电路Circuit->Chapter6 Capacitors and Inductors

Capacitors and Inductors

• 6.1 Storage Elements
• 6.2 Capacitors
• 6.3 Series and Parallel Capacitors
• 6.4 Inductors
• 1.v-i characteristics
• 2.Power
• 3.Energy
• 6.5 Series and Parallel Inductors
• 6.6 Application

6.1 Storage Elements

In contrast to resistors,which spend or dissipate energy irreversibly,capacitors and inductors do not dissipate but stores or releases energy,which can be retrieved at a later time(i.e.,have a memory).
For this reason,capacitors and inductors are called storge elements.

6.2 Capacitors

A capacitor consists of two conducting plates separated by an insulator(or dielectric).

这几个例子不重要,看不懂也不用查字典
Fixed capacitors:polyester capacitor;ceramic capacitor;electrolytic capacitor
Variable capacitors:trimmer capacitor;filmtrim capacitor

Capacitance is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates,measured in farads(F)
$C=\frac{q}{U}$C=Uq​


Circuit model of a nonideal capacitor(实际电容模型)

Important properties of a capacitor:

1. A capacitor is an open circuit to dc通直阻交
2. The voltage on a capacitor must be continuous.The voltage on a capacitor cannot change abruptly. The capacitor resists an abrupt change in the voltage across it.电压不能突变
3. The ideal capacitor does not dissipate energy.It takes power from the circuit when storing energy in its field and returns previously stored energy when delivering power to the circuit.不消耗能量
4. A real,nonideal capacitor has a parallel-model leakage resistance.The leakage resistance can be neglected for most practical applications.

6.3 Series and Parallel Capacitors

The series-parallel combination of resistive circuits can be extened to series -parallel connecs of capacitors.
How to replace capacitors in series-parallel connections by a single equivalent capacitor?
v-i characteristics


The equivalent capacitance of parallel-connected capacitors is the sum of the individual capacitances.
$C_{eq}=C_1+C_2+\cdots+C_N$Ceq​=C1​+C2​+⋯+CN​
The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.
$\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots+\frac{1}{C_N}$Ceq​1​=C1​1​+C2​1​+⋯+CN​1​

6.4 Inductors

An Inductor consists of a coil of conducting wire.
A practical inductor is usually formed into a cylindrical coil with many turns of conducting wire.

real inductors:solenoidal wound inductor;toroidal inductor;chip inductor

1.v-i characteristics

$u(t)=L\frac{di(t)}{dt}$u(t)=Ldtdi(t)​
Note:

1. Inductance is the property whereby an inductor exhibits opposition to the change of current flowing through it.
2. The voltage across an inductor depends on the time rate of change of the current.
3. When the current is constant,u=0.An inductor acts like a short circuit to dc.
4. The current through an inductor cannot change instantaneously.
   $i(t)=\frac{1}{L}∫_{-∞}^t udξ=\frac{1}{L}∫_{-∞}^{t_0} udξ+\frac{1}{L}∫_{t_0}^t udξ\\ ~\\ i(t)=i(t_0)+\frac{1}{L}∫_{t_0}^t udξ .$i(t)=L1​∫−∞t​udξ=L1​∫−∞t0​​udξ+L1​∫t0​t​udξ i(t)=i(t0​)+L1​∫t0​t​udξ.
   Note:
   i(t) is the total current for -∞ to t.
   i(t₀) is the total current for -∞ to t₀
   

2.Power

$u(t)=L\frac{di(t)}{dt}→p=ui=L\frac{di}{dt}·i$u(t)=Ldtdi(t)​→p=ui=Ldtdi​⋅i
passive sign convention

1. If i↑,p>0,takes power
2. if i↓,p>0,delivers power
   Inductor does not dissipate energy.The energy stored in it can be retrieved at a later time.The inductor takes power from the circuit when storing energy and delivers power to the circuit when returning previously stored energy.

3.Energy

$W_L=∫_{-∞}^t Li\frac{di}{dξ}dξ= \frac{1}{2}Li^2(ξ)|_{-∞}^t =\frac{1}{2}Li^2(t)-\frac{1}{2}Li^2(-∞)\\ W_L=\frac{1}{2}Li^2≥0\\ ΔW_L=\frac{1}{2}Li^2(t)-\frac{1}{2}Li^2(t_0)$WL​=∫−∞t​Lidξdi​dξ=21​Li2(ξ)∣−∞t​=21​Li2(t)−21​Li2(−∞)WL​=21​Li2≥0ΔWL​=21​Li2(t)−21​Li2(t0​)
Note:
1.The energy of an inductor depends on the current.
2.W_L≥0
电感储存的能量与电流有关
Circuit model of a nonideal inductor

6.5 Series and Parallel Inductors

1.Inductors in series

Equivalent inductor
$\Large u_1=L_1\frac{di}{dt}\\ \Large u_2=L_2\frac{di}{dt}\\ \Large u=u_1+u_2=(L_1+L_2)\frac{di}{dt}=L\frac{di}{dt}$u1​=L1​dtdi​u2​=L2​dtdi​u=u1​+u2​=(L1​+L2​)dtdi​=Ldtdi​
$L=L_1+L_2$L=L1​+L2​
Voltage division
$\Large u_1=L_1\frac{di}{dt}=\frac{L_1}{L}u=\frac{L_1}{L_1+L_2}u\\ \Large u_2=L_2\frac{di}{dt}=\frac{L_2}{L}u=\frac{L_2}{L_1+L_2}u$u1​=L1​dtdi​=LL1​​u=L1​+L2​L1​​uu2​=L2​dtdi​=LL2​​u=L1​+L2​L2​​u
2.Inductors in parallel
$L=\frac{1}{ \frac{1}{L_1}+\frac{1}{L_2}}=\frac{L_1L_2}{L_1+L_2}$L=L1​1​+L2​1​1​=L1​+L2​L1​L2​​

Equivalent inductor
$i_1=\frac{1}{L_1}∫_{-∞}^tu(ξ)dξ\\ i_2=\frac{1}{L_2}∫_{-∞}^tu(ξ)dξ\\ i=i_1+i_2=(\frac{1}{L_1}+\frac{1}{L_2})∫_{-∞}^tu(ξ)dξ=\frac{1}{L}∫_{-∞}^tu(ξ)dξ$i1​=L1​1​∫−∞t​u(ξ)dξi2​=L2​1​∫−∞t​u(ξ)dξi=i1​+i2​=(L1​1​+L2​1​)∫−∞t​u(ξ)dξ=L1​∫−∞t​u(ξ)dξ
Current division
$\Large i_1=\frac{L}{L_1}i=\frac{L_2i}{L_1+L_2}\\ i_2=\frac{L}{L_2}i=\frac{L_1i}{L_1+L_2}$i1​=L1​L​i=L1​+L2​L2​i​i2​=L2​L​i=L1​+L2​L1​i​
The equivalent inductance of series-connected inductors is the sum of the individual inductances.
$L_{eq}=L_1+L_2+\cdots+L_N$Leq​=L1​+L2​+⋯+LN​
The equivalent inductance of parallel inductors is the reciprocal of the sum of the reciprocal of the individual inductances.
$\frac{1}{L_{eq}}=\frac{1}{L_1}+\frac{1}{L_2}+\cdots+\frac{1}{L_N}$Leq​1​=L1​1​+L2​1​+⋯+LN​1​

6.6 Application

Integrator
Differentiator
Analog Computer