Structure of Equivalent EMF and Equivalent Internal Resistance for a Wheatstone-bridge-type Combination of Five Batteries

¿ ƒ½¨7HëH À Sae Mulli (The Korean Physical Society), Volume 54, Number 6, 2007¸ 6Z4, pp. 526∼529

I² Ž­ ŽÊ ] Ø ¬ ŽP U ] k ù  ¹ ÅU  Ž ì Å+ s Ç; c 6 ” X ¢ Ž Ö « M  ¹ ÅI í ÄÊ Ý Ž Ö « 5 ” ¼$ []  §8 ý  Œ º

ý

—

¡)o-> ^∗

 â

““§¹¢¤@/†<Ɠ§ õ†<Ɠ§¹¢¤õ, îߖ€ªœ 430-739

L

|<K*å^†

% ò

1lx@/†<Ɠ§ e”Z…n×¼™èáÔàÔJ?#Q†<Æõ, Øæ·¡¤ %ò1lx 370-701 (2007¸ 3Z4 6{9 ~ÃÎ6£§)

f

”

§>=õ #î§>=–Ð ìrK½+É Ãº \OH ©œ çߖéߖôÇ „t ƒ“ 6fàÔÛ¼—:r ÚÔot+þA „t ƒ (Fig. 1)_ (¿º éߖ a, b\ @/ôÇ) 1px l„§4 Eeqü< 1px ?/ÂÒ$†½Ó req\¦ ½¨ “¦ Th´evenin_ &ñoü< Norton_

&

ñ

o\¦ s6 x #Œ ½¨›¸\¦ ìr$3ôÇ. Eeqü< reqH —¸¿º ‘¿º ýa8£¤ „t („t 1, 2)_ #î§>=ƒ †½Ó+¿º ĺ 8

£

¤ „t („t 3, 4)_ #î§>=ƒ †½Ó+×怩œ „t („t 5)_ ´òõ Ÿí†<ʝ)a †½Ó’_ +þAI–Ð ³ð‰&³½+É Ãº e”“¦

¿

º Óüto|¾Ó_ [j P: †½Ó“Ér yŒ•yŒ• α bEeqü< α²breqs. #Œl\"f α ≡ (r2r3− r1r4)/[(r1+ r2)(r3+ r4)]s

“

¦ bEeqü< breqH yŒ•yŒ• ¿º éߖ c, d\ @/ôÇ (7£¤, ýa8£¤, ĺ8£¤, ×怩œ „t_ #î§>=ƒ\ @/ôÇ) 1px l„§4 õ

 1px ?/ÂÒ$†½Ós. "f çH+þA›¸|  α = 0 (7£¤, r2r3= r1r4)s ëߖ7á¤|¨c âĺ Eeqü< reqH ×怩œ „ t

ü< Áº›'a .

PACS numbers: 01.40.Fk, 07.50.-e, 82.47.Cb, 84.30.Bv Keywords: „t, 1px l„§4, 1px ?/ÂÒ$†½Ó, 6fàÔÛ¼—:r ÚÔot

„



tH l„§4õ ?/ÂÒ$†½Ós f”§>= ƒ)a כ ܼ–Ð —¸ +

þ

Ao½+É Ãº e”. ¿º éߖ\¦ ” e”__ „t ƒ“Ér Th´evenin_ &ño [1]\ _ #Œ 1px l„§4 Eeqü< 1px

 ?/ÂÒ$†½Ó req–Ð sÀÒ#Q” 1px „t–Ð @/u½+É Ãº e”



. ¢¸H sQôÇ „t ƒ“Ér Norton_ &ño [1]\  1

p

x „ÀÓ Ieqü< 1px ?/ÂÒ$†½Ó req #î§>= ƒ)a 1px



r–Ð\ _ #Œ @/u½+É Ãº e”.

„



t_ f”§>=ƒõ #î§>=ƒ\ @/ôÇ 1px „t (1px

l

„§4õ 1px ?/ÂÒ$†½Ó)H sp ¸ú˜ ·ú˜94R e”ܼ 9 [1–

3], f”§>=õ #î§>=`¦ ƒW&hܼ–Ð ›¸½+ˆ<Êܼ–Ð+‹ %3#QtH

™

è0A f”§>=-#î§>= ƒ (series-parallel combination)\ @/ ô



Ç 1px „t•¸ sQôÇ õ\¦ s6 x #Œ ~1> ½¨½+É Ãº e

”

. ÕªQ
 f”§>=õ #î§>=–Ð ìrK½+É Ãº \OH „t ƒ

•

¸ e”HX< s\ @/ôÇ ©œ çߖéߖôÇ \VH „t $Á >h–Ð s

ÀÒ#Q” 6fàÔÛ¼—:r ÚÔot+þA „t ƒ (Fig. 1)s.

Fig. 1\"f „t i (i = 1, · · · , 5)_ l„§4õ ?/ÂÒ$†½Ó

“ É

r yŒ•yŒ• Ei, ri “¦ „t\¦ •¸d”o l 0A #Œ ¼#_



©

œ Saslow [3]_ ³ðlZO`¦ G×þ˜ôÇ. ‘:r 7HëH\"fH s

„



t ƒ\ @/ôÇ 1px l„§4õ 1px ?/ÂÒ$†½Ó`¦ ½¨

“¦, Th´evenin_ &ñoü< Norton_ &ño\¦ 6 x #Œ ü@

∗E-mail: [email protected]

†E-mail: [email protected]

|

©œ 4Ÿ¤¸úšK ˜ÐsH õ 5Åq\ ?/F)a çߖ “¦ ½©gË:&h

“



 ½¨›¸\¦ {9ìøÍ 6fàÔÛ¼—:r ÚÔot\"f

H çH+þA ›¸

|

(balance condition)õ ›'aº #Œ ½©"î “¦ ôÇ.

y

Œ

• „t_ l„§4s 0 {9 âĺ\ 6fàÔÛ¼—:r ÚÔot+þA „ t

 ƒ“Ér $†½Ó[þt–Ð ½¨$í)a {9ìøÍ 6fàÔÛ¼—:r ÚÔotü< °ú 



. 6fàÔÛ¼—:r ÚÔot\"f $†½Ó[þts çH+þA›¸|  α ≡ (r2r3− r1r4)/[(r1+ r2)(r3+ r4)] = 0 (7£¤, r2r3= r1r4)`¦ ëߖ7á¤ôÇ



€ $†½Ó r5€ªœéߖ_ „0A  0 sÙ¼–Ð s $†½Ó\H „ÀÓ

 âìØÔt ·ú§H. sQôÇ ©œS!“Ér r5 éߖ|ÃÌ÷&
 éߖ‚

 )

a âĺü< ´òõ&hܼ–Ð °ú Ü¼Ù¼–Ð 1px $†½Ó req(α = 0)“Ér r₁, r₂_ #î§>=ƒõ r₃, r₄_ #î§>=ƒs "f–Ð f”§>= ƒ

 )

a âĺ ¢¸H r1, r3_ f”§>=ƒõ r2, r4_ f”§>=ƒs

"

f–Ð #î§>= ƒ)a âĺü< °ú :

req(α = 0) = r1||r2+ r3||r4= (r1+ r3)||(r2+ r4). (1) l

 ñ ||H ¿º „t #î§>= ƒ÷&%3`¦ M:\ l ñ „Êê\ Z



~sH Óüto|¾Ó_ 7áxÀÓ\  1px l„§4 ¢¸H 1px ?/ Â

Ò$†½Ó`¦ _p H כ ܼ–Ð &ñ_ôÇ. ÕªQÙ¼–Ð $†½Ó °úכ r1, r2\ @/ #Œ r1||r2≡ r1r2/(r1+ r2)s. çH+þA›¸| s ë

ß

–7á¤÷&t ·ú§H {9ìøÍ&h“ âĺ\ 6fàÔÛ¼—:r ÚÔot\ @/ ô



Ç 1px $†½Ó >íߖ“Ér sp sÀÒ#Q4R e” [3].

6

fàÔÛ¼—:r ÚÔot+þA „t ƒ\ @/ôÇ 1px l„§4õ 1

p

x ?/ÂÒ$†½Ó`¦ 1lxr\ ½¨ H ©œ lœí&h“ ~½ÓZO“Ér -526-

¿ ƒ½¨7HëH À 6fàÔÛ¼—:r ÚÔot+þA „t ƒ\ @/ôÇ 1px l„§4õ 1px ?/ÂÒ$†½Ó_ ½¨›¸ – <ª$3“ · þj]j%ò -527-

v

ØÔy ñáÔ_ ZOgË:`¦ s6 x #Œ Fig. 1\"f ÚÔot+þA „ t

 ƒ_ ¿º éߖ a, b\ ƒ)a ÂÒ $†½Ó R\ âìØԍH „ À

Ó I\¦ >íߖ H כ s. s õ\¦ I = Eeq/(R + req)ü<

q

“§ €

Eeq = A/C , (2) req = B/C (3)

\



¦ %3H. #Œl\"f A, B, CH yŒ•yŒ•

A = [r2(r3+ r4) + (r2+ r4)r5] E1+ [r1(r3+ r4) + (r1+ r3)r5] E2

+ [r4(r1+ r2) + (r2+ r4)r5] E3+ [r3(r1+ r2) + (r1+ r3)r5] E4+ (r2r3− r1r4)E5, (4) B = r1r2(r3+ r4) + (r1+ r2)r3r4+ (r1+ r3)(r2+ r4)r5

= r1r3(r2+ r4) + (r1+ r3)r2r4+ (r1+ r3)(r2+ r4)r5, (5) C = (r1+ r2)(r3+ r4) + (r1+ r2+ r3+ r4)r5 (6)

–

Ð ÅÒ#Q”.

Fig. 1_ r–ЍH 6£§õ °ú “Ér [j t ¨8Š T1, T2, T3\ @/ #Œ @/gA$í`¦ °úH:

T1 : 1 ⇔ 3, 2 ⇔ 4, E5→ −E5; (7) T2 : 1 ⇔ 2, 3 ⇔ 4, E5→ −E5; (8) T3 : 1 ⇔ 4, 2 ⇔ 3. (9)

#

Œl\"f i ⇔ jH „t iü< „t j çߖ_ l„§4õ ?/ÂÒ$

†

½

Ó_ “§¨8Š(Ei↔ Ej, ri ↔ rj)`¦ _pôÇ. d” (2), (3)“Ér d

”

 (4), (5), (6)`¦ :Ÿx #Œ sQôÇ @/gA$í[þts ¸ú˜



•

¸2Ÿ¤ ³ð‰&³÷&%3.

d

”

 (2), (3)\"f ]jr)a ÚÔot+þA „t ƒ_ 1px l

„



§4 Eeqü< 1px ?/ÂÒ$†½Ó req\¦ r5_ †<Êú–Ð ˜Ð“¦ †½Ó1px d

”

 f (r5) = f (0) +R_r5

0 (df /dr5) dr5\¦ s6 x €

Eeq = E1||E2+ E3||E4+ α bEeq, (10) req = r1||r2+ r3||r4+ α²breq (11) _

 +þAI–Ð r jþt ú e”. #Œl\"f E1||E2≡ (r2E1+ r1E2)/(r1+ r2)–Ð"f #î§>= ƒ)a ¿º „t 1, 2_ 1px l

„



§4`¦
·p. ¢¸ôÇ bEeqü< breqH yŒ•yŒ• Fig. 1\"f 6f à

ÔÛ¼—:r ÚÔot+þA „t ƒ_ ¿º éߖ a, b m c, d{9 âĺ_ 1px l„§4õ 1px ?/ÂÒ$†½Ó`¦
·p.

s

QôÇ „t ƒ“Ér éߖíHy [j 7áxÀÓ_ „t ÕªÒ¨[(i) „ t

 1õ 2_ f”§>=ƒ, (ii) „t 5, (iii) „t 3õ 4_ f”§>=

ƒ



]s "f–Ð #î§>= ƒ)a âĺs 9 „t[þt_ l„§4_

~

½

ӆ¾Ó`¦ “¦9 € Millman_ &ño [1]\ _ #Œ Ebeq = breq

µE2− E1

r1+ r2 +E3− E4

r3+ r4 +E5

r5

¶

, (12) b

req = (r1+ r2)(r3+ r4)r5/C (13) e

”

`¦ ·ú˜ ú e”. d” (10), (11)\"f ĺ_ [j P: †½Ó“Ér

—

¸¿º αH “ú\¦ t“¦ e”ܼ 9 „t 5H bEeqü< breq\¦ :

Ÿ

x #Œ s †½Ó\"fëߖ
èߖ. ÕªQÙ¼–Ð ?/ÂÒ$†½Ó[þts ç



H+þA›¸|  (α = 0)`¦ ëߖ7ᤠ€ 6fàÔÛ¼—:r ÚÔot+þA „t

ƒ



“Ér 1px l„§4õ 1px ?/ÂÒ$†½Ó —¸¿º „t 5ü< Áº

› '

a†<Ê`¦ ·ú˜ ú e”. \V\¦ [þt#Q, (E5, r5) = (0, 0), (0, ∞)“

¿

º âĺ\ @/ #Œ õ °ú Ü¼Ù¼–Ð d” (1)õ 8Ô¦#Q Eeq(α = 0) = E1||E2+ E3||E4= (E1+ E3)||(E2+ E4)

(14)

 $íwnôÇ. » ·¡­#Œ, „t 1, 2, 3, 4 e”_{9 M:\ „ t

 5 d” (12)_ F‹c ñ îߖ_ °úכs 0s ÷&H l„§4õ ?/ Â

Ò$†½Ó_ q\¦ °úH€ d” (10)\"f ĺ_ [j P: †½Ó s

 ”.

s

]j 1px l„§4õ 1px ?/ÂÒ$†½Ós d” (10), (11)õ

° ú

 s ³ð‰&³÷&H sÄ»\¦ Óüto&hܼ–Ð sKK ˜Ð.

Th´evenin_ &ño\ _ #Œ 6fàÔÛ¼—:r ÚÔot+þA „t ƒ



_ 1px l„§4“Ér ¿º éߖ a, b çߖ_ \P2;r–Ð éߖ„

· ú

šs. &h a, b, c, d\"f_ „0A\¦ yŒ•yŒ• Va, Vb, Vc(≡ 0), Vd . Vaü< Vb\¦ bEeq(= Vd− Vc)_ |t–Ð ½¨ €

Va = E1||E2+ µ r2

r1+ r2

¶

Ebeq, (15)

Vb = −(E3||E4) + µ r4

r3+ r4

¶

Ebeq (16)

s

. ÕªQ€ 6fàÔÛ¼—:r ÚÔot+þA „t ƒ_ 1px l„

§ 4

 Eeq(= Va− Vb)H d” (10)õ °ú s ³ð‰&³H†d`¦ ·ú˜ ú e”



.

Norton_ &ño\"f 6fàÔÛ¼—:r ÚÔot+þA „t ƒ_ 1

p

x „ÀÓ I_eqH ÚÔot+þA „tƒ_ ¿º éߖ\¦ éߖ|ÃÌr (



`¦ âĺ (Fig. 1_ r–Ð\"f ÂÒ $†½Ós R = 0“ â

-528- ôDzDGÓüto†<Ært “DhÓüto”, Volume 54, Number 6, 2007¸ 6Z4

R I

r₃

E₃

r₂

E₂ r₁

E₁

r₄

E₄ E₅ r₅

a b

c d

Fig. 1. A circuit for a Wheatstone-bridge-type combina- tion of five batteries with two terminals (points a and b) connected to a load resistor R. For convenience Saslow’s diagram [3] is adopted to denote a battery with an EMF Ei and an IR ri. The current I through the load resis- tor is calculated in order to obtain the equivalent EMF Eeqand the equivalent IR reqof the battery combination.

The battery combination with points c and d as two ter- minals is just the parallel combination of three groups of batteries [(i) battery 1 and 2 connected in series with opposite signs of EMF, (ii) battery 5, (iii) battery 3 and 4 connected in series with opposite signs of EMF] with the equivalent EMF bEeqand the equivalent IR breq.

Ä

º)_ éߖ|ÃÌ „ÀÓs. Th´evenin_ &ñoü< Norton_ &ño

\



¦ ½+Ë € 1px $†½Ó“Ér

req= Eeq/Ieq (17)

\



¦ 6 x #Œ ½¨½+É Ãº e”. „t ƒ\ @/ôÇ 1px ?/ÂÒ

$

†½Ó`¦ ½¨½+É M:\ yŒ• „t_ l„§4“Ér õ\ %ò†¾Ó`¦ ÅÒ t

 ·ú§Ü¼Ù¼–Ð e”__ °úכܼ–Ð ×þ˜K•¸ Áº~½Ó . d” (11)s d

”

 (10)õ Ä»ôÇ ½¨›¸e”`¦ “¦9 #Œ l„§4_ °úכ[þt`¦ E5= 0, Ek = (J/2)rk (k = 1, 2, 3, 4) (18)

–

Ð ×þ˜ . #Œl\"f JH e”__ €ªœ_ ©œÃºs. yŒ• „ t

 i (i = 1, · · · , 5)\ @/ #Œ l„§4_ ~½Ó†¾Óܼ–Ð âìØÔ





H „ÀÓ\¦ Ii . ÕªQ€ r1I1+ r3I3 = E1+ E3 = J(r1+r3)/2, I1−I3= I5 $íwnôÇ. s âĺ\ Fig. 2ü<

° ú

 s r5\¦ ÚÔot µ1Úܼ–Ð NS?/“¦ éߖ|Ã̂(R = 0)`¦ ÚÔo t

 r–Ð îߖܼ–Ð V,#Q r–Ð\¦ +þA . Dh–Ð ëߖ[þt#Q” ÚÔ o

t+þA „t ƒ“Ér éߖ|Ãܼ̂–Ð “ #Œ „t 1, 3_ #î§>=

ƒ



õ „t 2, 4_ #î§>=ƒ`¦ "f–Ð f”§>= ƒôÇ âĺ\ K

{©œ Ù¼–Ð „t 2, 3_ l„§4_ ~½Ó†¾Ós Ÿ÷¶ כ `¦ “¦9

€ 1px l„§4s 0s )a. ÕªQÙ¼–Ð $†½Ó r5\ âìØÔ

r₃ r5

I₅

-E₃ r₂

-E₂

r₁

E₁

r₄

E₄

I_eq

Fig. 2. The circuit in Fig. 1 with EMFs given by Eq. (18) and R = 0 can be deformed to a loaded Wheatstone- bridge-type combination of batteries with a center bat- tery replaced by a shorted wire. Ieq is the short current between two terminals (points a and b) of the original bridge-type combination of batteries in Fig. 1. The de- formed Wheatstone-bridge-type combination of batteries is the same as the series combination of two batteries 2, 4 in parallel with opposite signs of EMF and two bat- teries 1, 3 in parallel with opposite signs of EMF. Since the equivalent EMF of this combination is 0, the current I5through r5vanishes and thus all the batteries 1, 2, 3, 4 have the same magnitude J/2 of current, which leads to Ieq= J.





H „ÀÓ I5= 0e”`¦ ·ú˜ ú e”. "f I1= I3= J/2s

“

¦ Ä»ôÇ 7H_\ _ #Œ I2= I4 = J/2s. õ&hܼ

–

Ð „t[þt_ l„§4s d” (18)–Ð ÅÒ#QtH 6fàÔÛ¼—:r ÚÔo t

+þA „t ƒ\ @/ôÇ éߖ|ÃÌ „ÀӍH Ieq= I1+ I2= Js



. d” (18)`¦ d” (10)\ @/{9 “¦ d” (17)`¦ 6 x € d

”

 (11)s %3#Q”.

כ

¹€• €, ‘:r 7HëH\"f ĺoH d” (4), (5), (6)s @/{9

 )

a d” (2), (3)õ °ú s ü@|©œ Bĺ 4Ÿ¤¸úšôÇ +þAI–Ð ³ð‰&³÷&





H, 6fàÔÛ¼—:r ÚÔot+þA „t ƒ\ @/ôÇ 1px l„§4õ 1

p

x ?/ÂÒ$†½Ós Óüto&h ìr$3`¦ :Ÿx #Œ d” (10), (11)õ

° ú

 s Bĺ çߖéߖ “¦ ½©gË:&h“ ½¨›¸\¦ t“¦ e”6£§`¦ 7£x

"

î %i.

Y c

p wŠ ÃUØ ”ô

[1] R. L. Boylestad, Introductory Circuit Analysis, 10th ed. (Prentice-Hall, New York, 2003).

[2] D. H. Current, Am. J. Phys. 47(5), 463 (1979).

[3] W. M. Saslow, Electricity, Magnetism, and Light (Academic Press, New York, 2002).

¿ ƒ½¨7HëH À 6fàÔÛ¼—:r ÚÔot+þA „t ƒ\ @/ôÇ 1px l„§4õ 1px ?/ÂÒ$†½Ó_ ½¨›¸ – <ª$3“ · þj]j%ò -529-

Structure of Equivalent EMF and Equivalent Internal Resistance for a Wheatstone-bridge-type Combination of Five Batteries

Seok-In Hong^∗

Department of Science Education, Gyeongin National University of Education, Anyang 430-739

Je-Young Choi^†

Department of Embedded Software, Youngdong University, Youngdong, Chungbuk 370-701 (Received 6 March 2007)

As the simplest example of battery combinations that cannot be decomposed into series and parallel ones, the Wheatstone-bridge-type combination of five batteries (see Fig. 1) is considered.

We find its equivalent EMF, Eeq, and equivalent IR(internal resistance), req, (with respect to two terminals a and b) and analyze their structures by using Th´evenin’s theorem and Norton’s theorem.

Both of the Eeq and the req can be expressed in the form of ‘a parallel-combination term of two left batteries (batteries 1, 2)+a parallel-combination term of two right batteries (batteries 3, 4) + a term containing the effect of the center battery (battery 5)’. The third terms of these quantities are α bEeqand α²rbeqwith α ≡ (r2r3− r1r4)/[(r1+ r2)(r3+ r4)], respectively. Here, bEeqand breqare the equivalent EMF and the equivalent IR with respect to two terminals c and d (i.e., for the parallel combination of left, right, and center batteries), respectively. Hence, if the balance condition α = 0 (i.e., r2r3= r1r4) is satisfied, Eeqand reqare independent of the center battery.

PACS numbers: 01.40.Fk, 07.50.-e, 82.47.Cb, 84.30.Bv

Keywords: Battery, Equivalent EMF, Equivalent internal resistance, Wheatstone bridge

∗E-mail: [email protected] ^†E-mail: [email protected]