Electrical Circuit Formulae

7/23/2019 Electrical Circuit Formulae

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Electrical Circuit Formulae

Contents

- Notation- Resistance- Resistances in Series- Voltage Division by Series Resistances- Resistances in Parallel- Current Division by Parallel Resistances- Capacitance- Capacitances in Series- Voltage Division by Series Capacitances- Capacitances in Parallel- Charge Division by Parallel Capacitances

- Inductance- utual Inductance- Inductances in Series- Inductances in Parallel- !ime Constants- Po"er - Energy- #atteries- Voltmeter ultiplier - $mmeter Shunt- %heatstone #ridge

Notation

The library uses the symbol font for some of the notation and formulae. If the symbols for the

letters 'alpha beta delta' do not appear here [α β δ] then the symbol font needs to be installed

before all notation and formulae will be displayed correctly.

CEe&Ii

'(NP

capacitancevoltage sourceinstantaneous Econductancecurrentinstantaneous Icoefficient

inductancemutualinductancenumber of turnspower

[farads, F][volts, V][volts, V][siemens,][amps, !][amps, !]

[number][henrys, "][henrys, "][number][watts, #]

)*R!tV

v%

Ψ

ψ

chargeinstantaneous )resistancetime constantinstantaneoustimevoltage drop

instantaneous Venergymagnetic flu$magnetic lin%age

instantaneous Ψ

[coulombs,&][coulombs,&]

[ohms, Ω]

[seconds, s][seconds, s]

[volts, V][volts, V][oules, (][webers, #b][webers, #b][webers, #b]

Resistance

The resistance R of a circuit is e)ual to the applied direct voltage E divided by the resultingsteady current I*

R + E , I

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Resistances in Series

#hen resistances R, R., R/, ... are connected in series, the total resistance RS is*RS + R 0 R. 0 R/ 0 111

Voltage Division by Series Resistances

#hen a total voltage ES is applied across series connected resistances R and R., the currentIS which flows through the series circuit is*IS + ES , RS + ES , 2R 0 R.3

The voltages V and V. which appear across the respective resistances R and R. are*V + ISR + ESR , RS + ESR , 2R 0 R.3V. + ISR. + ESR. , RS + ESR. , 2R 0 R.3

In general terms, for resistances R, R., R/, ... connected in series*IS + ES , RS + ES , 2R 0 R. 0 R/ 0 1113Vn + ISRn + ESRn , RS + ESRn , 2R 0 R. 0 R/ 0 1113+ote that the highest voltage drop appears across the highest resistance.

Resistances in Parallel

#hen resistances R, R., R/, ... are connected in parallel, the total resistance RP is* , RP + , R 0 , R. 0 , R/ 0 111

!lternatively, when conductances &, &., &/, ... are connected in parallel, the total

conductance &P is*&P + & 0 &. 0 &/ 0 111where &n + , Rn

For two resistances R and R. connected in parallel, the total resistance RP is*RP + RR. , 2R 0 R.3RP product - sum

The resistance R. to be connected in parallel with resistance R to give a total resistance RP is*R. + RRP , 2R - RP3R. product - difference

Current Division by Parallel Resistances

#hen a total current IP is passed through parallel connected resistances R and R., thevoltage VP which appears across the parallel circuit is*VP + IPRP + IPRR. , 2R 0 R.3

The currents I and I. which pass through the respective resistances R and R. are*I + VP , R + IPRP , R + IPR. , 2R 0 R.3I. + VP , R. + IPRP , R. + IPR , 2R 0 R.3

In general terms, for resistances R, R., R/, ... with conductances &, &., &/, .../ connected inparallel*



VP + IPRP + IP , &P + IP , 2& 0 &. 0 &/ 0 1113In + VP , Rn + VP&n + IP&n , &P + IP&n , 2& 0 &. 0 &/ 0 1113where &n + , Rn

+ote that the highest current passes through the highest conductance with the lowestresistance/.

Capacitance

#hen a voltage is applied to a circuit containing capacitance, current flows to accumulatecharge in the capacitance*

) + ∫ idt + CV

!lternatively, by differentiation with respect to time*d*,dt + i + C dv,dt+ote that the rate of change of voltage has a polarity which opposes the flow of current.

The capacitance C of a circuit is e)ual to the charge divided by the voltage*

C + ) , V + ∫ idt , V

!lternatively, the capacitance C of a circuit is e)ual to the charging current divided by the rateof change of voltage*C + i , dv,dt + d*,dt , dv,dt + d*,dv

Capacitances in Series

#hen capacitances C, C., C/, ... are connected in series, the total capacitance CS is* , CS + , C 0 , C. 0 , C/ 0 111

For two capacitances C and C. connected in series, the total capacitance CS is*CS + CC. , 2C 0 C.3CS product - sum

Voltage Division by Series Capacitances

#hen a total voltage ES is applied to series connected capacitances C and C., the charge )Swhich accumulates in the series circuit is*

)S + ∫ iSdt + ESCS + ESCC. , 2C 0 C.3

The voltages V and V. which appear across the respective capacitances C and C. are*

V + ∫ iSdt , C + ESCS , C + ESC. , 2C 0 C.3

V. + ∫ iSdt , C. + ESCS , C. + ESC , 2C 0 C.3

In general terms, for capacitances C, C., C/, ... connected in series*

)S + ∫ iSdt + ESCS + ES , 2 , CS3 + ES , 2 , C 0 , C. 0 , C/ 0 1113

Vn + ∫ iSdt , Cn + ESCS , Cn + ES , Cn2 , CS3 + ES , Cn2 , C 0 , C. 0 , C/ 0 1113

+ote that the highest voltage appears across the lowest capacitance.



Capacitances in Parallel

#hen capacitances C, C., C/, ... are connected in parallel, the total capacitance CP is*CP + C 0 C. 0 C/ 0 111

Charge Division by Parallel Capacitances

#hen a voltage EP is applied to parallel connected capacitances C and C., the charge )P which accumulates in the parallel circuit is*

)P + ∫ iPdt + EPCP + EP2C 0 C.3

The charges ) and ). which accumulate in the respective capacitances C and C. are*

) + ∫ idt + EPC + )PC , CP + )PC , 2C 0 C.3

). + ∫ i.dt + EPC. + )PC. , CP + )PC. , 2C 0 C.3

In general terms, for capacitances C, C., C/, ... connected in parallel*

)P + ∫ iPdt + EPCP + EP2C 0 C. 0 C/ 0 1113

)n + ∫ indt + EPCn + )PCn , CP + )PCn , 2C 0 C. 0 C/ 0 1113

+ote that the highest charge accumulates in the highest capacitance.

Inductance

#hen the current changes in a circuit containing inductance, the magnetic lin%age changesand induces a voltage in the inductance*

dψ ,dt + e + ( di,dt

+ote that the induced voltage has a polarity which opposes the rate of change of current.

!lternatively, by integration with respect to time*

Ψ + ∫ edt + (I

The inductance ( of a circuit is e)ual to the induced voltage divided by the rate of change ofcurrent*

( + e , di,dt + dψ ,dt , di,dt + dψ ,di

!lternatively, the inductance ( of a circuit is e)ual to the magnetic lin%age divided by thecurrent*

( + Ψ , I

+ote that the magnetic lin%age Ψ is e)ual to the product of the number of turns N and the

magnetic flu$ *

Ψ + N + (I

utual Inductance



The mutual inductance of two coupled inductances ( and (. is e)ual to the mutuallyinduced voltage in one inductance divided by the rate of change of current in the otherinductance* + E.m , 2di ,dt3 + Em , 2di. ,dt3

If the self induced voltages of the inductances ( and (. are respectively Es and E.s for thesame rates of change of the current that produced the mutually induced voltages Em and E.m,then* + 2E.m , Es3(

+ 2Em , E.s3(.

&ombining these two e)uations* + 2EmE.m , EsE.s3

4 2((.34 + '2((.3

4

where ' is the mutual coupling coefficient of the two inductances ( and (..

If the coupling between the two inductances ( and (. is perfect, then the mutual inductance is* + 2((.3

4

Inductances in Series

#hen uncoupled inductances (, (., (/, ... are connected in series, the total inductance (S is*(S + ( 0 (. 0 (/ 0 111

#hen two coupled inductances ( and (. with mutual inductance are connected in series,the total inductance (S is*(S + ( 0 (. 5 .The plus or minus sign indicates that the coupling is either additive or subtractive, depending

on the connection polarity.

Inductances in Parallel

#hen uncoupled inductances (, (., (/, ... are connected in parallel, the total inductance (P is* , (P + , ( 0 , (. 0 , (/ 0 111

!ime Constants

Capacitance and resistanceThe time constant of a capacitance C and a resistance R is e)ual to CR, and represents thetime to change the voltage on the capacitance from 0ero to E at a constant charging current E , R which produces a rate of change of voltage E , CR across the capacitance/.

imilarly, the time constant CR represents the time to change the charge on the capacitancefrom 0ero to CE at a constant charging current E , R which produces a rate of change ofvoltage E , CR across the capacitance/.

If a voltage E is applied to a series circuit comprising a discharged capacitance C and aresistance R, then after time t the current i, the voltage vR across the resistance, the voltage

vC across the capacitance and the charge *C on the capacitance are*i + 2E , R3e - t , CR

vR + iR + Ee - t , CR



vC + E - vR + E2 - e - t , CR3*C + CvC + CE2 - e - t , CR3

If a capacitance C charged to voltage V is discharged through a resistance R, then after time tthe current i, the voltage vR across the resistance, the voltage vC across the capacitance andthe charge *C on the capacitance are*

i + 2V , R3e - t , CR

vR + iR + Ve - t , CR

vC + vR + Ve - t , CR

*C + CvC + CVe - t , CR

Inductance and resistanceThe time constant of an inductance ( and a resistance R is e)ual to ( , R, and represents thetime to change the current in the inductance from 0ero to E , R at a constant rate of change of current E , ( which produces an induced voltage E across the inductance/.

If a voltage E is applied to a series circuit comprising an inductance ( and a resistance R,then after time t the current i, the voltage vR across the resistance, the voltage v( across the

inductance and the magnetic lin%age ψ

( in the inductance are*i + 2E , R32 - e - tR , (3vR + iR + E2 - e - tR , (3v( + E - vR + Ee - tR , (

ψ( + (i + 2(E , R32 - e - tR , (3

If an inductance ( carrying a current I is discharged through a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage v( across the inductance and

the magnetic lin%age ψ( in the inductance are*

i + Ie - tR , (

vR + iR + IRe - tR , (

v( + vR + IRe - tR , (

ψ( + (i + (Ie - tR , (

Rise Time and Fall TimeThe rise time or fall time/ of a change is defined as the transition time between the 123 and423 levels of the total change, so for an e$ponential rise or fall/ of time constant !, the risetime or fall time/ t6-76 is*

t6-76 + ln2.4 5 ln2.1/! ≈ 6.6!

The half time of a change is defined as the transition time between the initial and 723 levelsof the total change, so for an e$ponential change of time constant !, the half time t86 is *

t86 + ln1.2 5 ln2.7/! ≈ 2.84!

+ote that for an e$ponential change of time constant !*

5 over time interval !, a rise changes by a factor - e - ≈ 2.89/ of the remaining change,

5 over time interval !, a fall changes by a factor e - ≈ 2.9:/ of the remaining change,

5 after time interval 9!, less than 73 of the total change remains,5 after time interval 7!, less than 13 of the total change remains.

Po"er

The power P dissipated by a resistance R carrying a current I with a voltage drop V is*P + V. , R + VI + I.R



imilarly, the power P dissipated by a conductance & carrying a current I with a voltage dropV is*P + V.& + VI + I. , &

The power P transferred by a capacitance C holding a changing voltage V with charge ) is*P + VI + CV2dv,dt3 + )2dv,dt3 + )2d*,dt3 , C

The power P transferred by an inductance ( carrying a changing current I with magnetic

lin%age Ψ is*

P + VI + (I2di,dt3 + Ψ2di,dt3 + Ψ2dψ ,dt3 , (

Energy

The energy % consumed over time t due to power P dissipated in a resistance R carrying acurrent I with a voltage drop V is*% + Pt + V.t , R + VIt + I.tR

imilarly, the energy % consumed over time t due to power P dissipated in a conductance & carrying a current I with a voltage drop V is*% + Pt + V.t& + VIt + I.t , &

The energy % stored in a capacitance C holding voltage V with charge ) is*% + CV. , . + )V , . + ). , .C

The energy % stored in an inductance ( carrying current I with magnetic lin%age Ψ is*

% + (I. , . + ΨI , . + Ψ

. , .(

#atteries

If a battery of open5circuit voltage E# has a loaded voltage V( when supplying load current I(,the battery internal resistance R# is*R# + 2E# - V(3 , I(

The load voltage V( and load current I( for a load resistance R( are*V( + I(R( + E# - I(R# + E#R( , 2R# 0 R(3I( + V( , R( + 2E# - V(3 , R# + E# , 2R# 0 R(3

The battery short5circuit current Isc is*Isc + E# , R# + E#I( , 2E# - V(3

Voltmeter ultiplier

The resistance RS to be connected in series with a voltmeter of full scale voltage VV and fullscale current drain IV to increase the full scale voltage to V is*RS + 2V - VV3 , IV

The power P dissipated by the resistance RS with voltage drop 2V - VV3 carrying current IV is*

P + 2V - VV3. , RS + 2V - VV3IV + IV.RS



$mmeter Shunt

The resistance RP to be connected in parallel with an ammeter of full scale current I$ and fullscale voltage drop V$ to increase the full scale current to I is*RP + V$ , 2I - I$3

The power P dissipated by the resistance RP with voltage drop V$ carrying current 2I - I$3 is*P + V$

. , RP + V$2I - I$3 + 2I - I$3.RP

%heatstone #ridge

The #heatstone ;ridge consists of two resistive potential dividers connected to a commonvoltage source. If one potential divider has resistances R and R. in series and the other

potential divider has resistances R/ and R9 in series, with R and R/ connected to one side ofthe voltage source and R. and R9 connected to the other side of the voltage source, then atthe balance point where the two resistively divided voltages are e)ual*R , R. + R/ , R9 If the value of resistance R9 is un%nown and the values of resistances R/, R. and R at thebalance point are %nown, then*R9 + R/R. , R

Electrical Circuit Formulae

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