CP2 Circuit Theory

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

• Problem set Thanks to • Synopsis and reading list Todd Huffman • 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. 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: Stored energy, RC, RL and LCR • Inductors: transient circuits. • AC theory: complex notation, phasor diagrams, RC, RL, LCR circuits, resonance, bridges… • Op amps: ideal operational amplifier circuits… Mathematics required

• Differential equations d2I R dI 1   I  0 dt2 L dt LC

jωt • Complex numbers V(t)=V0e V I  Z  R  jX Z • Linear equations

V0–I1R1–(I1–I2)R3 = 0

(I1–I2)R3–I2R2+2 = 0

Covered by Complex Nos & ODEs / Vectors & Matrices lectures Charge, voltage, current, power

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

[volts] Potential difference: V=VB–VA Energy to move unit charge from A to B in electric field B V   E  ds E  V A B W   QE  ds A Charge Q=e dQ Current: rate of flow of charge I   nAve dt

No. electrons/unit vol Drift velocity Cross-section area of conductor [amps] Power: work done per unit time

dW d QV P = = = IV dt dt

[watts = J/s] Ohm’s law

Voltage difference  current Resistor symbols:

L I R

A V

V  IR R=Resistance Ω[ohms] L R  ρ=Resistivity Ωm A 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

1 1 Conductance g  conductivity   [seimens] R [seimens/m] 

V2 Power dissipation by resistor: P  IV  I2R  R Voltage source

Ideal voltage source: supplies V0 independent of current V0 + – Rload

V0 V0 + Symbol: + or – Constant current source

Ideal current source: supplies I0 amps independent of voltage R I0 load

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 I 2 I1+I2–I3–I4=0 KCL In  0 I 3 I 4 (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! II Kirchoff’s voltage law: Around a closed loop the net change of potential is zero (Conservation of Energy)

V0 R V 0 2  n 

KVL

But what about the signs of Vn? Passive Sign Convention

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)? V  IR 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.

Power conservation ෍ 퐼푛푉푛 = 0 II Kirchoff’s voltage law: Around a closed loop the net change of potential is zero (Conservation of Energy)

I 1kΩ

V0 R V 0 2kΩ 2  n  5V 2kΩ

Calculate the voltage R3 across R2

Show on blackboard Passive Sign Convention

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

Kirchoff’s voltage law:  Vn  0

-V0+IR1+IR2+IR3=0

+IR1 5V=I(1+2+2)kΩ + – I=1mA

I 1kW + + –V0 2kW +IR – 2 V0 – 2kW

– + VR2=1mA×2kΩ=2V

+IR3 Series / parallel circuits

Show on blackboard Series / parallel circuits

R1 R2 R3 R T  Rn n

Resistors in series: RTotal=R1+R2+R3…

Resistors in parallel 1 1 R1 R2 R3   R T n Rn 1 1 1     R1 R2 R3

Two parallel R1R2 R T  resistors: R1  R2 Potential divider

V0R2 R2 R1  R2

USE PASSIVE SIGN CONVENTION!!! Show on blackboard Voltage source

Ideal voltage source: supplies V0 independent of current V0 + – Rload

Real voltage source: Rint V0 푉load = 푉0 − 퐼푅int Rload 푅푙표푎푑 푉load = 푉0 × 푅load + 푅int Constant current source

Ideal current source: supplies I0 amps independent of voltage R I0 load

Symbol: or

Real current source: 푉 Rint Rload I0 퐼load = 퐼0 − 푅int

푅int 퐼load = 퐼0 × 푅load + 푅int End of Lecture 1