Chap 9 Sinusoidal SteadyState Analysis Contents 9 1

Chap. 9 Sinusoidal Steady-State Analysis Contents 9. 1 The Sinusoidal Source 9. 2 The Sinusoidal Response 9. 3 The Phasor 9. 4 The Passive Circuit Elements in the Frequency Domain 9. 5 Kirchhoff’s Laws in the Frequency Domain 9. 6 Series, Parallel, and Delta-to-Wye Simplifications 9. 7 Source Transformations & Thévenin-Norton Equivalent Circuits 9. 8 The Node-Voltage Method 9. 9 The Mesh-Current Method 9. 10 The Transformer 9. 11 The Ideal Transformer 9. 12 Phasor Diagrams Objectives 1. 了解相量觀念，並藉此執行相量轉換及反相量轉換。 2. 利用相量觀念將時域之弦波電源電路轉換至頻域。 3. 利用下列電路分析技巧，解答頻域電路的相關問題： ◆ 克希荷夫定律 ◆ 串聯、並聯及Δ-Y 轉換 ◆ 分壓與分流 ◆ 戴維寧等效電路與諾頓等效電路 ◆ 節點電壓法 ◆ 網目電流法 4. 利用相量法分析含有線性變壓器之電路。 5. 了解理想變壓器之限制，並利用相量法分析含有線性變壓器之電路。 1

9. 1 The Sinusoidal Source 弦波電源(sinusoidal source)（不管是獨立或相依，電壓或電流源）: 用來產生一個隨時間呈弦波變化之電源(皆為週期性函數) 峰值 or 振幅 (peak or amplitude) 週期 (period) 角頻率(angular frequency): 頻率(frequency): (赫茲, Hz) <radians/second> <cycles/second> 相位角(phase angle): <通常用角度degree為單位> >0 2 2

均方根值(Root Mean Square, rms, Value): the square root of the mean value of the squared periodic function 弦波電源的rms� 或稱有效� (effective value)， 可經推導得： 3

EX 9. 1 Finding the Characteristics of a Sinusoidal Current A sinusoidal current has a maximum amplitude of 20 A. The current passes through one complete cycle in 1 ms. The magnitude of the current at zero time is 10 A. a) What is the frequency of the current in hertz (Hz)? b) What is the angular frequency in radians per second? c) Write the expression for i(t) using the cosine function. Express in degrees. d) What is the rms value of the current? a) b) c) d) & rms value = 4

EX 9. 2 Finding the Characteristics of a Sinusoidal Voltage A sinusoidal voltage is given by the expression a) What is the period of the voltage in milliseconds? b) What is the frequency in hertz? c) What is the magnitude of v at t = 2. 778 ms? d) What is the rms value of v ? & a) b) c) f= & 2. 778= d) 5

EX 9. 3 Translating a Sine Expression to a Cosine Expression Translate the sine function to the cosine function by subtracting 90◦ (π/2 rad) from the argument of the sine function. a) Verify the above translation. b) Express sin(ωt + 30◦) as a cosine function. a) Let & b) 6

EX 9. 4 Calculating the rms Value of a Triangular Waveform 7

EX 9. 5 Adding Cosines Using Phasors If and express y = y 1 + y 2 as a single sinusoidal function. a) Solve by using trigonometric identities. b) Solve by using the phasor concept. a) b) 12

9. 4 The Passive Circuit Elements in the Frequency Domain The V-I Relationship for a Resistor 時域 頻域 由於電阻本身為一純量， 故跨於電阻兩端之電壓相 量應與通過電阻之電流相 量同相(in phase)。 13 13

9. 5 The Kirchhoff’s Laws in the Frequency Domain KVL in the Frequency Domain 時域 =0 頻域 KCL in the Frequency Domain 時域 頻域 17 17

9. 6 The Series, Parallel, and Delta-to-Wye Simplifications Combining Impedances in Series and Parallel 等效阻抗等於各阻抗之總和 等效阻抗之倒數等於各阻抗倒數總和 見下一頁之定義 18 18

EX 9. 6 Combining Impedances in Series 750 cos(5000 t + 30◦) V a) Construct the frequency-domain equivalent circuit. b) Calculate the steady-state current i by the phasor method. a) b) 20

EX 9. 7 Combining Impedances in Series and in Parallel 8 cos 200, 000 t A a) Construct the frequencydomain equivalent circuit. b) Calculate the steady-state v, i 1, i 2, and i 3 by the phasor method. a) b) 21

EX 9. 8 Using a Delta-to-Wye Transform in the Frequency Domain 23

EX 9. 8 Using a Delta-to-Wye Transform (Contd. ) 24

EX 9. 9 Performing Source Transformations 26

EX 9. 10 Finding a Thévenin Equivalent KVL Also, 27

EX 9. 10 Finding a Thévenin Equivalent (Contd. ) 28

9. 8 The Node-Voltage Method EX 9. 11 Using the Node-Voltage Method in the Freq. Domain 節點 1： 節點 2： 控制變數： 節點 1： 節點 2： 29

9. 9 The Mesh-Current Method EX 9. 12 Using the Mesh-Current Method in the Freq. Domain 網目 1： 網目 2： 控制變數： 網目 1： 網目 2： 30

EX 9. 13 Analyzing a Linear Transformer (Frequency Domain) The parameters of a certain linear transformer are R 1 = 200 , R 2 = 100 , L 1 = 9 H, L 2 = 4 H, and k = 0. 5. The transformer couples an impedance consisting of an 800 resistor in series with a 1 µF capacitor to a sinusoidal voltage source. The 300 V (rms) source has an internal impedance of 500 + j 100 and a frequency of 400 rad/s. 33

EX 9. 13 Analyzing a Linear Transformer (Contd. ) c Open circuit: Thevenin’s Equiv. d 34

Exploring Limiting Values (Contd. ) Terminal Behavior of the ideal transformer 同理 理想變壓器： 1. 各線圈之每匝伏特數之 2. 絕對值相等，即 2. 各線圈之安培-匝數之 絕對值相等，即 Reflected Impedance 36

Determining the Voltage and Current Ratios & 理想變壓器的電壓關係 理想變壓器的電流關係 37

Three ways to show the Turns Ratio 匝數比(Turns Ratio): For a = 5 39

EX 9. 14 Analyzing an Ideal Transformer Ckt. (Frequency Domain) If vg = 2500 cos 400 t V, find the steady-state expressions for (a) i 1 ; (b) v 1 ; (c) i 2 ; and (d) v 2. & a) Also, 40

EX 9. 14 (Contd. ) b) c) d) 41

The Use of an Ideal Transformer for Impedance Matching 將理想變壓器二次側負載阻抗反射至一次側時，須乘上1/a 2。 42

9. 12 Phasor Diagrams A graphic representation of phasors. A phasor diagram shows the magnitude and phase angle of each phasor quantity in the complex-number plane. Phase angles are measured counterclockwise from the positive real axis, and magnitudes are measured from the origin of the axes. Two different magnitude scales are necessary, one for currents and one for voltages. The complex number − 7 − j 3. 43

EX 9. 15 Using Phasor Diagrams to Analyze a Circuit Find the value of R that will cause the current through that resistor, i. R , to lag the source current, is , by 45◦ when = 5 krad/s. By KCL 44

EX 9. 16 Analyzing Capacitive Loading Effects Use phasor diagrams to explore the effect of adding a capacitor across the terminals of the load on the amplitude of Vs if we adjust Vs so that the amplitude of VL remains constant. Utility companies use this technique to control the voltage drop on their lines. 45

EX 9. 16 Analyzing Capacitive Loading Effects (Contd. ) 46