Chapter 6 Inductance, Capacitance, and Mutual Inductance
6.1 The inductor 6.2 The capacitor 6.3 Series-parallel combinations of inductance and capacitance 6.4 Mutual inductance 6.5 Closer look at mutual inductance
1 Overview
In addition to voltage sources, current sources, resistors, here we will discuss the remaining 2 types of basic elements: inductors, capacitors.
Inductors and capacitors cannot generate nor dissipate but store energy.
Their current-voltage (i-v) relations involve with integral and derivative of time, thus more complicated than resistors.
2 Key points di Why the i-v relation of an inductor is Lv ? dt dv Why the i-v relation of a capacitor is Ci ? dt
Why the energies stored in an inductor and a capacitor are: 1 1 2 , CvLiw 2 , respective ly? 2 2
3 Section 6.1 The Inductor
1. Physics 2. i-v relation and behaviors 3. Power and energy
4 Fundamentals
An inductor of inductance L is symbolized by a solenoidal coil.
Typical inductance L ranges from 10 H to 10 mH.
The i-v relation of an inductor (under the passive sign convention) is:
di Lv , dt
5 Physics of self-inductance (1)
Consider an N1 -turn coil C1 carrying current I1 . The resulting magnetic field 1 )( INrB 11 (Biot-
Savart law) will pass through C1 itself, causing a
flux linkage 1 , where 1 rB )( N 111 , 1 1 )( INPsdrB 111 , S 1
P1 is the permeance. 2 111 INP 1.
6 Physics of self-inductance (2)
The ratio of flux linkage to the driving current is defined as the self inductance of the loop:
1 2 L1 1 PN 1, I1 which describes how easy a coil current can introduce magnetic flux over the coil itself.
7 Examples
Solenoidal & toroidal coils: ~2 cm
L = 270 H L = 36 H
RG59/U coaxial cable:
L = 351 nH/m.
8 The i-v relation
Faraday’s law states that the electromotive force v (emf, in units of volt) induced on a loop equals the time derivative of the magnetic flux linkage : d d d v , v LLi i. dt dt dt
Note: The emf of a loop is a non-conservative force that can drive current flowing along the loop. In contrast, the current-driving force due to electric charges is conservative. 9 Behaviors of inductors
di Lv dt
DC-current: inductor behaves as a short circuit.
Current cannot change instantaneously in an inductor, otherwise, infinite voltage will arise.
Change of inductor current is the integral of voltage during the same time interval: 1 t titi 0 )()( dv .)( L t0
10 Inductive effect is everywhere!
Nearly all electric circuits have currents flowing through conducting wires. Since it’s difficult to shield magnetic fields, inductive effect occurs even we do not purposely add an inductor into the circuit.
11 Example 6.1: Inductor driven by a current pulse
,0 t 0 ti )( 5t ,10 tte 0
The inductor voltage is: Inductor voltage can Memory-less di ,0 t 0 )( Ltv in steady 5t jump! dt ),51( tte 0 state.
12 Power & energy (1)
Consider an inductor of inductance L. The instantaneous power in the inductor is: di Livip . dt
Assume there is no initial current (i.e. no initial energy), i(t =0)=0, w(t =0)=0. We are interested in the energy W when the current increases from zero to I with arbitrary i(t).
13 Power & energy (2)
dw di W I p Li , , idiLdwdiLidw , dt dt 0 0
I i2 1 1 LW LI 2 , i.e. Liw 2 2 2 0 2
How the current changes with time doesn’t matter. It’s the final current I determining the final energy.
Inductor stores magnetic energy when there is nonzero current. 14 Example 6.3: Inductor driven by a current pulse
t <0.2, p>0, w, charging.
t >0.2, p<0, w, discharging.
In steady state (t ), i 0, v0, p0, w0 (no energy).
15 Example 6.3: Inductor driven by a voltage pulse
1 t ti )( idv )0()( L 0
p>0, w, always charging.
In steady state (t), i 2 A, v0, p0, w200 mJ (sustained current and With memory in steady state. constant energy).
16 Section 6.2 The Capacitor
1. Physics 2. i-v relation and behaviors 3. Power and energy
17 Fundamentals
A capacitor of capacitance C is symbolized by a parallel-plate .
Typical capacitance C ranges from 10 pF to 470 F. The i-v relation of an capacitor (under the passive sign convention) is:
dv Ci . dt
18 Physics of capacitance (1)
If we apply a voltage V12 between two isolated conductors, charge Q will be properly distributed over the conducting surfaces such that the resulting electric field rE )( satisfies: 2 rE )( 1 1 B1 VldrE 12 ,)( 2 which is valid for any integral path linking the two conducting E surfaces.
19 Physics of capacitance (2)
If 12 VV 12 , QQ while the spatial distribution of charge remains such that 1 ),()( 12 VldrEVrErE 12.)( 2 The ratio of the deposited charge to the bias voltage is defined as the capacitance of the conducting pair: Q C , V describing how easy a bias voltage can deposit charge on the conducting pair. 20 Examples
Ceramic disc & electrolytic:
RG59/U coaxial cable:
C = 53 pF/m.
21 The i-v relation
From the definition of capacitance: Q d d d C , tCvtq ),()( Cq , Civ v. V dt dt dt Note: Charge cannot flow through the dielectric between the conductors. However, a time-varying voltage causes a time-varying electric field that can slightly displace the dielectric bound charge. It is the time-varying bound charge contributing to the “displacement current”. 22 Polarization charge
23 Behaviors of capacitors
dv Ci dt
DC-voltage: capacitor behaves as an open circuit.
Voltage cannot change instantaneously in an capacitor, otherwise, infinite current will arise.
Change of capacitor voltage is the integral of current during the same time interval: 1 t tvtv 0 )()( di .)( C t0 24 Capacitive effect is everywhere!
A Metal-Oxide-Semiconductor (MOS) transistor has three conducting terminals (Gate, Source, Drain) separated by a dielectric layer with one another. Capacitive effect occurs even we do not purposely add a capacitor into the circuit.
(info.tuwien.ac.at)
charges
25 Power & energy (1)
Consider a capacitor of capacitance C. The instantaneous power in the capacitor is: dv Cvvip . dt
Assume there is no initial voltage (i.e. no initial energy), v(t =0)=0, w(t =0)=0. We are interested in the energy W when the voltage increases from zero to V with arbitrary v(t).
26 Power & energy (2)
dw dv W V p Cv , , vdvCdwdvCvdw , dt dt 0 0
2 V v 1 2 1 CW CV , i.e. Cvw 2 2 0 2 2 How the voltage increases with time doesn’t matter. It’s the final voltage V determining the final energy.
Capacitor stores electric energy when there is nonzero voltage. 27 Example 6.4: Capacitor driven by a voltage pulse
Capacitor current can jump!
t <1, p>0, w, charging.
t >1, p<0, w, discharging.
In steady state (t), i 0, v Memory-less in 0, p0, w0 (no energy). steady state.
28 Section 6.3 Series-Parallel Combinations
1. Inductors in series-parallel 2. Capacitors in series-parallel
29 Inductors in series
iiii 321 ,
vvvv 321 , di Lv j , jj dt di di di Lv L L 1 dt 2 dt 3 dt di L , eq dt n eq LL j j1
30 Inductors in parallel
1 t 321 , 321 , iiiiivvvv j j tidv 0 ),()( t 0 L j
3 t 3 t 3 1 1 i j tidv 0 )()( )( j tidv 0 )( t0 t0 j1 L j j1 L j j1 n 1 t 11 tidv 0 ),()( t 0 Leq eq j1 LL j 31 Capacitors in series
=
, iiiivvvv 321321 , n 11 1 t 1 t v j j 0 ),( vtvid tvid 0 ),( t0 t0 eq j1 CC j C j Ceq 32 Capacitors in parallel
=
, iiiivvvv , 321321 n dv dv eq CC j Ci , Ci , j1 jj dt eq dt
33 Section 6.4, 6.5 Mutual Inductance
1. Physics 2. i-v relation and dot convention 3. Energy
34 Fundamentals
Mutual inductance M is a circuit parameter between two magnetically coupled coils.
The value of M satisfies LLkM 21 , where
k 10 is the magnetic coupling coefficient.
The emf induced in Coil 2 due to time-varying
current in Coil 1 is proportional to 1 dtdiM .
35 The i-v relation (1)
Coil 1 of N1 turns is driven by a time-varying
current i1 , while Coil 2 of N2 turns is open. The flux components linking (1) only Coil 1, (2) both coils, and (3) total flux linking Coil 1 are: 11 1 )( INPsdrB , 211111 1 )( INPsdrB 1121 . S S 1 2
21111 , , PPPINPPINP 21111112111111 .
36 The i-v relation (2)
Faraday’s law states that the emf induced on
Coil 2 (when i2 remains constant) is: d d d d v N )( PNNiNPN Mi i . 2 dt 212 dt dt dt 1211211211212 One can show that the emf induced on Coil 1
(when i1 remains constant) is: d d PNNv Mi i . dt dt 212212211 For nonmagnetic media (e.g. air, silicon,
plastic), P21 =P12, M21 =M12= M=N1N 2P 21.
37 Mutual inductance in terms of self-inductance
The two self inductances and their product are: 2 2 11 , 221 PNLPNL 2 , 2 2 2 2 121 2 121 2 PPPPNNPPNNLL 12222111 P P 2 11 22 PNN 2121 ,11 if PP 1221 . P21 P21 1 P P M 2 Let 11 22 ,11 LL , LLkM . 2 21 2 21 k P21 P21 k
The coupling coefficient (1) k=0, if P21 21 =0
(i.e. no mutual flux), (2) k=1, if P11 =P22=0 (i.e. = =0, = = , no flux leakage). 11 22 1 2 21 38 Dot convention (1)
2 tiM )(
i2 leaves the dot of L2, the “+” polarity of 2 tiM )(
is referred the terminal of L1 without a dot.
The total voltage across L1 is: d d Lv 111 Mi i2. dt dt 39 Dot convention (2) 1 tiM )(
i1 enters the dot of L1, the “+” polarity of 1 tiM )(
is referred the terminal of L2 with a dot.
The total voltage across L2 is: d d Lv 222 Mi i1. dt dt 40 Example 6.6: Write a mesh current equation
d d 0)()H 8())( 5())( 20()H 4( 20()H 5())( iiiii 8())( ii 0)()H dt 1 21 1 g dt g 2 Self-inductance, Mutual-inductance,
passive sign convention ig i2 enters the dot of 16-H inductor 41 Example 6.6: Steady-state analysis
In steady state (t ), inductors are short,
the 3 resistors are in parallel (Req =3.75 ).
Let v2 =0. (1) v1 =(16A)(3.75 )=60 V. (2) i12 =
(60V)/(5) =12A, i1 =(16-12)=4 A (not zero!).
(3) i1'2 =(60V)/(20)=3 A, i22' =(12+3)=15 A.
4 A 1 2 60 V 1' 12 A 3 A
2' 42 Energy of mutual inductance (1)
Assume i1, i2 =0 initially. Fix i2 =0 while
increasing i1 from 0 to some constant I1 . The
energy stored in L1 becomes: I 1 1 2 1111 ILdiiLW 11 . 0 2 43 Energy of mutual inductance (2)
Now fix i1 =I1, while increasing i2 from 0 to I2. During this period, emf’s will be induced in
loops 1 and 2 due to the time-varying i2. The total power of the two inductors is:
di2 di2 )( 1 MItp 2 Lti 2 .)( dt dt
An extra energy of W12 +W2 is stored in the pair: I I 2 2 1 2 1212 2 222 21 ILIMIdiiLdiMIWW 22 . 0 0 2
44 Energy of mutual inductance (3)
The entire process contributes to a total energy
1 1 2 ILIMIILW 2 tot 2 11 21 2 22 for the two-inductor system.
Wtot only depends on the final currents I1 , I2
[independent of the time evolution of i1( t), i2( t)].
45 Key points di Why the i-v relation of an inductor is Lv ? dt dv Why the i-v relation of a capacitor is Ci ? dt
Why the energies stored in an inductor and a capacitor are: 1 1 2 , CvLiw 2 , respective ly? 2 2
46