9: capacitors and inductors

9: Capacitors and Inductors


• Capacitors

• Types of Capacitor

• Inductors

• Passive Components

• Series and ParallelInductors• Series and ParallelCapacitors

• Current/Voltage Continuity

• Average Current/Voltage

• Buck Converter

• Power and Energy

• Summary

E1.1 Analysis of Circuits (2015-7150) Capacitors and Inductors: 9 – 1 / 12

Capacitors


• Capacitors


• Inductors





• Buck Converter


• Summary


A capacitor is formed from two conducting plates separated by a thininsulating layer.

If a current i flows, positive change, q, willaccumulate on the upper plate. To preservecharge neutrality, a balancing negative chargewill be present on the lower plate.

Capacitors


• Capacitors


• Inductors





• Buck Converter


• Summary




There will be a potential energy difference (or voltage v) between the platesproportional to q.v = d

Aǫq where A is the area of the plates, d is their separation and ǫ is

the permittivity of the insulating layer (ǫ0 = 8.85 pF/m for a vacuum).

Capacitors


• Capacitors


• Inductors





• Buck Converter


• Summary







The quantity C = Aǫd

is the capacitance and is measured in Farads (F),hence q = Cv.

Capacitors


• Capacitors


• Inductors





• Buck Converter


• Summary







The quantity C = Aǫd

is the capacitance and is measured in Farads (F),hence q = Cv.

The current, i, is the rate of charge on the plate, hence the

capacitor equation: i = dqdt

= C dvdt

.

Types of Capacitor


• Capacitors


• Inductors





• Buck Converter


• Summary


Capacitor symbol represents the two separatedplates. Capacitor types are distinguished bythe material used as the insulator.

Types of Capacitor


• Capacitors


• Inductors





• Buck Converter


• Summary



Polystyrene: Two sheets of foil separated by athin plastic film and rolled up to save space.Values: 10 pF to 1 nF.

Types of Capacitor


• Capacitors


• Inductors





• Buck Converter


• Summary




Ceramic: Alternate layers of metal and ceramic(a few µm thick). Values: 1 nF to 1µF.

Types of Capacitor


• Capacitors


• Inductors





• Buck Converter


• Summary





Electrolytic: Two sheets of aluminium foilseparated by paper soaked in conductingelectrolyte. The insulator is a thin oxide layeron one of the foils. Values: 1µF to 10mF.

Types of Capacitor


• Capacitors


• Inductors





• Buck Converter


• Summary






Electrolytic capacitors are polarised: the foil with the oxide layer mustalways be at a positive voltage relative to the other (else explosion).Negative terminal indicated by a curved plate in symbol

Types of Capacitor


• Capacitors


• Inductors





• Buck Converter


• Summary






Electrolytic capacitors are polarised: the foil with the oxide layer mustalways be at a positive voltage relative to the other (else explosion).Negative terminal indicated by a curved plate in symbol or “-”.

Inductors


• Capacitors


• Inductors





• Buck Converter


• Summary


Inductors are formed from coils of wire, oftenaround a steel or ferrite core.

Inductors


• Capacitors


• Inductors





• Buck Converter


• Summary



The magnetic flux within the coil is Φ = µNAl

i where N is the number ofturns, A is the cross-sectional area of the coil and l is the length of the coil(around the toroid).

µ is a property of the material that the core is made from and is called itspermeability . For free space (or air): µ0 = 4π × 10−7 = 1.26 µH/m, forsteel, µ ≈ 4000µ0 = 5mH/m.

Inductors


• Capacitors


• Inductors





• Buck Converter


• Summary






From Faraday’s law: v = N dΦdt

= µN2Al

didt

= L didt

.

Inductors


• Capacitors


• Inductors





• Buck Converter


• Summary






From Faraday’s law: v = N dΦdt

= µN2Al

didt

= L didt

.

We measure the inductance, L = µN2Al

, in Henrys (H).

Passive Components


• Capacitors


• Inductors





• Buck Converter


• Summary


We can describe all three types of passive component by the relationshipbetween V and I using, in each case, the passive sign convention.

Passive Components


• Capacitors


• Inductors





• Buck Converter


• Summary



Resistor: v = Ri

Passive Components


• Capacitors


• Inductors





• Buck Converter


• Summary



Resistor: v = Ri

Inductor: v = L didt

Passive Components


• Capacitors


• Inductors





• Buck Converter


• Summary



Resistor: v = Ri


Capacitor: i = C dvdt

Passive Components


• Capacitors


• Inductors





• Buck Converter


• Summary



Resistor: v = Ri



Notes: (1) There are no minus signs anywhere whatever you were taught atschool.

(2) We use lower case, v, for time-varying voltages.

Series and Parallel Inductors


• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt

= (L1 + L2)didt



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt

= (L1 + L2)didt

Same equation as a single inductor of value L1 + L2



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt

= (L1 + L2)didt


didt

= d(i1+i2)dt



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt

= (L1 + L2)didt


didt

= d(i1+i2)dt

= di1dt

+ di2dt



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt

= (L1 + L2)didt


didt

= d(i1+i2)dt

= di1dt

+ di2dt

= vL1

+ vL2

= v(

1L1

+ 1L2

)



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt

= (L1 + L2)didt


didt

= d(i1+i2)dt

= di1dt

+ di2dt

= vL1

+ vL2

= v(

1L1

+ 1L2

)

v = 11

L1+ 1

L2

didt



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt

= (L1 + L2)didt


didt

= d(i1+i2)dt

= di1dt

+ di2dt

= vL1

+ vL2

= v(

1L1

+ 1L2

)

v = 11

L1+ 1

L2

didt

Same as a single inductor of value 11

L1+ 1

L2



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt

= (L1 + L2)didt


didt

= d(i1+i2)dt

= di1dt

+ di2dt

= vL1

+ vL2

= v(

1L1

+ 1L2

)

v = 11

L1+ 1

L2

didt


L1+ 1

L2

= L1L2

L1+L2



• Capacitors


• Inductors





• Buck Converter


• Summary


v = v1 + v2= L1didt

+ L2didt

= (L1 + L2)didt


didt

= d(i1+i2)dt

= di1dt

+ di2dt

= vL1

+ vL2

= v(

1L1

+ 1L2

)

v = 11

L1+ 1

L2

didt


L1+ 1

L2

= L1L2

L1+L2

Inductors combine just like resistors.

Series and Parallel Capacitors


• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt

= (C1 + C2)dvdt



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt

= (C1 + C2)dvdt

Same equation as a single capacitor of value C1 + C2



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt

= (C1 + C2)dvdt


dvdt

= d(v1+v2)dt



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt

= (C1 + C2)dvdt


dvdt

= d(v1+v2)dt

= dv1dt

+ dv2dt



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt

= (C1 + C2)dvdt


dvdt

= d(v1+v2)dt

= dv1dt

+ dv2dt

= iC1

+ iC2

= i(

1C1

+ 1C2

)



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt

= (C1 + C2)dvdt


dvdt

= d(v1+v2)dt

= dv1dt

+ dv2dt

= iC1

+ iC2

= i(

1C1

+ 1C2

)

i = 11

C1+ 1

C2

dvdt



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt

= (C1 + C2)dvdt


dvdt

= d(v1+v2)dt

= dv1dt

+ dv2dt

= iC1

+ iC2

= i(

1C1

+ 1C2

)

i = 11

C1+ 1

C2

dvdt

Same as a single capacitor of value 11

C1+ 1

C2



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt

= (C1 + C2)dvdt


dvdt

= d(v1+v2)dt

= dv1dt

+ dv2dt

= iC1

+ iC2

= i(

1C1

+ 1C2

)

i = 11

C1+ 1

C2

dvdt


C1+ 1

C2

= C1C2

C1+C2



• Capacitors


• Inductors





• Buck Converter


• Summary


i = i1 + i2= C1dvdt

+ C2dvdt

= (C1 + C2)dvdt


dvdt

= d(v1+v2)dt

= dv1dt

+ dv2dt

= iC1

+ iC2

= i(

1C1

+ 1C2

)

i = 11

C1+ 1

C2

dvdt


C1+ 1

C2

= C1C2

C1+C2

Capacitors combine just like conductances (i.e. parallel capacitors add).

Current/Voltage Continuity


• Capacitors


• Inductors





• Buck Converter


• Summary





• Capacitors


• Inductors





• Buck Converter


• Summary



For the voltage to change abruptly

dvdt

= ∞ ⇒ i = ∞.



• Capacitors


• Inductors





• Buck Converter


• Summary




dvdt

= ∞ ⇒ i = ∞.

This never happens so ...



• Capacitors


• Inductors





• Buck Converter


• Summary




dvdt

= ∞ ⇒ i = ∞.


The voltage across a capacitor never changes instantaneously.



• Capacitors


• Inductors





• Buck Converter


• Summary




dvdt

= ∞ ⇒ i = ∞.


The voltage across a capacitor never changes instantaneously.Informal version: A capacitor “tries” to keep its voltage constant.



• Capacitors


• Inductors





• Buck Converter


• Summary




dvdt

= ∞ ⇒ i = ∞.






• Capacitors


• Inductors





• Buck Converter


• Summary




dvdt

= ∞ ⇒ i = ∞.




For the current to change abruptly

didt

= ∞ ⇒ v = ∞.



• Capacitors


• Inductors





• Buck Converter


• Summary




dvdt

= ∞ ⇒ i = ∞.





didt

= ∞ ⇒ v = ∞.




• Capacitors


• Inductors





• Buck Converter


• Summary




dvdt

= ∞ ⇒ i = ∞.





didt

= ∞ ⇒ v = ∞.


The current through an inductor never changes instantaneously.



• Capacitors


• Inductors





• Buck Converter


• Summary




dvdt

= ∞ ⇒ i = ∞.





didt

= ∞ ⇒ v = ∞.


The current through an inductor never changes instantaneously.Informal version: An inductor “tries” to keep its current constant.

Average Current/Voltage


• Capacitors


• Inductors





• Buck Converter


• Summary


For a capacitor i = C dvdt

.



• Capacitors


• Inductors





• Buck Converter


• Summary



. Take the average of both sides:

1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt



• Capacitors


• Inductors





• Buck Converter


• Summary




1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt= C

t2−t1

∫ v(t2)

v(t1)dv



• Capacitors


• Inductors





• Buck Converter


• Summary




1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt= C

t2−t1

∫ v(t2)

v(t1)dv

= Ct2−t1

[v]v(t2)v(t1)



• Capacitors


• Inductors





• Buck Converter


• Summary




1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt= C

t2−t1

∫ v(t2)

v(t1)dv

= Ct2−t1

[v]v(t2)v(t1)

= Ct2−t1

(v(t2)− v(t1))



• Capacitors


• Inductors





• Buck Converter


• Summary




1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt= C

t2−t1

∫ v(t2)

v(t1)dv

= Ct2−t1

[v]v(t2)v(t1)

= Ct2−t1

(v(t2)− v(t1))

(1) If v(t1) = v(t2) then the averagecurrent exactly equals zero.



• Capacitors


• Inductors





• Buck Converter


• Summary




1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt= C

t2−t1

∫ v(t2)

v(t1)dv

= Ct2−t1

[v]v(t2)v(t1)

= Ct2−t1

(v(t2)− v(t1))


(2) If v is bounded then the average current→ 0 as (t2 − t1) → ∞.



• Capacitors


• Inductors





• Buck Converter


• Summary




1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt= C

t2−t1

∫ v(t2)

v(t1)dv

= Ct2−t1

[v]v(t2)v(t1)

= Ct2−t1

(v(t2)− v(t1))



The average current through a capacitor is zero



• Capacitors


• Inductors





• Buck Converter


• Summary




1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt= C

t2−t1

∫ v(t2)

v(t1)dv

= Ct2−t1

[v]v(t2)v(t1)

= Ct2−t1

(v(t2)− v(t1))



The average current through a capacitor is zero and, likewise, the averagevoltage across an inductor is zero.



• Capacitors


• Inductors





• Buck Converter


• Summary




1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt= C

t2−t1

∫ v(t2)

v(t1)dv

= Ct2−t1

[v]v(t2)v(t1)

= Ct2−t1

(v(t2)− v(t1))



The average current through a capacitor is zero and, likewise, the averagevoltage across an inductor is zero. The circuit symbols remind you of this.



• Capacitors


• Inductors





• Buck Converter


• Summary




1t2−t1

∫ t2

t1idt = 1

t2−t1

∫ t2

t1C dv

dtdt= C

t2−t1

∫ v(t2)

v(t1)dv

= Ct2−t1

[v]v(t2)v(t1)

= Ct2−t1

(v(t2)− v(t1))



The average current through a capacitor is zero and, likewise, the averagevoltage across an inductor is zero. The circuit symbols remind you of this.

“Average” can either be over an exact number of periods of a repetitivewaveform or else the long-term average (provided v and i remainbounded).

“v is bounded” means |v| always stays less than a predefined maximumvalue.

Buck Converter


• Capacitors


• Inductors





• Buck Converter


• Summary


A buck converter converts a highvoltage, V , into a lower one, Y .

The switch, S, closes for a fraction aof the time. a is the duty cycle andis 1

3 in this example.

Buck Converter


• Capacitors


• Inductors





• Buck Converter


• Summary




3 in this example.

When S is closed, x = v, and acurrent iL flows.

Buck Converter


• Capacitors


• Inductors





• Buck Converter


• Summary




3 in this example.


When S opens, the current iL cannotchange instantly and so it mustflow through the diode (weassume the diode is ideal).

Buck Converter


• Capacitors


• Inductors





• Buck Converter


• Summary




3 in this example.



The average value of x is aV ⇒ the average value of y must also be aV .

Buck Converter


• Capacitors


• Inductors





• Buck Converter


• Summary




3 in this example.




The average current through R is aVR

so, since the average current through

C must be zero, the average current iL must also be aVR

.

Buck Converter


• Capacitors


• Inductors





• Buck Converter


• Summary




3 in this example.







.

C dydt

= iL − iR ⇒ if C is large, then the variations in y will be very small.

Buck Converter


• Capacitors


• Inductors





• Buck Converter


• Summary


[Do not memorize this circuit]



3 in this example.







.

C dydt

= iL − iR ⇒ if C is large, then the variations in y will be very small.

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary


Electrical power absorbed by any component at the instant t is v(t)× i(t).

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary



So total energy absorbed between times t1 and t2 is W =∫ t2

t=t1vi dt.

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary




t=t1vi dt.


, so

W = C∫ t2

t=t1v dvdtdt

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary




t=t1vi dt.


, so

W = C∫ t2

t=t1v dvdtdt= C

∫ v(t2)

v=v(t1)vdv

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary




t=t1vi dt.


, so

W = C∫ t2

t=t1v dvdtdt= C

∫ v(t2)

v=v(t1)vdv

= C[

12v

2]v(t2)

v(t1)

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary




t=t1vi dt.


, so

W = C∫ t2

t=t1v dvdtdt= C

∫ v(t2)

v=v(t1)vdv

= C[

12v

2]v(t2)

v(t1)= 1

2C(

v2(t2)− v2(t1))

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary




t=t1vi dt.


, so

W = C∫ t2

t=t1v dvdtdt= C

∫ v(t2)

v=v(t1)vdv

= C[

12v

2]v(t2)

v(t1)= 1

2C(

v2(t2)− v2(t1))

If v(t1) = v(t2) then there has been nonett energy absorbed: all the energyabsorbed when the voltage rises isreturned to the circuit when it falls.

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary




t=t1vi dt.


, so

W = C∫ t2

t=t1v dvdtdt= C

∫ v(t2)

v=v(t1)vdv

= C[

12v

2]v(t2)

v(t1)= 1

2C(

v2(t2)− v2(t1))


The energy stored in a capacitor is 12Cv2

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary




t=t1vi dt.


, so

W = C∫ t2

t=t1v dvdtdt= C

∫ v(t2)

v=v(t1)vdv

= C[

12v

2]v(t2)

v(t1)= 1

2C(

v2(t2)− v2(t1))


The energy stored in a capacitor is 12Cv2 and likewise in an inductor 1

2Li2.

Power and Energy


• Capacitors


• Inductors





• Buck Converter


• Summary




t=t1vi dt.


, so

W = C∫ t2

t=t1v dvdtdt= C

∫ v(t2)

v=v(t1)vdv

= C[

12v

2]v(t2)

v(t1)= 1

2C(

v2(t2)− v2(t1))


The energy stored in a capacitor is 12Cv2 and likewise in an inductor 1

2Li2.

If v and i remain bounded, then the average power absorbed by acapacitor or inductor is always zero.

Summary


• Capacitors


• Inductors





• Buck Converter


• Summary


• Capacitor:

◦ i = C dvdt

Summary


• Capacitors


• Inductors





• Buck Converter


• Summary


• Capacitor:

◦ i = C dvdt

◦ parallel capacitors add in value

Summary


• Capacitors


• Inductors





• Buck Converter


• Summary


• Capacitor:

◦ i = C dvdt


◦ average i is zero, v never changes instantaneously.

Summary


• Capacitors


• Inductors





• Buck Converter


• Summary


• Capacitor:

◦ i = C dvdt



◦ average power absorbed is zero

Summary


• Capacitors


• Inductors





• Buck Converter


• Summary


• Capacitor:

◦ i = C dvdt




• Inductor:

◦ v = L didt

Summary


• Capacitors


• Inductors





• Buck Converter


• Summary


• Capacitor:

◦ i = C dvdt




• Inductor:

◦ v = L didt

◦ series inductors add in value (like resistors)

Summary


• Capacitors


• Inductors





• Buck Converter


• Summary


• Capacitor:

◦ i = C dvdt




• Inductor:

◦ v = L didt


◦ average v is zero, i never changes instantaneously.

Summary


• Capacitors


• Inductors





• Buck Converter


• Summary


• Capacitor:

◦ i = C dvdt




• Inductor:

◦ v = L didt




Summary


• Capacitors


• Inductors





• Buck Converter


• Summary


• Capacitor:

◦ i = C dvdt




• Inductor:

◦ v = L didt




For further details see Hayt et al. Chapter 7.

9: capacitors and inductors

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