Resistance

Review of Resistors

• The resistance is an intrinsic property of a material which impedes the flow of charge requiring a pd to be applied so that there can be current flow.

Review of Resistors

• The resistance is an intrinsic property of a material which impedes the flow of charge requiring a pd to be applied so that there can be current flow.

• From ohm’s law, the resistance of a device is the ratio of the potential difference across it to the current flowing through it.

I

VR

• The unit of the resistor is the ohm ( ).

RC Circuits

RC Circuits

• The current in the previous circuits are time independent once the emf of the source is time independent.

RC Circuits

• The current in the previous circuits are time independent once the emf of the source is time independent.

• However we may have circuits which are time dependent.

• An example is an RC circuit.

• A RC circuit consists of a resistor R connected in series with a capacitor C.

• The following circuit can be use the test the charging and discharging of the capacitor through the resistor.

• Consider charging:

• Consider charging:•

• Initially the capacitor is uncharged.

• Consider charging:•

• Initially the capacitor is uncharged.

• When in the charging position current flows and the capacitor charges.

• From Kirchoff’s law: crbat VVV

• Which can be written as: C

tqRtIVbat

• Which can be written as:

• Since

• We can rewrite the equation as,

C

tqRtIVbat

dt

dqtI

C

tqR

dt

dqVbat

• Which can be written as:

• Since

• We can rewrite the equation as,

• Doing some algebra,

C

tqRtIVbat

dt

dqtI

C

tqR

dt

dqVbat

CVqRCdt

dqbat

• Which can be written as:

• Since

• We can rewrite the equation as,

• Doing some algebra,

• We must separate the variables so that we can integrate and find the final charge on the capacitor.

C

tqRtIVbat

dt

dqtI

C

tqR

dt

dqVbat

CVqRCdt

dqbat

• Separating variables,

dtRCCVq

dqbat

11

• Separating variables,

• Integrating, dt

RCCVqdq

bat

11

dtRC

dqCVq

tq

bat

00

11

• Separating variables,

• Integrating, dt

RCCVqdq

bat

11

dtRC

dqCVq

tq

bat

00

11t

dtRC 0

1

• Separating variables,

• Integrating,

• Which gives,

dtRCCVq

dqbat

11

dtRC

dqCVq

tq

bat

00

11t

dtRC 0

1

RC

t

CV

CVq

bat

bat

ln

• Taking the antilog and simplifying we get,

RC

t

bat eCVtq 1

• Taking the antilog and simplifying we get,

RC

t

bat eCVtq 1

VbatCq(t)

t

• The product RC in the previous equation is called the time constant.

• Has units of time.

• Time taken for the charge to increase from zero to 63% of its final value.

RC

• The pd across the capacitor

• Which gives

RC

t

batc eVV 1

C

tqVc

VbatVc

t

• The current for the charging

• Which gives RC

tbat eR

VI

Vbat/RI(t)

t

dt

dqI

• Consider discharging:

• Consider discharging:

• For the discharge position, the battery is no longer in the circuit.

crbat VVV 0

0

C

tqRtI

0

C

tqRtI

• Since

• We can write that

0

C

tqRtI

dt

dqtI

0

C

tqR

dt

tdq

• Since

• We can write that

• Separating variables,

0

C

tqRtI

dt

dqtI

0

C

tqR

dt

tdq

• Since

• We can write that

• Separating variables,

• Which in separated form is,

0

C

tqRtI

dt

dqtI

0

C

tqR

dt

tdq

RC

tq

dt

tdq

dt

RCq

tdq 1

• Integrating,

tq

q

dtRCq

tdq

0

1

0

• Integrating,

• We get

• Which after simplification is,

tq

q

dtRCq

tdq

0

1

0

t

qq RC

tq

00

ln

RC

t

eqq

0

• This can be written as, , noting that the initial charge is CVbat.

RC

t

bat eCVq

• This can be written as, , noting that the initial charge is CVbat.

• Differentiating gives the current,

• The voltage across the capacitor is,

RC

t

bat eCVq

RC

t

eRC

q

dt

dqi

0 RC

tbat eR

V

C

tqtVc RC

t

bateV

• Limiting conditions:

1. At t=0, q= CVbat.

2. At t=inf, q= 0.

RC

t

bat eCVq

CVbat

q

t

Vbat

R

Vbat

I(t)

t

t

Power, Energy

Power

• The net rate of energy transfer from the source (battery) P is given by,

• Power is in watts(W) or joules/second

• The rate at which energy is dissipated through through the resistor is,

• The energy lost is in the form of thermal energy.

• The power supplied to the capacitor is,

batIVP

Rr IVP RI 2

CC IVP

Energy

• The total energy supplied by the battery in a time t is given by,

• The total energy dissipated in a time t,

• The total energy supplied to the capacitor in time t,

ft

batbat dtVtIE0

ft

RR dtVtIE0

ft

CC dtVtIE0

Energy

• From the conservation of energy,

CRbat EEE

Resistance in Series and Parallel

• Series:

• From the conservation of energy,321 VVVVV bat

• From the conservation of energy,

• where,

321 VVVVV bat

11 IRV 22, IRV 33, IRV

• From the conservation of energy,

• where,

321 IRIRIRVbat

11 IRV 22, IRV 33, IRV

321 VVVVV bat

eqIR

• From the conservation of energy,

• where,

321 IRIRIRVbat

11 IRV 22, IRV 33, IRV

321 VVVVV bat

eqIR

321 RRRReq

• In general,

n

i

ieq RR1

• Parallel:

• From the conservation of charge,

321 IIII

• From the conservation of charge,

• where,

321 IIII

11 R

VI

22, R

VI

33, R

VI

• From the conservation of charge,

• where,

321 IIII

11 R

VI

22, R

VI

33, R

VI

321 R

V

R

V

R

VI

• From the conservation of charge,

• where,

321 IIII

11 R

VI

22, R

VI

33, R

VI

321 R

V

R

V

R

VI

eq

bat

R

V

eqR

V

• From the conservation of charge,

• where,

321 IIII

11 R

VI

22, R

VI

33, R

VI

321 R

V

R

V

R

VI

eq

bat

R

V

eqR

V

321

1111

RRRReq

• In general,

n

i ieq RR1

11