Sinusoidal Functions, Complex Numbers, and Phasors Discussion D9.2 Sections 4-2, 4-3 Appendix A 3/26/2006 1 Sinusoids A sinusoid is a signal that has the form of the sine or cosine function. XM x(t) 0 2 3 2 XM x(t) 2 t 3 2 0 2 2 x(t ) X M cos t x(t ) X M sin t period T 2 t sin 2 1 cos 2 0 2 Sinusoids In general: x(t ) X M sin t x(t ) X M cos t Note trig identities: sin t sin t cos cos t sin cos t cos t cos sin t sin 3 A sinusoidal current or voltage is usually referred to as an alternating current (or AC) or voltage and circuits excited by sinusoids are called ac circuits. Sinusoids are important for several reasons: 1. 2. 3. 4. 5. Nature itself is characteristically sinusoidal (the pendulum, waves, etc.). A sinusoidal current or voltage is easy to generate. Through Fourier Analysis, any practical periodic signal (one that repeats itself with a period T) can be represented by a sum of sinusoids. A sinusoid is easy to handle mathematically; its derivative and integral are also sinusoids. When a sinusoidal source is applied to a linear circuit, the steady-state response is also sinusoidal, and we call the response the sinusoidal steady-state response. 4 Complex Numbers What is the solution of X2 = -1 X 1 j Complex Plane j imaginary 1 -1 j 2 1 j 1 Note: real 1 j j 2 j j jj j -j 5 Complex Numbers A x jy imag A A cos j jA sin j y A sin j A x2 y 2 j tan 1 A y x j e jj cos j j sin j e j j j measured positive counter-clockwise A Ae jj Note: real x A cos j Euler's equation: Complex Plane e e cos j j sin j jj j 6 Sinusoidal Functions and Phasors t t=0 0 0 e t jt cos t j sin t http://www.kineticbooks.com/physics/trialp se/33_Alternating%20Current%20Circuits/1 3/sp.html 7 Relationship between sin and cos Im Im cos(t ) Re Re t=0 cos t=0 sin(t ) sin 0 Note that 0 sin t cos t 2 or Phasor projection on the real axis t sin t cos t 2 t 8 Relationship between sin and cos Im Im cos(t ) Re Re t=0 cos t=0 sin 0 0 sin(t ) sin t cos t 2 sin t sin t cos t sin t 2 Phasor projection on the real axis cos t cos t t t sin t sin t cos cos t sin cos t cos t cos sin t sin 9 Comparing Sinusoids -sin t cos t cos t Im -cos t Re cos t 2 Note: positive angles are counter-clockwise Im 45 135 45 cos t sin t 2 sin t cos t 45 sin t sin t sin t cos t sin t Re cos t cos t leads sin t by 90 cos t lags - sin t by 90 sin t 45 cos t 135 leads cost by 45 and leads sint by 135 10 R i(t) VM cos t + KVL + VR AC - VM cos t Ri (t ) L - + VL - L di (t ) dt This is a differential equation we must solve for i(t). How? Guess a solution and try it! Note that Re VM e jt Re VM cos t jVM sin t VM cos t It turns out to be easier to use VM e jt as the forcing function rather than VM cos t and then take the real part of the solution. de jt je jt This is because dt which will allow us to convert the differential equation to an algebraic equation. Let's see how. 11 Solve the differential equation for i(t). i(t) VM cos t + R + VR AC - + VL - di (t ) VM cos t Ri (t ) L dt L Instead, solve VM e jt Ri (t ) L di (t ) dt (1) and take the real part of the solution Guess that i(t ) I M e j t and substitute in (1) VM e jt RI M e j t j LI M e j t e jt RI M e j j LI M e j Divide by e j t VM RI M e j j LI M e j 12 Solve the differential equation for i(t). i(t) VM cos t + R VM e jt Ri (t ) L - + + VR AC L VL - - Rearrange (3) Recall that di (t ) dt i(t ) I M e j t VM RI M e j j LI M e j VM IM e R j L j R j L R 2 2 L2 e (1) (2) (3) (4) L j tan 1 R Therefore (4) can be written as I M e j VM R 2 2 L2 e L j tan 1 R (5) 13 Solve the differential equation for i(t). i(t) VM cos t + R + VR VM e jt Ri (t ) L - + AC L VL - i(t ) I M e j t I M e j Therefore, from (5) VM R 2 2 L2 IM di (t ) dt L j tan 1 R e VM 2 2 (2) (5) L tan R 1 R L 2 (1) (6) Substituting (6) in (2) and taking the real part i(t ) L cos t tan 1 2 2 2 R R L VM 14 Phasors A phasor is a complex number that represents the amplitude and phase of a sinusoid. X M e jj X M j X XM j j t Recall that when we substituted i(t ) I M e j t in the differential equations, the e cancelled out. We are therefore left with just the phasors 15 Solve the differential equation for i(t) using phasors. i(t) VM cos t + R + VR - + AC L VL - di (t ) dt (1) i(t ) I M e jt Ie jt (2) V VM 0 (3) VM e jt Ri (t ) L - Substitute (2) and (3) in (1) Ve jt RIe jt j LIe jt (4) Divide by e j t and solve for I I V I M R j L i(t ) L tan 1 R R 2 2 L2 VM L cos t tan 1 2 2 2 R R L VM (5) 16 By using phasors we have transformed the problem from solving a set of differential equations in the time domain to solving a set of algebraic equations in the frequency domain. The phasor solutions are then transformed back to the time domain. A cos t A A sin t A 90 17 Impedance Impedance Z i(t) VM cos t + AC - V phasor voltage I phasor current R + VR I - + VL - L Units = ohms V R j L V Z R j L Z z I Z R jX R resistance Z R2 2 L2 X reactance 1 L z tan R Note that impedance is a complex number containing a real, or resistive component, and an imaginary, or reactive, component. 18 Admittance 1 I phasor current Admittance Y = = Z V phasor voltage i(t) VM cos t + Units = siemens R + VR AC - conductance G V I R j L - + VL - R R 2 2 L2 L I 1 R j L Y= G jB 2 V R j L R 2 L2 susceptance B L R 2 2 L2 19 Capacitor Circuit R i(t) VM cos t + + VR VM e jt Ri (t ) -+ C VC AC - - Ve RIe (1) i(t ) I M e jt Ie jt (2) V VM 0 (3) Substitute (2) and (3) in (1) jt 1 i (t )dt C jt 1 Ie jt jC (4) Divide by e j t and solve for I V I R 1 jC i (t ) I M VM 1 R 2 2 C 2 VM R2 1 2C 2 1 tan 1 RC 1 1 cos t tan RC (5) 20 Impedance V phasor voltage Z I phasor current Impedance i(t) VM cos t + R + VR -+ VC AC - - 1 Z R 2 2 C 2 z tan 1 1 RC V I R C Z 1 jC V 1 1 R R j Z z I jC C Z R jX R resistance X reactance 21 Im V V I I Re I in phase with V V Im I V Re I lags V V I Im I V I Re I leads V 22 Expressing Kirchoff’s Laws in the Frequency Domain KVL: Let v1, v2,…vn be the voltages around a closed loop. KVL tells us that: v1 v2 ...vn 0 Assuming the circuit is operating in sinusoidal steady-state at frequency we have: VM 1 cos t 1 VM 2 cos t 2 ... VMn cos t n 0 or Re VM 1e j1 e jt Re VM 2e j2 e jt ... Re VMn e jn e jt 0 j Phasor Vk VMk e k Since e jt 0 Re V1 V2 ... Vn e jt 0 V1 V2 ... Vn 0 Which demonstrates that KVL holds for phasor voltages. 23 KCL: Following the same approach as for KVL, we can show that I1 I 2 ... I n 0 Where Ik is the phasor associated with the kth current entering a closed surface in the circuit. Thus, both KVL and KCL hold when working with phasors in circuits operating in sinusoidal steady-state. This implies that all of the circuit analysis methods (mesh and nodal analysis, source transformations, voltage & current division, Thevenin equivalent, combining elements, etc,) work in the same way we found for resistive circuits. The only difference is that we must work with phasor currents & voltages and the impedances &/or admittances of the elements. 24 I Find Zin DC Z1 V Z2 I1 Z3 I2 Z4 Zin Z1 Z2 Z3 Z3 Z3 I1 V Z3 Z 4 I 2 0 Z1 Z2 Z3 I1 Z3I2 V Z3I1 Z3 Z4 I 2 0 25 Z1 Z2 Z3 I1 Z3I2 V Z3I1 Z3 Z4 I 2 0 I DC Z1 V Z2 I1 Z3 I2 Z4 Zin Z3 I2 I1 Z3 Z 4 I I1 Z32 I1 V Z1 Z2 Z3 I1 Z3 Z 4 2 Z Z Z Z Z Z 1 2 3 3 4 3 Z3 Z 4 I1 V Z3Z 4 V Zin Z1 Z 2 I Z3 Z 4 26 I Z3Z 4 Zin Z1 Z 2 Z3 Z 4 DC Z1 V Z2 I1 Z3 I2 Z4 Zin We see that if we replace Z by R the impedances add like resistances. Impedances in series add like resistors in series Impedances in parallel add like resistors in parallel 27 Voltage Division Z1 + V1 DC V I Z2 + V2 - V I Z1 Z 2 But V1 Z1I V2 Z2I Therefore Z1 V1 V Z1 Z 2 Z2 V2 V Z1 Z 2 28