ins umen a ion and Measu emen . This is he au ho 's e sion which has no been ully edi ed and
con en may change p io o inal publica ion. Ci a ion in o ma ion: DOI 10.1109/TIM.2025.3542674
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MANUSCRIPT ID NUMBER TIM-24-06142
sec ion (ε=0.01). In dashed line, he esul wi h a classic
esis i e o RR Whea s one B idge; and in a con inuous line,
he esul o his RC Whea s one B idge opology exci ing wi h
a s ep signal.
III.
E
XPERIMENT
The good pe o mance o he p oposed ci cui has been e i ied
by expe imen ing wi h wo Whea s one B idge ci cui s, Fig. 4a and
4b. Fo he B idges, shel componen s ha e been used, bu hey
ha e been measu ed wi h an ohmme e wi h an addi ional
capaci ance measu emen op ion and hey ha e been chosen so ha
he di e ences be ween hem we e less han 0.1%. Fou equal 10
kΩ esis o s and wo equal 100 nF capaci o s ha e been chosen.
Thus, he ime cons an in he B idge wi h wo RC b anches, Fig.
4b, is (τ=1.0 ms). Fo he same imbalance in he wo Whea s one
B idges, ano he 100 kΩ esis o (which leads o app oxima ely a
10% imbalance) has been connec ed in pa allel o only one o he
10 kΩ esis o s, 10 kΩ//100 kΩ= 9.09 kΩ, ε=0.1).
In addi ion, he powe supply o he b idge is connec ed h ough
a P-channel MOSFET ha is con olled by he Ga e wi h an
op ocoupled s anda d signal. The RDS_ON (Resis ance be ween
D ain and Sou ce o he MOSFET) is o he o de o ens o
milliOhm which does no in luence he measu emen .
Thus, bo h Whea s one B idges ha e been supplied wi h a di ec
ol age V
dc
=10 V, see Fig. 4a and 4b.
Fig. 4a. Classic measu emen ci cui wi h a esis i e o RR
Whea s one B idge ha measu es he imbalance o esis o s. In
p esen expe imen , an imbalance o one o he esis o s (R
1
).
Fig. 4b. P oposed measu emen ci cui wi h RC Whea s one
B idge ha imp o es in a sui able ime ame he sensi i i y o he
classic ci cui (Fig. 4a). In p esen expe imen , also an imbalance
is caused in one o he esis o s (R
2
).
The opology in Fig. 4b is sligh ly di e en om ha in Fig. 4a.
I he opology in Fig. 4b we e he same as he opology in Fig. 4a,
he capaci o s would show cha ge e en i P
0
was OFF. Then, a
pa h mus be p o ided o discha ge hem. In his case, h ough he
N-channel MOSFET go e ned by P
1
and he p o ec ion esis o o
he R
p
ci cui .
The R
p
esis o limi s he maximum cu en ha would low
h ough he ansis o s i he wo MOSFETs we e d i ing a he
same ime. I R
p
is small compa ed o R
1
o R
2
, he discha ging
ime is sligh ly longe han he cha ging ime. In he expe imen
R
p
= 180 Ω.
The R&S RTB2004 oscilloscope has been used o gene a e he
op ocoupled signal pa e ns ha ha e exci ed he MOSFETs (P
0
in
Fig. 4a, and P
0
and P
1
in Fig. 4b). The MOSFETs ha e ac ed as
swi ches and hese signals (P
0
and P
1
) ha e p ope ly sequenced
(see Fig. 5) he ON/OFF s a es. In addi ion, his oscilloscope has
measu ed he imbalance in he B idges, ol ages V
a
and V
b
(Fig.
4a and 4b).
Bo h ci cui s ha e been exci ed wi h a epe i i e con ol pa e n
in 20 s eps wi h a du a ion o 1 ms each s ep. This s a egy esul s
in a pe iod ha epea s e e y 20 ms and is equi alen o a squa e
signal equency o 50 Hz in he B idge. Wi h balanced B idges,
he di e ence be ween V
a
and V
b
is lowe han he sensi i i y o
he oscilloscope, excep in he ansi ions o he RC Whea s one
B idge (Fig. 4b), wi h gli ches below 5 mV.
Fig. 5 illus a es he P
0
and P
1
pa e ns o he RC Whea s one
B idge. I mus be ensu ed ha bo h swi ches (P
0
and P
1
) on he
RC Whea s one B idge a e no closed a he same ime (no gli ch
ansi ions). Fig. 5 shows he ON/OFF pa e n o he cha ge (P
0
)
and he discha ge (P
1
), which wi h he gua d ime s eps (1 and 11)
in which he wo MOSFETs a e OFF, ensu es ha bo h MOSFETs
a e no ON a he same ime. As men ioned, i is no possibili y ha
bo h swi ches a e ON a he same ime, no e en in he ime o a
gli ch.
Because o hese sa e y ma gins, P
0
has 9 s eps in ON and 11
s eps in OFF. The e o e, he signal ha exci es he RR Whea s one
B idge is no symme ical (du y cycle sligh ly below 50%) as can
be seen in Fig. 6a.
Fig. 5. Powe and g ound connec ion and disconnec ion pa e ns o
he RC Whea s one B idge, wi h an OFF-OFF sa e y ime be ween
he wo swi ches.
IV.
R
ESULTS
Fig. 6a shows he measu emen o he ol ages V
a
, V
b
( om Fig.
4a o he RR Whea s one B idge) wi h he oscilloscope. I also
shows hei di e ence (V
a
– V
b
). The measu emen co esponds o
R
2
=R
3
=R
4
=10 kΩ and R
1
=9.09 kΩ, i.e., 10% unbalanced (ε=0.1).
The oscilloscope ime base has been se o 5 ms/di and e ical
This a icle has been accep ed o publica ion in IEEE T ansac ions on Ins umen a ion and Measu emen . This is he au ho 's e sion which has no been ully edi ed and
con en may change p io o inal publica ion. Ci a ion in o ma ion: DOI 10.1109/TIM.2025.3542674
This wo k is licensed unde a C ea i e Commons A ibu ion 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by/4.0/
4
MANUSCRIPT ID NUMBER TIM-24-06142
sensi i i y o channels 1 and 2 (which ha e measu ed V
a
and V
b
)
has been se o 1 V/di ( he scale on he le in Fig. 6a). Finally, he
e ical sensi i i y o he ma hema ical channel (which has d awn
he di e ence be ween V
a
and V
b
) has been se o 0.1 V/di ( he
scale on he igh in Fig. 6a).
The blue and o ange aces co espond o he ol ages V
a
and V
b
wi h a alue o app oxima ely hal o he supply ol age (V
dc
/2).
And he g ay ace co esponds o he ma hema ical channel wi h
he calcula ion di e ence be ween V
a
and V
b
. Applying (1), a
heo e ical alue o he ollowing is ob ained:
|𝑉𝑉|𝑉∗𝜀410 𝑉∗0.1
40.25 𝑉
Which co esponds o he ampli ude alue o he g ay ace in
Fig. 6a.
Fig. 6a. Measu ed alues V
a
and V
b
in he esis i e o RR Whea s one B idge, (blue and o ange aces); and measu ed alue o
hei di e ence (g ay ace) o an imbalance o 10% (ε=0.1).
Fig. 6b shows he esul o measu ing wi h he oscilloscope he
ol ages in he RC Whea s one B idge o he ci cui in Fig. 4b: V
a
,
V
b
and hei di e ence (V
a
– V
b
) when he e is an imbalance equal
o he p e ious case o he unbalanced esis i e o RR Whea s one
B idge by 10% (i.e., ε=0.1). Fo example, wi h bo h capaci o s o
C = 100 nF, R
1
= 10 kΩ and R
2
= 9.09 kΩ.
On he RC Whea s one B idge (Fig. 4b ci cui ), he oscilloscope
ime base has been se again o 5 ms/di . The e ical sensi i i y o
channels 1 and 2 (which ha e measu ed he V
a
and V
b
ol ages)
has been se o 2 V/di . Finally, he e ical sensi i i y o he
ma hema ical channel (which has d awn he di e ence be ween V
a
and V
b
) has been se o 0.2 V/di .
The blue and o ange aces co espond o he ol ages V
a
and
V
b
. In his case, when he swi ching ime is 5 imes lowe han
he ime cons an (τ=RC=1.0 ms) o he RC Whea s one B idge,
he ol ages V
a
and V
b
each a alue close o V
dc
=10 V. And
he g ay ace, co esponds o he di e ence calcula ion
be ween V
a
and V
b
. Applying (2b), o imes close o he ime
cons an (τ=RC=1.0 ms) o he RC Whea s one B idge, a
heo e ical maximum alue o he ollowing is ob ained:
|𝑉𝑉|𝑉∗𝜀𝑒10 𝑉∗0.1
2.72 0.37 𝑉
Which co esponds o he maximum ampli ude alue o he g ay
ace in Fig. 6b.
In summa y, a he ins an o maximum ( =τ=RC) he RC
Whea s one B idge opology has a highe sensi i i y, 46%
highe han he RR Whea s one B idge opology
(0.37/0.25=1.46).
Maximum sensi i i y is ob ained when measu ed a he
ins an =τ a e each swi ching o he squa e signal. Howe e ,
in he b anch ha causes he imbalance, τ is no known wi h
ce ain y because i depends on he alue o he physical
a iable ha is being measu ed. Fo small a ia ions in he
alue o he senso elemen , wo s a egies can be aken:
This a icle has been accep ed o publica ion in IEEE T ansac ions on Ins umen a ion and Measu emen . This is he au ho 's e sion which has no been ully edi ed and
con en may change p io o inal publica ion. Ci a ion in o ma ion: DOI 10.1109/TIM.2025.3542674
This wo k is licensed unde a C ea i e Commons A ibu ion 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by/4.0/
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MANUSCRIPT ID NUMBER TIM-24-06142
1) Measu e a he ins an =τ o he e e ence b anch ha
does no con ain any senso elemen . Fig. 3 shows
gen le slopes o he cu e a ound he maximum. Wi h
his, a ce ain loss o sensi i i y can be assumed,
al hough each case mus be s udied in pa icula .
2) Measu e a se e al poin s o e a pe iod and choose he
alue o he maximum (which will be close a ime =τ).
This second s a egy can also be used o la ge
a ia ions o he sensing elemen in he RC Whea s one
B idge.
In addi ion o measu ing ol ages, i should be measu ed in
ime ( =τ=RC) since he las swi ching o he RC Whea s one
B idge powe supply o maximize sensi i i y as men ioned. I
he e is only one sensing elemen (o he ou possible in a
Whea s one B idge) and p e ious measu emen s ha e been
s o ed in an equilib ium si ua ion, a di ec in e ace si ua ion
could be conside ed [4, 5]. In his si ua ion, he IA would no
longe be needed (bu he Sample&Hold ci cui (S&H) and
Analog o Digi al Con e e (ADC) would be needed). This
si ua ion will be s udied in a u u e.
Fig. 6b. Measu ed alues V
a
and V
b
in he RC Whea s one B idge, (blue and o ange aces); and measu ed alue o hei di e ence
(g ay ace) o he same imbalance o 10% (ε=0.1).
V.
C
ONCLUSION
Fo many a iables o in e es ha a e measu ed h ough
small changes in he elec ical p ope ies o senso s, he
Whea s one B idge is a widely used ci cui ha allows hem o
be measu ed wi h su icien accu acy and esolu ion. This
a icle has p esen ed a new RC Whea s one B idge opology
( he b anches o he B idge ha e been made om capaci o s and
esis o s). Fo squa e al e na ing signal exci a ion (o u( ) ype
s ep exci a ion) and a combina ion o esis i e o capaci i e
senso s, his opology imp o es sensi i i y by 46% compa ed
o he classic con igu a ion o he esis i e o RR Whea s one
B idge (wi h he same exci a ion ampli ude). The acquisi ion o
he measu emen mus be pe o med a a ime close o he ime
cons an (τ = RC) o he b anches o he RC Whea s one B idge
opology.
C
ONTRIBUTION
Fi s au ho : Concep ion o he idea, heo e ical
de elopmen s, design o he expe imen and w i ing. Second
au ho : Design o he expe imen and ealiza ion o he same.
A
CKNOWLEDGMENT
Au ho s hank he kindness and willingness o help o he
lib a y s a o he UPC – Ba celona TECH BRGF: Mon se
A agües, Ma a Ga cía, Ru h Iñigo, Raquel Lo ca, Josep
Ma ínez and Cla a Po eda.
This a icle has been accep ed o publica ion in IEEE T ansac ions on Ins umen a ion and Measu emen . This is he au ho 's e sion which has no been ully edi ed and
con en may change p io o inal publica ion. Ci a ion in o ma ion: DOI 10.1109/TIM.2025.3542674
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MANUSCRIPT ID NUMBER TIM-24-06142
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EFERENCES
[1] E.O. Doebelin, Measu emen Sys ems: Applica ion and Design, e ised
ed., McG aw-Hill, Inc., New Yo k, 1975, pp. 605–612.
[2] R. Pallás-A eny, J.G. Webs e , Senso s and Signal Condi ioning, John
Wiley and Sons, Inc., New Yo k, 1991, pp. 92–100.
[3] E.O. Doebelin, Measu emen Sys ems: Applica ion and Design, e ised
ed., McG aw-Hill, Inc., New Yo k, 1975, pp. 246–257.
[4] J.E. Gai án-Pi e, M. Gasulla, and R. Pallàs-A eny, “Analysis o a Di ec
In e ace Ci cui o Capaci i e Senso s,” IEEE T ansac ions on
Ins umen a ion and Measu emen , ol. 58, no. 9, pp. 2931-2937, May 2009,
doi: 10.1109/TIM.2009.2016782.
[5] T. Miyazaki, S. Nakagawa, and H. Ishiku o, "High-Resolu ion Au o-
Balancing Whea s one-B idge wi h Successi e App oxima ion o ΔΣ-
Modula ed Digi ally Con olled Va iable Resis o ," 26 h IEEE In e na ional
Con e ence on Elec onics, Ci cui s and Sys ems (ICECS), pp. 474-477,
No embe 2019, doi: 10.1109/ICECS46596.2019.
[6] H. V. Malms ad , C. G. Enke, and E. C. To en, Elec onics o Scien is s.
P inciples and Expe imen s o Those Who Use Ins umen s, 2
nd
p in ing, wi h
co ec ions, W. A. Benjamin, Inc., New Yo k, 1963, chap e 6.
[7] G. Weiss, "Whea s one B idge Sensi i i y," IEEE T ansac ions on
Ins umen a ion and Measu emen , ol. 18, no. 1, pp. 2-6, Ma ch 1969, doi:
10.1109/TIM.1969.4313752.
[8] J. E. Maisel, "Op imiza ion o he Whea s one B idge Sensi i i y," IEEE
T ansac ions on Ins umen a ion and Measu emen , ol. 26, no. 1, pp. 17-21,
Ma ch 1977, doi: 10.1109/TIM.1977.4314476.
[9] E. Takagishi, "Commen s on "Op imiza ion o he Whea s one B idge
Sensi i i y", IEEE T ansac ions on Ins umen a ion and Measu emen , ol.
27, no. 3, pp. 304-306, Sep embe 1978, doi: 10.1109/TIM.1978.4314692.
[10] E. Takagishi, "On he Balance o an AC Whea s one B idge," IEEE
T ansac ions on Ins umen a ion and Measu emen , ol. 29, no. 2, pp. 131-
136, June 1980, doi: 10.1109/TIM.1980.4314886.
Josep M. To en s ecei ed Mas e
Telecommunica ion Enginee ing and PhD
deg ees om he UPC – Ba celona TECH,
Ba celona, Spain, in 1989 and 1996,
espec i ely.
He is an Associa e P o esso o
Elec onic Enginee ing Depa men UPC –
Ba celona TECH, and he has augh
cou ses in se e al a eas o elec onics in
EETAC, Cas ellde els (Ba celona), ETSEIB, Ba celona, FNB,
Ba celona and ETSETB, Ba celona. Cu en ly, he eaches in
FNB and ETSETB. F om 1998 o 1999, he was a Visi ing
Schola a No hwes e n Uni e si y, E ans on, IL. His esea ch
includes ins umen a ion me hods and senso s based on
elec ical impedance and magne ic measu emen s, mainly
applied in ce amic ma e ials.
ORCID 0000-0002-8289-6706.
Julio Ga cia-Cal e e ecei ed Mas e
Indus ial Enginee ing and PhD deg ees
om he UPC – Ba celona TECH,
Ba celona, Spain, in 2005 and 2011,
espec i ely.
He was Assis an P o esso o Elec ical
Enginee ing Depa men UPC – Ba celona
TECH, and he augh cou ses in ETSEIT,
Te assa (Ba celona). He also wo ked o
ENDESA om 1982 o 2023. He holds se e al Pa en s ela ed
wi h assis an de ices o handicap people. He is now an
independen esea che .
ORCID 0009-0006-4169-9374.
This a icle has been accep ed o publica ion in IEEE T ansac ions on Ins umen a ion and Measu emen . This is he au ho 's e sion which has no been ully edi ed and
con en may change p io o inal publica ion. Ci a ion in o ma ion: DOI 10.1109/TIM.2025.3542674
This wo k is licensed unde a C ea i e Commons A ibu ion 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by/4.0/
