- 1. e, Phasor, and Sinusoidal Steady-State Analysis Chien-Jung Li Department of Electronic Engineering National Taipei University of Technology
- 2. Department of Electronic Engineering, NTUT Compound Interest • 複利公式: 本金P, 年利率r, 一年複利n次, t年後本金加利息之總和為 1 nt r S P n • Let P=1, r=1, and t=1 1 1 n S n When n goes to infinite, S converges to 2.718… (= e) Let P=10萬, r/n=10%/12, t=1 S=11,0471 Let P=10萬, r/n=10%, and n=36, t=1 S=3,091,268 2/33
- 3. Department of Electronic Engineering, NTUT Development of Logarithm • Michael Stifel (1487-1567) • John Napier (1550-1617) • 利用對數而將乘法變成加法的特性，刻卜勒成功 計算了火星繞日的軌道。 2 52 5 7 m m m m 7 7 4 3 4 m m m m 2 2 3 1 3 1m m m mm 3 2 1 0 1 2 3 , , , , 1, , , ,m m m m m m m 3/33
- 4. Department of Electronic Engineering, NTUT Definition of dB (分貝) • , where • Power gain • Voltage gain • Power (dBW) • Power (dBm) • Voltage (dBV) • Voltage (dBuV) 10 logdB G aG b 2 1 10 log P P 2 1 20 log V V 10 log 1-W P 10 log 1-mW P 20 log 1-Volt V 20 log 1- V V 相對量 (比例, 比值, 無單位, dB) 絕對量 (因相對於一絕對單位, 因此可表示一絕對量. 有單位, 單位即為dBW, dBm, dBV…) 4/33
- 5. Department of Electronic Engineering, NTUT In some textbooks, phasor may be represented as Euler’s Formula • Euler’s Formula cos sinjx e x j x cos Re Re j t j j t p p pv t V t V e V e e def j p pV V e V • Phasor (相量) Don’t be confused with Vector (向量) which is commonly denoted as A (How it comes?) 取實部 (即cosine部分) phasor Consider a real signal v(t) that can be represented as: V V 5/33
- 6. Department of Electronic Engineering, NTUT Definition of e lim 1 n x n x e n 2 3 lim 1 1 1! 2! 3! n x n x x x x e n x jx 2 3 1 1! 2! 3! jx jx jxjx e • Euler played a trick let , where 1j 1 lim 1 n n e n 6/33
- 7. Department of Electronic Engineering, NTUT • Since , , , How It Comes… 1j 2 1j 3 1j 4 1j 2 4 3 5 1 2! 4! 3! 5! x x x x j x 2 4 cos 1 2! 4! x x x 3 5 sin 3! 5! x x x x cos sinjx e x j x cos sinjx e x j x cos 2 jx jx e e x sin 2 jx jx e e x j 2 3 1 1! 2! 3! jx jx jxjx e • Use and we have (姊妹式) 7/33
- 8. Department of Electronic Engineering, NTUT Coordinate Systems x-axis y-axis x-axis y-axis P(r,θ) θ r P(x,y) 2 2 r x y 1 tan y x cosx r siny r Cartesian Coordinate System (笛卡兒座標系, 直角座標系) Polar Coordinate System (極坐標系) (x,0) (0,y) cos ,0r 0, sinr Projection on x-axis Projection on y-axis 8/33
- 9. Department of Electronic Engineering, NTUT Sine Waveform x-axis y-axis P(x,y) x y r θ θθ y θ 0 π/2 π 3π/2 2π Go along the circle, the projection on y-axis results in a sine wave. 9/33
- 10. Department of Electronic Engineering, NTUT x θ 0 π/2 π 3π/2 Cosine Waveform x-axis y-axis θ Go along the circle, the projection on x-axis results in a cosine wave. Sinusoidal waves relate to a Circle very closely. Complete going along the circle to finish a cycle, and the angle θ rotates with 2π rads and you are back to the original starting-point and. Complete another cycle again, sinusoidal waveform in one period repeats again. Keep going along the circle, the waveform will periodically appear. 10/33
- 11. Department of Electronic Engineering, NTUT Complex Plan (I) It seems to be the same thing with x-y plan, right? • Carl Friedrich Gauss (1777-1855) defined the complex plan. He defined the unit length on Im-axis is equal to “j”. A complex Z=x+jy can be denoted as (x, yj) on the complex plan. (sometimes, ‘j’ may be written as ‘i’ which represent imaginary) Re-axis Im-axis Re-axis Im-axis P(r,θ) θ r P(x,yj) 2 2 r x y 1 tan y x cosx r siny r (x,0j) (0,yj) cos ,0r 0, sinr 1j 11/33
- 12. Department of Electronic Engineering, NTUT Complex Plan (II) Re-axis Im-axis 1 Every time you multiply something by j, that thing will rotate 90 degrees. 1j 2 1j 3 1j 4 1j 1*j=j j j*j=-1 -1 -j -1*j=-j -j*j=1 (0.5,0.2j) (-0.2, 0.5j) (-0.5, -0.2j) (0.2, -0.5j) • Multiplying j by j and so on: 12/33
- 13. Department of Electronic Engineering, NTUT Sine Waveform Re-axis Im-axis P(x,y) x y r θ θθ y=rsinθ θ 0 π/2 π 3π/2 2π To see the cosine waveform, the same operation can be applied to trace out the projection on Re-axis. 13/33
- 14. Department of Electronic Engineering, NTUT Phasor Representation (I) – Sine Basis sin Im Imj j t j j sv t A t Ae e Ae e Re-axis Im-axis P(A,ф) y=Asin ф θ 0 π/2 π 3π/2 2π ф t Given the phasor denoted as a point on the complex-plan, you should know it represents a sinusoidal signal. Keep this in mind, it is very very important! time-domain waveform 14/33
- 15. Department of Electronic Engineering, NTUT Phasor Representation (II) – Cosine Basis cos Re Rej j t j j sv t A t Ae e Ae e Re-axis Im-axis P(A,ф) y=Acos ф θ 0 π/2 π 3π/2 2π ф t time-domain waveform 15/33
- 16. Department of Electronic Engineering, NTUT Phasor Representation (III) 1 1 1 1 1sin Im j j t v t A t A e e Re-axis Im-axis P(A1,ф1) ф1 P(A2,ф2) P(A3,ф3) θ 0 π/2 π 3π/2 2π t A1sin ф1 2 2 2 2 2sin Im j j t v t A t A e e 3 3 3 3 3sin Im j j t v t A t A e e A2sin ф2 A3sin ф3 16/33
- 17. Department of Electronic Engineering, NTUT Mathematical Operation j t j tde j e dt 1j t j t e dt e j 0 1 t v t i t dt C 0 1 1t j t j t j t Ve Ie dt I e C j C 1 CV I Z I j C di t v t L dt j t j t j t d Ie Ve L j LI e dt LV j L I Z I 1 1 CZ j C sC LZ j L sL • L and C: from time-domain to phasor-domain analysis (s is the Laplace operator) , here let 0s j 17/33
- 18. Department of Electronic Engineering, NTUT Phasor is what you always face with • 電路學、電子學: Phasor 常見為一個固定值 (亦可為變量) • 電磁學、微波工程: Phasor 常見為變動量, 隨傳播方向變化 • 通訊系統: Phasor 常見為變動量, 隨時間變化 此變動的phasor也經常被稱作複數波包(complex envelope)、波包 (envelope)，或帶通訊號的等效低通訊號(equivalent lowpass signal of the bandpass signal)。Phasor如果被拆成正交兩成分，常稱作I/Q訊 號，而在數位通訊裡表示I/Q訊號的複數平面(座標系)也被稱為星座 圖(constellation)。 • Don’t be afraid of phasor, you will see it many times in your E.E. life. It just appears with different names, and it is just a representation or an analysis technique. • Keep in mind that a phasor represents a signal, it’s like a head on your body. 18/33
- 19. Department of Electronic Engineering, NTUT Simple Relation Between Sine and Cosine • Sine Cosine π/2 π 3π/2 2π sinθ θ 0 cosθ • Negative sine or cosine cos sin 90 sin cos 90 cos cos 180 sin sin 180 Try to transform into sine-form:cos cos sin 90 sin 270 sin 90 19/33
- 20. Department of Electronic Engineering, NTUT Cosine as a Basis cos Re j t pv t V t Ve 0pV V sin cos Re 2 j t p pv t V t V t Ve 90pV V cos cos Re j t p pv t V t V t Ve 180pV V sin cos Re 2 j t p pv t V t V t Ve 90pV V cosine sine negative cosine negative sine Phasor Phasor Phasor Phasor 20/33
- 21. Department of Electronic Engineering, NTUT Sine as a Basis sin Im j t pv t V t Ve 0pV V cos sin Im 2 j t p pv t V t V t Ve 90pV V sin sin Im j t p pv t V t V t Ve 180pV V cos sin Im 2 j t p pv t V t V t Ve 90pV V Phasor Phasor Phasor Phasor cosine sine negative cosine negative sine 21/33
- 22. Department of Electronic Engineering, NTUT Addition of Sinusoidal A basic property of sinusoidal functions is that the sum of an arbitrary number of sinusoids of the same frequency is equivalent to a single sinusoid of the given frequency. It must be emphasized that all sinusoids must be of the same frequency. sinpv t V t 1 1 1pV V 2 2 2pV V n pn nV V 1 2 nV V V V 1 1 2 2sin sin sinp p pn nv t V t V t V t 1v t 2v t nv t 22/33
- 23. Department of Electronic Engineering, NTUT Example 0 1 2v t v t v t 1 20cos 100 120v t t 2 15sin 100 60v t t 1 20 30 17.3205 10V j 2 15 120 7.5 12.9904V j 0 17.3205 10 7.5 12.9904V j j 0 25sin 100 66.87v t t 9.8205 22.9904 25 66.87j 1 20 120 10 17.321V j 2 15 150 12.9904 7.5V j 0 10 17.321 12.9904 7.5V j j 22.9904 9.8205 25 23.13j 0 25cos 100 23.13v t t 25sin 100 66.87t Choose the basis you like, and the results are identical. and For calculate use sine function as a basis use cosine function as a basis 23/33
- 24. Department of Electronic Engineering, NTUT Steady-state Impedance V Z R jX I • Steady-state impedance resistance reactance I Y G jB Z • Steady-state admittance conductance susceptance 30 40Z j 30R 40X 1 0.012 0.016 30 40 Y j j 0.012G S 0.016X S 24/33
- 25. Department of Electronic Engineering, NTUT Conversion to Phasor-domain i t v t V I RR i t v t i t v t C L 1 j C V I j LV I V R I 1 V I j C V j L I V I V I V I V and I are in-phase V lags I by 90o V leads I by 90o R C L 25/33
- 26. Department of Electronic Engineering, NTUT Frequency Response Frequency-independent All pass Frequency-dependent High-pass Frequency-dependent Low-pass V I R 1 j C V I j LV I Z R jX R 1 Z R jX C 2 f 2 f 2 f Z R jX L 26/33
- 27. Department of Electronic Engineering, NTUT Calculate the Impedance (I) 1 j C V • Calculate the impedance of a 0.01-uF capacitor at (a) f=50Hz (b) 1kHz (c) 1MHz 6 1 0 318.309 k 2 50 0.01 10 Z R jX j j 318.309 kX 318.309 kZ I (a) f = 50 Hz 3 6 1 0 15.92 k 2 1 10 0.01 10 Z R jX j j 15.92 kX 15.92 kZ (b) f = 1 kHz 6 6 1 0 15.92 2 1 10 0.01 10 Z R jX j j 15.92X 15.92Z (c) f = 1 MHz 0.01 μFC 27/33
- 28. Department of Electronic Engineering, NTUT Calculate the Impedance (II) • Calculate the impedance of a 100-mH inductor at (a) f=50Hz (b) 1kHz (c) 1MHz 3 0 2 50 100 10 31.42Z R jX j j 31.42X 31.42Z (a) f = 50 Hz 3 3 0 2 1 10 100 10 628.32Z R jX j j 628.32X 628.32Z (b) f = 1 kHz 6 3 0 2 1 10 100 10 628.32 kZ R jX j j 628.32 kX 628.32 kZ (c) f = 1 MHz j LV I 100 mHL 28/33
- 29. Department of Electronic Engineering, NTUT Calculate the Impedance (III) • Calculate the impedance of following circuit at (a) f=50Hz (b) 1kHz (c) 1MHz 6 1 200 0.2 318.309 k 2 50 0.01 10 Z R jX j j 318.309 kZ (a) f = 50 Hz 3 6 1 200 0.2 15.92 k 2 1 10 0.01 10 Z R jX j j 15.92 kZ (b) f = 1 kHz 6 6 1 200 200 15.92 2 1 10 0.01 10 Z R jX j j 200.63Z (c) f = 1 MHz 1 j C 0.01 μFC R 200R 318.309k 89.96Z 15.92k 89.26Z 200.63 -4.55Z 29/33
- 30. Department of Electronic Engineering, NTUT Calculate the Impedance (IV) • Calculate the impedance of following circuit at (a) f=50Hz (b) 1kHz (c) 1MHz 3 200 2 50 100 10 200 31.42Z R jX j j 202.45Z (a) f = 50 Hz 3 3 200 2 1 10 100 10 200 628.32Z R jX j j 659.38Z (b) f = 1 kHz 6 3 200 2 1 10 100 10 0.2 628.32 kZ R jX j j 628.32 kZ (c) f = 1 MHz j L 100 mHL R 200R 202.45 8.93Z 659.38 72.34Z 628.32 k 89.98Z 30/33