Electric Machinery 王嘉帷 MA320108

1. 22 The magnetic circuit of Fig. 1 1.22 The magnetic circuit of Fig. 1.32 has two windings and two air gaps. The core can be assumed to be of infinite permeability. The core dimensions are indicated in the figure. 圖1.32的磁路具有兩個繞組和兩個空隙。鐵芯可以被假定為無窮滲透性。 Figure 1.32

a. Assuming coil 1 to be carrying a current Il and the current in coil 2 to be zero, calculate (i) the magnetic flux density in each of the air gaps, (ii) the flux linkage of winding l, and (iii) the flux linkage of winding 2. 假設線圈1帶電流I1，線圈2的電流為零，計算（i）在每個空隙磁通密度，（ⅱ）繞組1的磁通鏈，（iii）繞組2的磁通鏈。

b. Repeat part (a), assuming zero current in winding 1 and a current I2 in winding 2. 重複（a）部分，假設在繞組1電流為零，繞組2電流I2。

c. Repeat part (a), assuming the current in winding 1 to be I1 and the current in winding 2 to be I2. 重複（a）部分，假設繞組1電流是I1，繞組2電流為I2。

d. Find the self-inductances of windings 1 and 2 and the mutual inductance between the windings. 找出繞組1和2的自感，和繞組之間的互感的。

1. 23 The symmetric magnetic circuit of Fig. 1. 33 has three windings 1.23 The symmetric magnetic circuit of Fig. 1.33 has three windings. Windings A and B each have N turns and are wound on the two bottom legs of the core. The core dimensions are indicated in the figure. 圖1.33的對稱磁路具有三個繞組。繞組A和B各自有N個線匝，並纏繞在鐵芯的兩個底腿。 Figure 1.33

a. Find the self-inductances of each of the windings. 找到每個繞組的自感。

b. Find the mutual inductances between the three pairs of windings. 找出三對繞組之間的互感。

c. Find the voltage induced in winding 1 by time-varying currents iA(t) and iB (t) in windings A and B. Show that this voltage can be used to measure the imbalance between two sinusoidal currents of the same frequency. 找出感應繞組1隨時間變化繞組A和B中電流iA(t)和iB(t)的電壓。該電壓可以用來測量相同頻率的兩個正弦電流之間的不平衡。

1. 28 The coils of the magnetic circuit shown in Fig. 1 1.28 The coils of the magnetic circuit shown in Fig. 1.36 are connected in series so that the mmf's of paths A and B both tend to set up flux in the center leg C in the same direction. The coils are wound with equal turns, Nl = N2 = 100. 圖1.36所示的線圈磁路串聯連接，以使磁動勢的通道A和B兩者的傾向於在中間相同方向建立磁通量。線圈匝數以等比纏繞，N1= N 2= 100。 Figure 1.36

The dimensions are: Cross-section area of A and B legs = 7 cm 2 Cross-section area of C legs = 14 cm 2 Length of A path = 17 cm Length of B path = 17 cm Length of C path = 5.5 cm Air gap = 0.4 cm The material is M-5 grade, 0.012-in steel, with a stacking factor of 0.94. Neglect fringing and leakage. A和B的截面積=7平方公分、C的截面積=14平方公分 A長17cm、B長17cm、C長5.5cm 空隙=0.4cm 材質M-5級，0.012的鋼，0.94的疊層係數 忽視邊緣和洩漏

a. How many amperes are required to produce a flux density of 1 a. How many amperes are required to produce a flux density of 1.2 T in the air gap? From the M-5 magnetization curve, for B = 1.2 T, Hm = 14 A/m. Similarly, Hg = B/μ0 = 9.54 × 105 A/m. Thus, with I1 = I2 = I 要多少安培才能在空隙中產生的1.2T的磁通密度在空隙？ 從M-5的磁化曲線，for B = 1.2 T, Hm = 14 A/m. 同理，Hg = B/μ0 = 9.54 × 105 A/m. Thus, with I1 = I2 = I

b. Under the condition of part (a), how many joules of energy are stored in the magnetic field in the air gap? 在(a)的狀況下，在空隙中的磁場儲存多少焦耳能量？

c. Calculate the inductance. 計算電感