Question

2.42 Show that the circuit in Fig. P2.42 is all pass. Specifically: (a) Show that its gain is unity at all frequencies.

Do without using Laplace

Answer 1

(4) -loswot Va I + Valt) - TV Write in phosor wilt) = Coswt) V = 1 LO & V₂ H) = Volt) - Vy(t) write into phasor form V = Va Vo ZR=R & capacitor write in form impedance Ze='+jxe - - terminal (e-d) and terminal (F-E) paralld, the voltage across terminal (o-d) and terminal (F-F) are same v vez. Óz, vz Vi=V2 =V because parallel in voltage same m each arms tence ved=V&E = V, = 1 Lov ® Now find voltage to using voltage division rule Vo = ved 2 = 1 SER V ER

het V-V-7 - Pyjoe = Vo - Vo = 14 l RC - WRC Ve=V(jakso Hlin) 12 = 1-jWRC 1 1+jwc imput Gain 1 + (WRC) ² 1+(WRC)2 V₂ = output V, = imput Hljw) = framster function = output [How) = 1 + (WRCA - 1+ (wrc) - 1 J1+ (WRC) ² - It (RWC)2 - | Hjw) = Gain = 1 this gan value Dall frequency because independ to frequenc 6 from equation write phase angle [H(jw) = tan" | WRC) - tan (WRC) A lost wat) - HW input Y CH) = 1 + (jw) w hos luotto) ! where $ =/H(jw) ... I w=w. imput -Hliw) - Y(t) = H ( w/ w: Sm (Wot) = output 1 Coswot)

VC4) = Hiw) | www.Smw.t) =/(ww.cos(wat - 909) where $ = - 90=/ Hiw|ww. compare equation on she H(jw) / w-5-90 = tan 1 - WRC) - tan (WRC) . TW=W. -90" = - tan ( W.Rc) - tan' (wore) - 90º = -2tan (WRC) 45 = tan' | Wo Rc) Worc = tan 45º = 1 Two - 1 e rad/sec] this frequency input output signal phase difference go