Kevin D. Donohue, University of Kentucky

1

Energy Storage Elements

Capacitance/Inductance and RC Op Amp Circuits

Kevin D. Donohue, University of Kentucky

2

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.

An inductor can store the energy in a magnetic field from the current used to create lines of flux around it.

L+v(t)

-

i(t)

C

+v(t)

-

i(t)

2

1)( , )( , )(

1)()( , , 2

0

0

Cvtwdt

dvCvtpdxxi

Ctvtv

dt

dvCiCvq

t

t

20 2

1)( , )( , )(

1)()( , ,

0

Litwdt

diLitpdxxv

Ltiti

dt

diLvLi

t

t

Kevin D. Donohue, University of Kentucky

3

Examples

Solve for voltages, currents, charge, power, and energy in simple circuits containing inductors and capacitors.

Kevin D. Donohue, University of Kentucky

4

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

5

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 lumped-parameter models

C Rleak

L

Rleak

Kevin D. Donohue, University of Kentucky

6

Capacitor Combinations

Series capacitors can be combined according to the following formula:

Parallel capacitors can be combined according to the following formula:

C1 C2 CN Ceq

C1 C2 CN Ceq

N

kk

eq

C

C

1

1

1

…

…

N

k

keq CC

1

Kevin D. Donohue, University of Kentucky

7

Inductor Combinations

Series inductors can be combined according to the following formula:

Parallel inductors can be combined according to the following formula:

L1 L2 LN Leq

L1 L2 LN Leq

N

kk

eq

L

L

1

1

1

…

…

N

k

keq LL

1

Kevin D. Donohue, University of Kentucky

8

Examples

Simplify circuits with series and parallel combinations of inductor and capacitors.

Kevin D. Donohue, University of Kentucky

9

Application - Integrator Circuit

Show that this circuit integrates the input signal vs(t) according to the equation below for time greater than 0:

t

soo dxxvRC

vtv

0

)(1

)0()(

+

vo(t)-

vs(t)

R

C

Kevin D. Donohue, University of Kentucky

10

Application - Differentiator Circuit

Show that this circuit differentiates the input signal vs(t) according to the equation below:

dt

tdvRCtv s

o)(

)(

+

vo(t)-

vs(t)

R

C