CP2 Circuit TheoryProf. Todd Huffman todd.huffman@physics.ox.ac.uk

http://www-pnp.physics.ox.ac.uk/~huffman/

Aims of this course:Understand basic circuit components (resistors, capacitors, inductors, voltage and current sources, op-amps)

Analyse and design simple linear circuits

+

– ++

Circuit Theory: Synopsis

Basics: voltage, current, Ohm’s law…Kirchoff’s laws: mesh currents, node voltages…Thevenin and Norton’s theorem: ideal voltage and

current sources…Capacitors:Inductors:AC theory: complex notation, phasor diagrams, RC,

RL, LCR circuits, resonance, bridges…Op amps: ideal operational amplifier circuits…

Op-amps are on the exam syllabus

Stored energy, RC and RL transient circuits

Reading List• Electronics: Circuits, Amplifiers and Gates, D V Bugg, Taylor and

FrancisChapters 1-7

• Basic Electronics for Scientists and Engineers, D L Eggleston, CUPChapters 1,2,6

• Electromagnetism Principles and Applications, Lorrain and Corson, FreemanChapters 5,16,17,18

• Practical Course Electronics Manual http://www-teaching.physics.ox.ac.uk/practical_course/ElManToc.htmlChapters 1-3

• Elementary Linear Circuit Analysis, L S Bobrow, HRWChapters 1-6

• The Art of Electronics, Horowitz and Hill, CUP

Why study circuit theory?

• Foundations of electronics: analogue circuits, digital circuits, computing, communications…

• Scientific instruments: readout, measurement, data acquisition…

• Physics of electrical circuits, electromagnetism, transmission lines, particle accelerators, thunderstorms…

• Not just electrical systems, also thermal, pneumatic, hydraulic circuits, control theory

Mathematics required

• Differential equations

• Complex numbers

• Linear equations

0ILC1

dtdI

LR

dtId2

2

V(t)=V0ejωt

jXRZ ZV

I

V0–I1R1–(I1–I2)R3 = 0

(I1–I2)R3–I2R2+2 = 0

Covered by Complex Nos & ODEs / Vectors & Matrices lectures (but with fewer dimensions)

Charge, voltage, current

Potential difference: V=VA–VB

Energy to move unit charge from A to B in electric field

B

AV dsE VE

B

AQW dsE

Charge: determines strength of electromagnetic force quantised: e=1.62×10-19C [coulombs]

[volts]

Power: rate of change of work

IV

dtdQ

VdtdV

QQVdtd

dtdW

P

nAvedtdQ

I

No. electrons/unit vol

Cross-section area of conductor

Drift velocity

Charge Q=e

Current: rate of flow of charge

[amps]

[watts]

Ohm’s lawVoltage difference current

I

L

AV

AL

R

IRV

R=Resistance Ω[ohms]

ρ=Resistivity Ωm

Resistor symbols:

R

Resistivities

Silver 1.6×10-8 Ωm

Copper 1.7×10-8 Ωm

Manganin 42×10-8 Ωm

Distilled water 5.0×103 Ωm

PTFE (Teflon) ~1019 Ωm

R1

g Conductance[seimens]

conductivity[seimens/m]

1

Power dissipation by resistor: RV

RIIVP2

2

Voltage source

V0

Ideal voltage source: supplies V0 independent of current

V0Rint

Real voltage source:

Vload=V0–IRint

Rload

Rload

+–

battery cell

Constant current sourceIdeal current source: supplies I0 amps independent of voltage

Real current source:

Rload

Rload

I0

RintI0

int0 R

VII load

Symbol: or

AC and DC

DC (Direct Current): Constant voltage or current

Time independent

V=V0

AC (Alternating Current): Time dependent

Periodic

I(t)=I0sin(ωt)T2

f2

50Hz power, audio, radio…

+–

+

-

RMS values

AC Power dissipation R

VVIP

2RMS

RMSRMS

2

VV 0

RMS

Why ? 2

T

0

2RMS dttV

T1

V

Square root of mean of V(t)2

Passive Sign ConventionPassive devices ONLY - Learn it; Live it; Love it!

IRV

R=Resistance Ω[ohms]

Which way does the current flow, left or right?

Two seemingly Simple questions:

Voltage has a ‘+’ side and a ‘-’ side (you can see it on a battery)on which side should we put the ‘+’? On the left or the right?

Given V=IR, does it matter which sides for V or whichdirection for I?

Kirchoff’s LawsI Kirchoff’s current law: Sum of all currents at a

node is zero

I1

I3

I2

I4

I1+I2–I3–I4=0

0In

(conservation of charge)

Here is a cute trick:It does not matter whether you pick “entering”or “leaving” currents as positive.

BUT keep the same convention for all currents on one node!

II Kirchoff’s voltage law: Around a closed loop the net change of potential is zero(Conservation of Energy)

V0

I

R1

R2

R3

0Vn

5V

3kΩ

4kΩ

Calculate the voltage across R2

1kΩ

Kirchoff’s voltage law:

V0

I 1kW

3kW

4kW

-V0+IR1+IR2+IR3=0

0Vn

+

+

+

+

–

–

–

–

–V0

+IR1

+IR2

+IR3

5V=I(1+3+4)kΩ

mA625.08000

V5I

VR2=0.625mA×3kΩ=1.9V

Series / parallel circuitsR1 R2 R3

Resistors in series: RTotal=R1+R2+R3…

n

nT RR

R1 R2 R3

Resistors in parallel

321

n nT

R1

R1

R1

R1

R1

Two parallel resistors: 21

21T RR

RRR

Potential divider

R1

R2

V0

21

20

RRRV

USE PASSIVE SIGN CONVENTION!!!Show on blackboard

Mesh currents

9V 2V+ +

R1 R2

R3

I1 I2I3

I1 I2

R1=3kΩR2=2kΩR3=6kΩ

Second job: USE PASSIVE SIGN CONVENTION!!!

First job: Label loop currents in all interior loops

++

+

Third job: Apply KCL to elements that share loop currents

Define: Currents Entering Node are positive

I1–I2–I3 = 0 I3 = I1-I2

Mesh currents

9V 2V+ +

R1 R2

R3

I1 I2I3

I1 I2

R1=3kΩR2=2kΩR3=6kΩ

Last job: USE Ohm’s law and solve equations.

Fourth job: Apply Kirchoff’s Voltage law around each loop.

++

+

Mesh currents

9V 2V+ +

R1 R2

R3

I1 I2I3

I1 I2

R1=3kΩR2=2kΩR3=6kΩ

-9V+I1R1+I3R3 = 0 I3 =I1–I2

–I3R3+I2R2+2V = 0

9V/kΩ = 9I1–6I2

-2V/kΩ = -6I1+8I2

I2=1 mA VIRV

mAmAI

43

2I 35

333

31

++

+Solve simultaneous equations

Node voltages

9V 2V+

+

R1 R2

R3

I1 I2I3

R1=3kΩR2=2kΩR3=6kΩ

VX

0V

-

-

+

-

Step 1: Choose a ground node!

Step 2: Label voltages on all unlabled nodes

Step 3: Apply KCL and ohms law using the tricks

Node voltages

9V 2V+

+

R1 R2

R3

I1 I2I3

R1=3kΩR2=2kΩR3=6kΩ

VX

0V

132

132

9)2(0

0

R

VV

R

V

R

VV

III

XXX

USE PASSIVE SIGN CONVENTION!!!

-

-

+

-

Only one equation,Mesh analysis would give two.

All currents leave all labeled nodesAnd apply DV/R to each current.

V2V mA2k1V

mA2k1

31

61

21

V

mA2mA1mA3RV2

RV9

R1

R1

R1

V

XX

X

21132X

mA 3

1

6

2

2mA 2

22mA

3

7

3

92

3

21

k

VI

k

VVI

k

VVI

2V9V+

+

R1 R2

R3

I1 I2I3

VX

0V132

132

9)2(0

0

R

VV

R

V

R

VV

III

XXX

Thevenin’s theorem

VeqReq

Any linear network of voltage/current sources and resistors

Equivalent circuit

In Practice, to find Veq, Req…

RL (open circuit) IL0 Veq=VOS

RL0 (short circuit) VL0SS

OSeq I

VR

V0

+R1

RL

I1

Iss

R2

I2 VL

21

20 RR

RVVos

21

21

1

0

21

20

eq RRRR

RV

RRR

VR

R2

I2

Req = resistance between terminals when all voltages sources shorted – Warning! This is not always obvious!

1

0

R

VI ss

Norton’s theorem

Req

Any linear network of voltage/current sources and resistors

Equivalent circuit

I0

V0R1

ReqeqIRload Rload

VL=V0–ILR1

LLeqeqeq

Leq

LLReq

VIRIR

IRV

III

1eq

eq

0eq

RR

RV

I

V0R

R RV0

IR IL

using Thevenin’s theorem:VOC =Veq= 2.0V

2V+

+

R1=3kΩ R2=2kΩ

R3=6kΩ

A

B

9V

2V+

+

A

B

V9

R2=2kΩma

k

VI 3

3

91

kR 31

kI

VR

maIII

mak

VI

SS

OCeq

SS

0.1

2

12

2

21

2

+

A

B

2.0V

1.0kΩ

Iss

I1 I2

using Norton’s theorem:

2V+

+

R1=3kΩ R2=2kΩ

R3=6kΩ

A

B

9V 1kΩ2mA

A

B

Same procedure: Find ISS and VOC

IEQ = ISS and REQ = VOC/ISS

Superposition

9V 2V+

+

R1 R2

R3

I1 I2I3

9V+

R1R2

R3

IA IBIC

2V+

R1 R2

R3

IE IFIG

=

+

I1=IA+IE I2=IB+IF I3=IC+IG

Important: label Everything the same directions!

9V+

R1R2

R3

IA IBIC

9V+

R1

R2R3

IA

IBIC

k5.1

RRRR

32

32

R1=3kΩR2=2kΩR3=6kΩ

k5.4

RRRR

R32

321

mA2k5.4V9

I A

mA5.1I mA5.0I

V35.45.1

V9RIRI

BC

2B3C

Example:Superposition

2V+

R1 R2

R3

IE IFIG

2V+

R2

R3

IF

IEIG

R1

R1=3kΩR2=2kΩR3=6kΩ

k2RR

RR

31

31

k4

RRRR

R31

312

mA5.0k4V2

IF

mA

31

I mA61

I

V142

V2RIRI

EG

1E3G

9V 2V+

+

R1 R2

R3

I1 I2I3

9V+

R1R2

R3

IA IBIC

2V+

R1 R2

R3

IE IFIG

=

+

I1=IA+IE I2=IB+IF I3=IC+IG

mA37

mA31

2I1

mA2

mA5.05.1I2

mA31

mA61

5.0I3

Matching: maximum power transfer

V0Rin

Rload

L2

Lin

2L2

0L

2L

R1

RR

RV

RV

P

Find RL to give maximum power in load

0R2RR

0RR

RRR2RRV

dRdP

LLin

4Lin

LinL2

Lin20

L

Lin RR

Maximum power transfer when RL=Rin

Note – power dissipated half in RL and half in Rin

Circuits have Consequences

• Problem: – My old speakers are 60W

speakers. – Special 2-4-1 deal at El-

Cheap-0 Acoustics on 120W speakers!! • (“Offer not seen on TV!”)

– Do I buy them?

• Depends! 4W, 8W, or 16W speakers?

• Why does this matter?

And Now for Something Completely Different