CP2 Circuit TheoryRob Smith robert.smith@physics.ox.ac.uk

https://www2.physics.ox.ac.uk/contacts/people/robertsmith (‘Teaching’ tab):

• Problem set

• Synopsis and reading list

• Lecture summaries

• Slides

But do make your own notes because: (i) it is helpful for you to learn, (ii) I will say extra things, (iii) I will do some stuff on the blackboard.

Thanks to Todd Huffman

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, vacuum, control theory

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

Analyse and design simple linear circuits (considering both DC, AC and transient response)

+

– +

+

Circuit Theory: Synopsis

• Basics: voltage, current, Ohm’s law, ideal voltage and current sources…

• Kirchoff’s laws and tricks for solving: mesh currents, node voltage, Thevenin and Norton’s theorem, superposition…

• Capacitors:

• Inductors:

• AC theory: complex notation, phasor diagrams, RC, RL, LCR circuits, resonance, bridges…

• Op amps: ideal operational amplifier circuits…

Stored energy, RC, RL and LCR transient circuits.

Mathematics required

• Differential equations

• Complex numbers

• Linear equations

0ILC

1

dt

dI

L

R

dt

Id2

2

V(t)=V0ejωt

jXRZ Z

VI

V0–I1R1–(I1–I2)R3 = 0

(I1–I2)R3–I2R2+2 = 0

Covered by Complex Nos & ODEs / Vectors & Matrices lectures

Charge, voltage, current, power

Potential difference: V=VB–VA

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

B

AV dsE VE

B

AQW dsE

Charge: determines strength of electromagnetic forcequantised: e=1.602×10-19C [coulombs]

[volts]

Power: work done per unit time

nAvedt

dQI

No. electrons/unit vol

Cross-section area of conductor

Drift velocity

Charge Q=e

Current: rate of flow of charge

[amps]

[watts = J/s]

P =dW

dt=d QV

dt= IV

Ohm’s law

Voltage difference current

I

L

AV

A

LR

IRV

R=Resistance Ω[ohms]

ρ=Resistivity Ωm

Resistor symbols:

R

Resistivities

Silver 1.6×10-8 Ωm

Copper 1.7×10-8 Ωm

Manganese 144×10-8 Ωm

Distilled water 5.0×103 Ωm

PTFE (Teflon) ~1019 Ωm

R

1g Conductance

[seimens]conductivity[seimens/m]

1

Power dissipation by resistor:R

VRIIVP

22

Voltage source

V0

Ideal voltage source: supplies V0

independent of currentRload

+–

+–

V0

Symbol: or

V0 +

Constant current source

Ideal current source: supplies I0 amps independent of voltage

RloadI0

Symbol: or

Circuits

Out of these components we can make (arbitrarily complicated) circuits:

But how do we work out what they do…

Kirchoff’s Laws

I 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)

It does not matter whether you pick “entering”or “leaving” currents as positive.

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

KCL

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

But what about the signs of Vn?

KVL

Passive Sign Convention

IRV

V0 + Sources have a + sign on the terminal the current normally leaves

Where do we put the + sign on a resistor (or other passive component)?

Procedure

• Choose direction of current you are defining as positive.

• For any passive component make a + sign on the side of that component that the current is entering.

• When applying KVL, as you go round a loop a – to + component has a negative voltage and a + to –component has a positive voltage.

Learn it; Live it; Love it!

Passive Sign Convention

The ‘convention’ is related to how power input/output from a circuit is defined:

• Power flowing out of a circuit into an electrical component is defined as positive.

• Power flowing into a circuit from an electrical component is defined as negative.

𝐼𝑛𝑉𝑛 = 0Power conservation

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

2kΩ

2kΩ

Calculate the voltage across R2

1kΩ

Show on blackboard

Passive Sign Convention

IRV

V0 + Sources have a + sign on the terminal the current normally leaves

Where do we put the + sign on a resistor (or other passive component)?

Procedure

• Choose direction of current you are defining as positive.

• For any passive component make a + sign on the side of that component that the current is entering.

• When applying KVL, as you go round a loop a – to + component has a negative voltage and a + to –component has a positive voltage.

Learn it; Live it; Love it!

Kirchoff’s voltage law:

V0

I 1kW

2kW

2kW

-V0+IR1+IR2+IR3=0

0Vn

+

+

+

+

–

–

–

–

–V0

+IR1

+IR2

+IR3

5V=I(1+2+2)kΩ

VR2=1mA×2kΩ=2V

I=1mA

Series / parallel circuits

Show on blackboard

Series / parallel circuits

R1 R2 R3

Resistors in series: RTotal=R1+R2+R3…

n

nT RR

R1 R2 R3

Resistors in parallel

321

n nT

R

1

R

1

R

1

R

1

R

1

Two parallel resistors: 21

21T

RR

RRR

Potential divider

R1

R2

V0

21

20

RR

RV

USE PASSIVE SIGN CONVENTION!!!

Show on blackboard

Voltage source

V0

Ideal voltage source: supplies V0

independent of current

Real voltage source:

Rload

V0Rint

Rload

+–

𝑉load = 𝑉0 ×𝑅𝑙𝑜𝑎𝑑

𝑅load + 𝑅int

𝑉load = 𝑉0 − 𝐼𝑅int

Constant current source

Ideal current source: supplies I0 amps independent of voltage

Real current source:

RloadI0

RloadRintI0

Symbol: or

𝐼load = 𝐼0 −𝑉

𝑅int

𝐼load = 𝐼0 ×𝑅int

𝑅load + 𝑅int

End of Lecture 1