p212c27: 1

Direct Current Circuits

Two Basic Principles:Conservation of ChargeConservation of Energy

Resistance Networks

V IR

RV

I

ab eq

eqab

Req

Ia

b

p212c27: 2

Resistors in seriesConservation of Charge

I = I1 = I2 = I3

Conservation of Energy Vab = V1 + V2 + V3

RV

I

V V V

IV

I

V

I

V

I

V

I

V

I

V

I

R R R R

eqab

eq

1 2 3

1 2 3 1

1

2

2

3

3

1 2 3

a

b

I

R1

V1=I1R1

R2

V2=I2R2

R3

V3=I3R3

Voltage Divider:V

V

R

Rab eq

1 1

p212c27: 3

Resistors in parallelConservation of Charge

I = I1 + I2 + I3

Conservation of Energy Vab = V1 = V2 = V3

1

1 1 1 1

1 2 3

1 2 3 1

1

2

2

3

3

1 2 3

R

I

V

I I I

V

I

V

I

V

I

V

I

V

I

V

I

V

R R R R

eq ab ab

ab ab ab

eq

aI

R1

V1=I1R1

R2

V2=I2R2

R3

V3=I3R3

b

Current Divider:I

I

R

Req1

1

p212c27: 4

R1=4

R2=3 R3=6

=18V

Example: Determine the equivalent resistance of the circuit as shown.Determine the voltage across and current through each resistor.Determine the power dissipated in each resistorDetermine the power delivered by the battery

p212c27: 5

Kirchoff’s Rules

Some circuits cannot be represented in terms of series and parallel combinations

1

2

R1

R2

R3

R1

R3

R2

R4

R5

Kirchoff’s rules are based upon conservation of energy and conservation of charge.

p212c27: 6

I1 I2

I3

Conservation of Charge: Junction ruleThe algebraic sum of the currents into any

junction is zero.

I = 0 = + I1 I2 I3

I = 0 = I1 I2 I3

R1

R2

R3

Alternatively: The sum of the currents into a junction is equal to the sum of the currents out of that junction.

I1 I2 =I3

p212c27: 7

Conservation of Energy: Loop ruleThe algebraic sum of the potential

differences around any closed loop is zero.

I1

I2

I3

1

2

R1

R2

R3

p212c27: 8

Potential Differences in the direction of travel

+

direction of travel

V =

+

direction of travel

V = +

I

+

direction of travel

V =IR

I

+

direction of travel

V = + IR

p212c27: 9

I1

I2

I3

1

2 R2

R1

R3

I1 I2 I3 =01 + I1 R1 I2 R2 + 2 =0 I3 R3 I2 R2 + 2 =0 I3 R3 I1 R11 =0

p212c27: 10

I1

I2

I3

1 =12V

2 =5V R2 =1

R1 =2

R3 =3

I1 I2 I3 =012 I1 2I2 1 + 5 =0 I3 3 I2 1 + 5 =0

p212c27: 11

Standard linear form:

I1 1 I2 1 I3 = 0 2 I1 1 I2 0 I3 = 7 0 I1 1 I2 3 I3 = 5

I1 I2 I3 =012 I1 2I2 1 + 5 =0 I3 3 I2 1 + 5 =0

TI-89: [2nd] [MATH] [4](selects matrix) [5] (selects simult)

Entry line:

simult([1,1,-1;2,-1,0;0,1,3],[0;7;5])

I1= 3

I2= 1

I3= 2

2

1

3

5

7

0

,

310

012

111

simult

p212c27: 12

R1 R2 R3

I 3A (-)1A 2AVP

P1

P2

I1

I2

I3

1 =12V

2 =5V R2 =1

R1 =2

R3 =3

p212c27: 13

Electrical InstrumentsGalvanometer:

Torque proportional to current IRestoring torque proportional to angle=> (Equilibrium) angle proportional to angle

Maximum Deflection =“Full Scale Deflection”Ifs = Current to produce Full Scale DeflectionRc = Resistance (of coil)Vfs = Ifs Rc

example: Ifs = 1.0 mA, Rc = 20.0 Vfs = 20 mV = .020 V

G

p212c27: 14

Wheatstone Bridge

Circuit Balanced: Vab = 0

R2

b

R1

R3 R4

a G

4

2

3

1

44

22

33

11

4

2

3

1

4321

4231

0

R

R

R

R

RI

RI

RI

RI

V

V

V

V

VVVV

IIII

IG

p212c27: 15

AmmeterMeasures current through device

Measure currents larger than Ifs by bypassing the galvanometer with another resistor (shunt Resistor Rs).

Ia = ammeter full scale

When I = Ia , we need Ic = Ifs VG = Vsh => Ifs Rc = (Ia -Ifs)Rsh

G

Rsh

I

Ic

I-Ic

afs I

I

Design a 50 mA ammeter from example galvanometerIfs = 1.0 mA, Rc = 20.0 Vfs = 20 mV = .020 V

I

p212c27: 16

VoltmeterMeasures potential difference across device

Measure V larger than Vfs by adding a resistor (Rs) in series to the galvanometer.

Vm = Voltmeter full scaleWhen V = Vm , we need Ic = Ifs

Ic = Is , Vm = Vs + VG => Vm = Ifs Rs + Ifs Rc

Rs = (Vm / Ifs )- Rc

GRs

I

V

fs fs m

I

I

V

V

V

Design a 10 V voltmeter from example galvanometerIfs = 1.0 mA, Rc = 20.0 Vfs = 20 mV = .020 V

p212c27: 17

Ohmmeter

Measures resistance across terminalsMeter supplies an EMF, resistance is determined

by response.

When R = 0 , we need Ic = Ifs = (Ifs Rs + Ifs Rc)=> Vm = Ifs Rs + Ifs Rc

Rs = (E / Ifs )- Rc

GRs

I

R

fs fs

s c

s c

II

R RR R R

R

p212c27: 18

p212c27: 19

p212c27: 20

Resistance Capacitance Circuits

Charging Capacitor

i

RC

switch

+q -q

vc = q/C vR = iR

Capacitor is initially uncharged.Switch is closed at t=0 Apply Loop rule:

-q/C-iR=0

p212c27: 21

tRCC

Ctq

uduu

CqudtRCCq

dq

dtRCCq

dq

CqRCdt

dqdtdq

RCq

Ri

iRCq

ttq

1))((ln

ln1

,1

)(

1)(

)(1

=-

0=--

0

)(

0

p212c27: 22

i

RC

switch

+q -q

vc = q/C vR = iR

//

/

/

/

)1

(

"ntTimeConsta"

)1(

)1()(

1))((ln

to

RCt

RCt

tfinal

RCt

eIeR

eRC

Cdtdq

i

RC

eQ

eCtq

tRCC

Ctq

0 0.2 0.4 0.6 0.8 1

t

q

0 0.2 0.4 0.6 0.8 1

t

i

p212c27: 23

Resistance Capacitance Circuits

Discharging Capacitor

i

RC

switch

+q -q

vc = q/C vR = iR

Capacitor has an initial charge Qo.Switch is closed at t=0 Apply Loop rule:

q/C-iR=0

p212c27: 24

tRCQ

tq

dtRCq

dq

dtRCq

dq

qRCdt

dqdt

dq

RC

qi

iRC

q

o

ttq

Qo

1)(ln

1

1

1

=

0=

0

)(

p212c27: 25

/

/

/

/

Constant" Time" same

)(

1)(ln

to

RCto

to

RCto

o

eI

eRC

Q

dt

dqi

RC

eQ

eQtq

tRCQ

tq

0 0.2 0.4 0.6 0.8 1

t

q

i

RC

switch

+q -q

vc = q/C vR = iR

0 0.2 0.4 0.6 0.8 1

t

i