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CP2 Circuit TheoryRob Smith [email protected]
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
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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
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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)
+
– +
+
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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.
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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
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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]
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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
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Ohm’s law
Voltage difference current
I
L
AV
A
LR
IRV
R=Resistance Ω[ohms]
ρ=Resistivity Ωm
Resistor symbols:
R
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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
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Voltage source
V0
Ideal voltage source: supplies V0
independent of currentRload
+–
+–
V0
Symbol: or
V0 +
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Constant current source
Ideal current source: supplies I0 amps independent of voltage
RloadI0
Symbol: or
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Circuits
Out of these components we can make (arbitrarily complicated) circuits:
But how do we work out what they do…
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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
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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
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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!
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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
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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
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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!
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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
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Series / parallel circuits
Show on blackboard
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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
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Potential divider
R1
R2
V0
21
20
RR
RV
USE PASSIVE SIGN CONVENTION!!!
Show on blackboard
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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
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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
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End of Lecture 1