hapte 33 Altenating uent icuits Most of the electical enegy is poduced by electical geneatos in the fom of sinusoidal altenating cuent. Why do we use the sinusoidal electic potential but neithe the tiangula no the squae wes? 33. A Souces Φ m BA, d Φ db da d cos m Acos θ + B cosθ + BA θ What s the best and the cheapest way to make the flux change? Φ m NBAcosθ if the angula velocity is --> Φm NBAcos ( t + δ ) dφm NBAsin t + ( t + δ ) sin( δ ) 33. esistos in an A icuit ( δ ) sin t + et π δ --> sin ( t + δ ) cos( t), cos ( t) Applying Ohm s law (What is the Ohm s law? Why do we use this? --> inea elation between and )
,, cos( t) --> cos( t) cos( t) oot Mean Squae alues ( instantaneous ), cos( t) instantane ous ( ) ( cos ( t) ) instantaneous. 77 Example: Find the eage cuent and the cuent fo the sawtooth wefom. n the egion <t<, the cuent is given by t t. ( ) t --> 3 3 33.3 nductos in an A icuit d Apply Kichhoff s loop ule: d d --> cos ( t)
d cos ( t) --> sin t cos ( t) d π cos t --> ( ) ( ) -->, sin t, -->, compae with --> esistance like --> impedance --> inductive eactance he esistance is not changed with the fequency but the impedance may change with the fequncy., --> nstantaneous owe Deliveed: ( ( t) ) sin( t) ( ) sin( t) cos he eage of dissipated enegy is: sin( t) dissipated in an inducto.. --> No enegy is Example: nductive eactance he potential dop acoss a 4-mH inducto is sinusoidal with a potential dop of. Find the inductive eactance and the cuent when the fequency is 6 Hz. 3 ( π )( 6)( 4 ) 5.Ω, 33.4 apacitos in an A icuit Q Q, 3
--> cos ( t) dq d Q --> cos( t) sin ( ) sin( t) sin( t) ( t) cos t + π --> --> Example: apacitive eactance A -µf capacito is placed acoss an ac geneato that applies a potential dop with an amplitude ( value) of. Find the capacitive eactance and the cuent amplitude when the fequency is 6 Hz. 33Ω 6 π 6, ( )( ) 33.5 he Seies icuit hasos he cuent in a steady-state ac cicuit vaies with time as cos( t) --> he voltage dop acoss a esistance is cos( t). --> he voltage dop acoss an inducto is sin( t) π cos t d. π cos t + 4
--> he voltage dop acoss a capacito is sin( t) π cos t π cos t Seies icuit app cos ( t) d Q --> Diff. Eq.: cos ( t) d Q dq cos( t) Q (foced oscillation d x dx m + b + kx F cos( t) ) Foced Oscillations --> Foce he Oscillato ibate with he Fequency Q Asin t δ et ( ) t + A t δ A t δ A t δ --> cos( ) sin( ) cos( ) sin( ) cos A ( t) + A( sin( t) cosδ cos( t) sinδ ) ( cos( t) cosδ + sin( t) sinδ ) A( sin( t) cosδ cos( t) sinδ ) cos ( t) : Asinδ A cosδ + Asinδ sin ( t) : cosδ sinδ cosδ --> tan δ --> A + ( ) ( ) ( + ) ( ) + 5
Q sin t + ( ) + ( ) Z ( δ ), cos t + ( ) ( δ ) otal eactance: mpedance: ( ) Z + haso: app cos( t) & cos ( t δ ) --> cos ( t δ ) app d +, + + π sin,, + ( t δ ) cos ( t δ ) cos ( t δ ) + ( ) app,,,,,,,,, + π app --> + ( ) Z (Z: impedance) + tan δ Example: A esisto and capacito ae in seies with a geneato. he geneato applies a potential dop acoss the combination given by cos( t) app. app, Find the potential dop acoss the capacito as a functional of fequency. cos t ( + δ ) ( + δ ) cos t 6
cos t π ( + δ / ) app + ( ) app app, + / ( ) + / ( ) + app, out / out, ( ) out, app, + aallel icuit + / 33.6 owe in an A icuit he instantaneous powe dissipation: cos ( t) he eage powe dissipation: ( ) ( cos ( t) ) ( cos ( t) ) What s the time eage? What s the spatial eage? ( ) time _ λ ( ) spatial _ dx λ π cos t ( cos ( t) ) π + cos t --> 7
he eage powe deliveed by the geneato is: ( ) ( cos ( t) ) ( ) cos( t) cos( t), --> Fo the cicuit: ( sinφ) ( t) cos( t φ) cos ( t) cosφ cos( t) sin( t) cos + g cos φ Z 33.7 esonance in a Seies icuit Z ( ) + --> Z + ( ) + --> Z + + ( ) he powe supplied to the esisto is. app, app, Z ( ) + he dissipated powe has a maximum value at the esonance fequency. Full wih at half maximum: Quality Facto: Q / 33.8 he ansfome and owe 8
ansmission Assume that the magnetic flux though a single tun of coil is dφm oop : N --> dφ m N oop : (he coils behe as a voltage souce.) dφ N m N N φ m. Enegy conseved: the powe input fom the geneato is equal to the powe output, -->. f the input voltage is and the output voltage is, which coppe wie of the coils is thick? (the input coils o the output coils) 33.9 ectifies and Filtes 9
ow ass Filte High ass Filte