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Chap. 6 Inductance, Capacitance,� and Mutual Inductance

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Contents

6.1 The Inductor

6.2 The Capacitor

6.3 Series-Parallel Combinations of Inductance

and Capacitance

6.4 Mutual Inductance

6.5 A Closer Look at Mutual Inductance

Objectives

  1. 認識並能使用電感(容)器的電壓、電流、功率與能量方程式。
  2. 能解解定電流流經電感器(定電壓加在電容器)的行為方式和電流(電壓)必須維持連續的要求。
  3. 能將具有初始條件之串聯和並聯電感(容)器，結合成具初始條件的單一等效電感(容)器。
  4. 瞭解互感的基本觀念，並能對一個包含磁耦線圈的電路以黑點標示慣用法寫出它的網目──電流方程式。

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6.1 The Inductor

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  1. 電感器是電子元件的一種，由線圈纏繞著磁性或非磁性的心材組成，其動作行為基於磁場現象。
  2. 電感器的兩端電壓與流經電感器的電流變化率成正比。
  3. 如果流經一理想電感器的電流為定值時，則跨於電感器上的電壓為零。
  4. 電感器中的電流無法瞬間改變(即需維持連續)。

when someone opens the switch on an inductive circuit in an actual system, the current initially continues to flow in the air across the switch, a phenomenon called arcing.

The arc across the switch prevents the current from dropping to zero instantaneously.

Inductor v -i Equation

Differential Form

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EX 6.1 Determining the Voltage, Given the Current, at the � Terminals of an Inductor

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b) At what instant of time is the current maximum?

a) Sketch the current waveform.

c) Express the voltage across the terminals of the 100 mH

inductor as a function of time.?

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EX 6.1 Contd.

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d) Sketch the voltage waveform.

e) Are the voltage and the current at a maximum at the same time?

f) At what instant of time does the voltage change polarity?

No; the voltage is proportional to di/dt , not i .

At 0.2 s, which corresponds to the moment when di/dt is passing through zero and changing sign.

g) Is there ever an instantaneous change in voltage across the

inductor? If so, at what time?

Yes, at t = 0. Note that the voltage can change

instantaneously across the terminals of an inductor.

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Current in an Inductor in Terms of the Voltage Across the Inductor

Integral Form

Inductor i - v Equation

when to = 0

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Differential

Form

初始電流

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EX 6.2 Determining the Current, Given the Voltage, at the � Terminals of an Inductor

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b) Find and sketch the inductor current as a function of time.

a) Sketch the voltage as a function of time.

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Power and Energy in the Inductor

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Or,

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EX 6.3 Determining the Current, Voltage, Power,� and Energy for an Inductor

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An increasing energy curve indicates that energy is being stored. Thus energy is being stored in the time interval 0 to 0.2 s.

Note that this corresponds to the interval when p > 0.

27.07 mJ

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6.2 The Capacitor

Integral Form

Capacitor i - v Equation

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  1. 電容器是電子元件的一種，由絕緣體或介電材料所隔開的二個導體組成，其動作行為基於電場現象。
  2. 電容器的電流與電容器兩端電壓變化率成正比。
  3. 如果橫跨在電容器兩端的電壓為定值時，則電容器之電流為零。
  4. 電容器的兩端電壓無法瞬間改變(即需維持連續)。

Capacitor v -i Equation

Differential Form

CAPACITOR POWER EQUATION

Or,

CAPACITOR ENERGY EQUATION

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EX 6.4 Determining the Current, Voltage, Power,� and Energy for a Capacitor

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= 0.5μF

a) Derive the expressions for the capacitor current, power, and energy.

b) Sketch the voltage, current, power, and energy as functions of time.

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EX 6.4 Contd.

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c) Specify the interval of time when energy is being stored in the capacitor.

d) Specify the interval of time when energy is being delivered by the capacitor.

Energy is being delivered by the capacitor whenever the power is negative. Thus energy is being delivered for all t>1 s.

Energy is being stored in the capacitor whenever the power is positive. Hence energy is being stored in the interval 0–1 s.

e) Evaluate the integrals:

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6.3 Series-Parallel Combinations of Inductance and Capacitance

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KVL

Combining Inductors in Series

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Combining Inductors in Parallel

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Equivalent Inductance

& Initial Current

KCL

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Combining Capacitors in Series

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Equivalent Capacitance & Initial Voltage

+ KVL

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Combining Capacitors in Parallel

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Equivalent Capacitance

+ KCL

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6.4 Mutual Inductance

What about the polarities?

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自感(Self-Inductance)： L1 and L2

在同一電感器電路中，電壓對應時變電流之參數。

​

互感(Mutual-Inductance) ： M

在以磁場相交連的二個電感器電路中，第二個電路感應

的電壓對應第一個電路的時變電流之參數。

For the left coil,

Self-induced voltage:

Mutually induced voltage:

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Dot Convention

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Passive sign convention: the self-induced voltage is a voltage drop in the direction of the current producing the voltage.

當一電流的參考方向為「流入」線圈黑點端時，在另一線圈所感應到的電壓，以黑點端代表參考極性為「正」。

當一電流的參考方向為「流出」線圈黑點端時，在另一線圈所感應到的電壓，以黑點端代表參考極性為「負」。

互感電壓之

極性判定

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The Procedure for Determining Dot Markings

決定黑點的步驟：

​

  1. 任意選取線圈的一個端點並標

上黑點（如圖D 端）。

2) 指定電流方向為流入黑點端並標示為iD。

3) 依右手定則決定由iD 所建立的磁通方向並標示為φD。

4) 從另一個線圈中任意選取一個端點，並指定測試電流iA 流入此端點（如圖A端）。

5) 依右手定則決定由iA 所建立的磁通方向並標示為φA 。

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6) 比較φD 和φA 兩個磁通方向，若一致則在第二個線圈的測試電流iA 流入處標上黑點(磁通方向相同，黑點標示在A端)。如果磁通方向不同，則在第二個線圈的測試電流iA 流出處B端標上黑點。

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Experimental Setup for Determining Polarity Markings

  1. 將連接至直流電壓源正極端之線圈端點標示黑點。
  2. 當開關閉合時，福特計指針瞬間增加刻度，則連接至伏特計正極端之線圈端點標示黑點。
  3. 若福特計指針瞬間減少刻度，則將連接至伏特計負極端之線圈端點標示黑點。

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  • 陰影區域無法直接觀察
  • R用以限制直流電壓源供給之電流大小

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EX 6.6 Finding Mesh-Current Equations for a Circuit � with Magnetically Coupled Coils

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i1 mesh:

i2 mesh:

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6.5 A Closer Look at Mutual Inductance

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A Review of Self-Inductance

法拉第定律(Faraday’s Law )

λ ：磁通鏈(Flux Linkage; weber-turns)

φ ：磁通(Magnetic Flux; Wb)

Coil current

P：Permeance

Assume that the core material, the space containing the flux, is nonmagnetic.

  • The permeance is constant.
  • a linear relationship between φ and i .
  1. 當i增加時，di/dt 為正，電壓v 亦為正，能量被用來建立磁場，能量儲存率(功率)為vi。
  2. 當磁場開始減弱時，di/dt為負，感應電壓v的極性變成抵抗磁場的變化，線圈磁場的減弱表示能量還回給電路。

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The Concept of Mutual Inductance

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The flux φ1 produced by the current i1 can be divided into two components, labeled φ11 (linking only the N1 turns) and φ21 (linking the N2 turns and the N1 turns).

Also,

M21

Self-inductance

Mutual-inductance

due to current i1

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The Concept of Mutual Inductance (Contd.)

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Also,

M12

Self-inductance

Mutual-inductance

due to current i2

For nonmagnetic materials, the permeances P12 and P21 are equal.

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Mutual Inductance in terms of Self-Inductance

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OR

k ：coefficient of coupling

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Energy Calculation

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