1. Lecture #9
OUTLINE
The capacitor
The inductor
Chapter 3
Reading

2. The Capacitor Q = CVab Two conductors (a,b) separated by an insulator:
difference in potential = Vab
=> equal & opposite charge Q on conductors
Q = CVab
where C is the capacitance of the structure,
positive (+) charge is on the conductor at higher potential
(stored charge in terms of voltage)
Parallel-plate capacitor:
area of the plates = A
separation between plates = d
dielectric permittivity of insulator = 
=> capacitance

3. Units: Farads (Coulombs/Volt)
Symbol:
Units: Farads (Coulombs/Volt)
Current-Voltage relationship: C C
(typical range of values: 1 pF to 1 mF) ic + vc –
Note: Q (vc) must be a continuous function of time

4. Voltage in Terms of Current

5. Op-Amp Integrator C ic
– vC + R in
vin – + + vn – vo + vp –

6. Stored Energy
You might think the energy stored on a capacitor is QV, which has the dimension of Joules. But during charging, the average voltage across the capacitor was only half the final value of V for a linear capacitor.
Thus, energy is
Example: A 1 pF capacitance charged to 5 Volts has ½(5V)2 (1pF) = 12.5 pJ

7. A more rigorous derivationic + vc –

8. Example: Current, Power & Energy for a Capacitor
v (V)
v(t) + –
10 mF 1
t (ms) 1 2 3 4 5
i (mA)
vc must be a continuous
function of time; however,
ic can be discontinuous.
t (ms) 1 2 3 4 5
Note: In “steady state”
(dc operation), time
derivatives are zero
 C is an open circuit

9. p (W)
i(t)
v(t) + –
10 mF
t (ms) 1 2 3 4 5
w (J)
t (ms) 1 2 3 4 5

10. Capacitors in Parallel+
v(t) –
i(t) C1 C2 +
v(t) –
i(t)
Ceq
Equivalent capacitance of capacitors in parallel is the sum

11. Capacitors in Series C + = + v1(t) – + v2(t) – + v(t)=v1(t)+v2(t) – C1
Ceq 2 1 C eq + =

12. Capacitive Voltage Divider
Q: Suppose the voltage applied across a series combination of capacitors is changed by Dv. How will this affect the voltage across each individual capacitor?
DQ1=C1Dv1
Note that no net charge can
can be introduced to this node.
Therefore, -DQ1+DQ2=0
Q1+DQ1 +
v1+Dv1 – C1
-Q1-DQ1
v+Dv + –
Q2+DQ2 +
v2(t)+Dv2 – C2
-Q2-DQ2
DQ2=C2Dv2
Note: Capacitors in series have the same incremental charge.

13. Application Example: MEMS Accelerometer
Capacitive position sensor used to measure acceleration (by measuring force on a proof mass) g1 g2
FIXED OUTER PLATES

14. Sensing the Differential Capacitance
Begin with capacitances electrically discharged
Fixed electrodes are then charged to +Vs and –Vs
Movable electrode (proof mass) is then charged to Vo
Circuit model Vs C1 Vo C2
–Vs

15. Practical Capacitors
A capacitor can be constructed by interleaving the plates with two dielectric layers and rolling them up, to achieve a compact size.
To achieve a small volume, a very thin dielectric with a high dielectric constant is desirable. However, dielectric materials break down and become conductors when the electric field (units: V/cm) is too high.
Real capacitors have maximum voltage ratings
An engineering trade-off exists between compact size and high voltage rating

16. The Inductor
An inductor is constructed by coiling a wire around some type of form.
Current flowing through the coil creates a magnetic field and a magnetic flux that links the coil: LiL
When the current changes, the magnetic flux changes
 a voltage across the coil is induced: +
vL(t) iL _
Note: In “steady state” (dc operation), time
derivatives are zero  L is a short circuit

17. Units: Henrys (Volts • second / Ampere)
Symbol:
Units: Henrys (Volts • second / Ampere)
Current in terms of voltage: L
(typical range of values: mH to 10 H) iL + vL –
Note: iL must be a continuous function of time

18. ò p ( t ) = v ( t ) i ( t ) = w ( t ) = p ( t ) d t = 1 1 w ( t ) = Li
Stored Energy
Consider an inductor having an initial current i(t0) = i0 p ( t ) = v ( t ) i ( t ) = t ò w ( t ) = p ( t ) d t = t 1 1 2 w ( t ) = Li 2 - Li 2 2

19. Equivalent inductance of inductors in series is the sum
+ v1(t) –
+ v2(t) – +
v(t)=v1(t)+v2(t) – L1 L2
i(t)
i(t) + – + –
v(t)
v(t)
Leq
Equivalent inductance of inductors in series is the sum

20. Inductors in Parallel +
v(t) – +
v(t) – i1 i2
i(t)
i(t)
Leq L1 L2

21. Summary Capacitor v cannot change instantaneously
i can change instantaneously
Do not short-circuit a charged
capacitor (-> infinite current!)
n cap.’s in series:
n cap.’s in parallel:
Inductor
i cannot change instantaneously
v can change instantaneously
Do not open-circuit an inductor with current (-> infinite voltage!)
n ind.’s in series:
n ind.’s in parallel: