CP2 Circuit Theory
Revision Lecture
Basics,
Kirchoff’s
laws,
Thevenin
and Norton’s
theorem, Capacitors, Inductors
•
June 2004 q7
•
September 2010 q7
AC
theory, complex notation, LCR circuits
•
September 2005 q8
•
June 2002 q7
Op

amps
Sam Henry (Tony Weidberg
) www.physics.ox.ac.uk/users/weidberg/teaching
V
0
I
R
1
R
2
R
3
V
0
–
IR
1
–
IR
2
–
IR
3
=0
+
+
+
+
–
–
–
–
–
V
0
+IR
1
+IR
2
+IR
1
I
1
I
3
I
2
I
4
I
1
+I
2
–
I
3
–
I
4
=0
Kirchoff’s laws
V
eq
R
eq
R
eq
Thevenin
and Norton theorem
In Practice, to find V
eq
, R
eq
…
R
L
(open circuit) I
L
0 V
eq
=V
L
R
L
0 (short circuit) V
L
0
R
eq
= resistance between terminals
when all voltages sources shorted
•
Capacitors
C
+Q

Q
Q = CV
Capacitors resist change in voltage
•
Capacitors in series
•
Capacitors in parallel
•
Stored energy
stored in form of electric field
C
1
C
2
C
N
•
Inductors
•
Inductors in series
•
Inductors in parallel
•
Stored energy
stored in form of magnetic field
L
“back
emf”
Self

inductance
Inductors resist a
change of current
L
1
L
2
+
V
0
R
+
V
0
R
L
C
RC and RL circuits
CP2 June 2004
Check
Energy conservation:
Total energy dissipated
by resistor = Energy
delivered by source
CP2 September 2010
AC circuit theory
•
Voltage represented by complex exponential
•
Impedance relates current and voltage V=ZI
in complex notation:
Resistance
R
Inductance
j
L
Capacitance
1/j
C
and combinations thereof
•
Impedance has magnitude and phase
represented by real component of
easily shown from
Q=VC
•
Current is given by
•
So Z gives the ratio of magnitudes of V and I, and
give the
phase difference by which current lags voltage
CP2 September 2005
The complex impedance is given by the voltage in a circuit
divided by the current when both are expressed in complex
form. The magnitude of the complex impedance is the ratio
of the magnitudes of the voltage and current; the phase is
the phase lag between the voltage and current.
Physics 3 June 2002
An oscillating voltage of frequency
ω
, V(t)=V
0
cos(
ω
t) is represented
in complex notation by V
0
e
j
ω
t
. The current in the circuit is then given
by I=V/Z, where Z is the complex impedance Z=Z
e
j
φ
, where Z is
the ratio of the magnitude of the voltage to the magnitude of the
current, and
φ
is the phase by which the current leads the voltage.
The real component of Z is the resistance, and the imaginary
component is the reactance
where C the capacitance and
L is the inductance of the circuit. Complex impedances allow circuits
when inductance and capacitance to be analysed using linear circuit
theory in the same way as resistance
The sum of these peak voltages
is not equal to V
0
as V
R
, V
C
and
V
L
all have different phases.
When the complex voltages are
summed V
R
+V
C
+V
L
=V
0
LCR circuit
overdamped solution
critically damped solution
underdamped (oscillatory) solution
+
V
0
R
C
L
I(t)
Op

amps
Gain is very large (A
)
Inputs draw no current (Z
IN
=
)
Feedback
v
+
=v
–
V
OUT
+
–
v
+
v
–
V
IN
R
1
R
2
Non

Inverting
Amplifier Circuit
+
V
OUT
V
IN
R
1
Inverting Amplifier Circuit
R
2
–
v
–
v
+
i
i
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