Capacitance/Inductance and RC Op
Amp Circuits
Kevin D. Donohue, University of Kentucky
1
Analogous to a spring storing the energy used to compress
it or a flywheel storing the energy used rotate it …
Ø
a capacitor can store energy in an electric field from the voltage
used to move charge into it.
i(t)
t
q = Cv , i = C
dv
1
, v(t ) = v(t 0 ) +
dt
C
ò
i ( x)dx , p (t ) = Cv
dv
1
, w(t ) = Cv 2
dt
2
C
-
t0
Ø
+
v(t)
An inductor can store the energy in a magnetic field from the
current used to create lines of flux around it.
i(t)
t
1
di
λ = Li , v = L , i (t ) = i (t 0 ) +
dt
L
ò
1
di
v( x)dx , p(t ) = Li , w(t ) = Li 2
2
dt
t0
Kevin D. Donohue, University of Kentucky
L
+
v(t)
2
Solve for voltages, currents, charge, power,
and energy in simple circuits containing
inductors and capacitors.
Kevin D. Donohue, University of Kentucky
3
Ø
Ø
Ø
Ø
What happens if current changes instantaneously
in an ideal inductor?
What happens if voltage changes instantaneously
in a ideal capacitor?
What would be an equivalent model for an ideal
inductor in a DC circuit?
What would be an equivalent model for an ideal
capacitor in a DC circuit?
Kevin D. Donohue, University of Kentucky
4
Ø
Ø
A small amount of current leaks through
the dielectric in an actual capacitor. A
practical model can be constructed from
2 ideal lumped-parameter models
The coils used to construct an inductor
may have a significant resistance
component. A practical model can be
constructed from 2 ideal lumpedparameter models
Kevin D. Donohue, University of Kentucky
C
Rleak
L
Rleak
5
Ø
Series capacitors can be combined according to
the following formula:
C1
C2
…
CN
⇔
Ceq =
Ceq
1
N
å
k =1
Ø
1
Ck
Parallel capacitors can be combined according to
the following formula:
N
…
C1
C2
CN
⇔
Ceq
Ceq =
Kevin D. Donohue, University of Kentucky
åC
k
k =1
6
Ø
Series inductors can be combined according to the
following formula:
N
…
L1
Ø
L2
LN
⇔
Leq
Leq =
åL
k
k =1
Parallel inductors can be combined according to
the following formula:
1
Leq =
…
L1
L2
LN
⇔
Leq
N
å
k =1
Kevin D. Donohue, University of Kentucky
1
Lk
7
Simplify circuits with series and parallel
combinations of inductor and capacitors.
Kevin D. Donohue, University of Kentucky
8
Show that this circuit integrates the input signal vs(t)
according to the equation below for time greater than 0:
t
C
1
vo (t ) = vo (0) −
RC
R
+
vs(t)
ò
v s ( x) dx
0
vo(t)
-
Kevin D. Donohue, University of Kentucky
9
Show that this circuit differentiates the input signal vs(t)
according to the equation below:
R
dv s (t )
vo (t ) = − RC
dt
C
+
vs(t)
vo(t)
-
Kevin D. Donohue, University of Kentucky
10