Download Basic EE Definitions, Units, and Conventions

Energy Storage Elements
Capacitance/Inductance and RC Op
Amp Circuits
Kevin D. Donohue, University of Kentucky
1
Energy Storage in Electric Circuits
Analogous to a spring storing the energy used to compress
it or a flywheel storing the energy used rotate it …

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(t0 ) 
dt
C

i ( x)dx , p(t )  Cv
dv
1
, w(t )  Cv 2
dt
2
C
-
t0

+
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
di
1
  Li , v  L , i (t )  i (t0 ) 
dt
L

di
1
v( x)dx , p(t )  Li , w(t )  Li 2
dt
2
t0
Kevin D. Donohue, University of Kentucky
L
+
v(t)
2
Examples
Solve for voltages, currents, charge, power,
and energy in simple circuits containing
inductors and capacitors.
Kevin D. Donohue, University of Kentucky
3
Ideal and Practical Models




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
Ideal and Practical Models


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
Capacitor Combinations

Series capacitors can be combined according to
the following formula:
C1
C2
…
CN

Ceq 
Ceq
1
N

k 1

1
Ck
Parallel capacitors can be combined according to
the following formula:
N
…
C1
C2
CN

Ceq
Ceq 
Kevin D. Donohue, University of Kentucky
C
k
k 1
6
Inductor Combinations

Series inductors can be combined according to the
following formula:
N
…
L1

L2
LN

Leq
Leq 
L
k
k 1
Parallel inductors can be combined according to
the following formula:
1
Leq 
…
L1
L2
LN

Leq
N

k 1
Kevin D. Donohue, University of Kentucky
1
Lk
7
Examples
Simplify circuits with series and parallel
combinations of inductor and capacitors.
Kevin D. Donohue, University of Kentucky
8
Application - Integrator Circuit
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)

vs ( x)dx
0
vo(t)
-
Kevin D. Donohue, University of Kentucky
9
Application - Differentiator
Circuit
Show that this circuit differentiates the input signal vs(t)
according to the equation below:
R
dvs (t )
vo (t )   RC
dt
C
+
vs(t)
vo(t)
-
Kevin D. Donohue, University of Kentucky
10