Lecture 5 omplex Variables II (Applicatios i Physics) (See hapter i Boas) To see why complex variables are so useful cosider first the (liear) mechaics of a sigle particle described by Newto s equatio with viscous dampig (as appeared first i Lecture ) a liear restorig force ad a drivig force mx bx kx F t (5) Alog the same lie recall Kirchoff s equatio describig a (series) RL circuit with a voltage source With the curret I as the free variable (coordiate) we have dt IR I LI V t (5) where the first term is the voltage across the resistor the secod term is the voltage across the capacitor ( V Q ) the third term is the voltage across the iductor ad the right-had-side is the applied voltage Writte i terms of the charge o the capacitor Kirchoff s equatio is idetical i form to Newto Q LQ RQ V t (53) ie i both situatios we have a liear secod order ihomogeeous differetial equatio (due to the drivig term o the RHS) Such a situatio with multiple similar equatios is simplified by applicatio of the rule of Feyma (oe of may rules with the same ame) which i this case states that the same equatios have the same solutios Oly the ames of the variables ad costats have chaged i goig from Eq (5) to Eq (53) m L b R k ad F V Here we will study Eq (53) but the discussio applies also to Eq (5) With oly a small loss of geerality (as we will see later i the course) we ca assume that the drivig voltage V t V t with period ad we expad the time is a periodic fuctio of time depedece i a Fourier Series (more about this later) V t V cos t where (the fudametal frequecy) This set of fuctios (with both a magitude ad phase to be specified for each term) costitutes a complete set of fuctios with the required periodicity ay fuctio with this periodicity ca be represeted as such a sum! We will prove this essetial poit later i the course Physics 7 Lecture 5 Autum 8
Note that the = terms allows for the possibility of a costat term while the phases are equivalet to icludig both sies ad cosies We focus first o the fudametal frequecy term which we ca rewrite as i it it V t V cos t Re Ve e Re V e (54) i We have defied the complex costat V Ve which carries the iformatio o both the magitude V ad phase of the fudametal compoet of the drivig voltage By the rule of Feyma we ca apply exactly the same decompositio to the drivig force i the oscillator problem (assumig that it is also periodic) F t Fcos t The correspodig Asatz (ie educated guess) for the charge o the capacitor is cos Q t Q t where each term requires us to solve for a i magitude Q ad a phase ie for the complex costat Q Qe Re i t Q t Q e The correspodig curret is the give by Re i t I t Q t i Qe (Note the symbols for the phases of the applied voltage ad of the charge o the capacitor) Magic Poit #: The essetial feature here is that both Eq (5) ad Eq (53) are liear equatios i the sese that the free dyamic variable (x ad Q respectively) appears liearly (to the power uity) i each term o the left-had-side As a result we ca use liear superpositio to solve the geeral equatio We break up the right-had-side of the equatio ito bite-sized pieces (idividual frequecies) as we are discussig here solve the equatio correspodig to each piece of the righthad-side the sum up all of these idividual solutios to fid the particular solutio to the origial equatio The geeral solutio to the origial equatio plus iitial coditios Qt xt ad Q t x t ca be foud by summig the particular solutio ad the solutio to the homogeeous problem with zero right-had-side (Uderstad this result ad you will have mastered much of this course!) The ext big step is to rewrite the real differetial equatio as a complex equatio We ca always take the real part i the ed to fid the desired physical solutio For Physics 7 Lecture 5 Autum 8
ow we focus o a sigle form of the time depedece ie a sigle frequecy for the drivig voltage We fid after switchig to complex otatio takig derivatives i t ad cacelig commo factors that (recall our Asatz is Qt Re Q e ) Q LQ RQ V t i t d d ReQe L Re Q e R Re Q e Re V e i t it it dt dt i t d i t d i t Q e it L Q e R Q e V e dt dt i t i t i t Qe it L Q e irq e V e Q L Q irq i t V cacelig the e factor (55) it it Magic Poit #: By usig complex otatio cost Re e e ad the t t special feature of the expoetial fuctio de dt e we have succeeded i covertig the origial differetial equatio ito a algebraic equatio which is solvable by elemetary meas (ie arithmetic) This is a major step i simplifyig our task (Recall that we are lazy ad smart!) It is ow a simple matter to solve for the complex form of the charge ad the curret V Q L ir I t Q t I i Q I i V V i L ir R il (56) This last expressio plus our previous experiece with D circuits I V R suggests that we defie a complex frequecy depedet impedace via Physics 7 Lecture 5 3 Autum 8
Z R il R il i (57) where Z R L phasez Z ta L R (58) These expressios suggest that the frequecy L where Z R ad Z must play a special role This is just the atural frequecy of the L circuit whe there is o real resistace ( R ) i the same way that the atural frequecy of the udamped oscillator i Eq (5) ( b ) is HO km I terms of this parameter we ca write L Z L R L L phasez Z ta L R (59) The startig with the complex form of the applied voltage of frequecy the correspodig complex charge ad curret are V V I Q Z iz (5) I particular for the case of a sigle fudametal frequecy we have Physics 7 Lecture 5 4 Autum 8
Q Q e e V i Z I iq e Z i V Z i Z (5) where Z Z R L L L ta L R (5) Thus for the sigle drivig voltage V t V cos t the use of complex variables (ad the bits of magic oted above) allows us to quickly solve for the correspodig curret i a geeral RL circuit We fid it i V I t ReI e cos t (53) Z Z ie the resultig curret differs from the applied voltage i magitude by over the modulus of the complex impedace (evaluated at the drivig frequecy) i phase from the applied voltage by the egative of the phase of the complex impedace (evaluated at the drivig frequecy) Note that the correspodig charge (o the capacitor) differs from the curret by a factor of i magitude ad i phase ( the charge lags behid the curret by 9 ie it takes time for the curret to build up the charge) Physics 7 Lecture 5 5 Autum 8
V Re Z Z i t i Q t Q e cos t V t Z Z si (54) Also ote that if the drivig voltage is give istead by a sie fuctio ie the resultig curret i Eq (53) is also give by a sie fuctio ad the charge i Eq (54) will be give by mius a cosie fuctio (with all the fuctios t ) havig the same argumet Z Lookig at these results we observe first that as expected the respose of the circuit has its maximum amplitude whe the drivig frequecy equals the atural frequecy ( L this is how radio tuers work) where the impedace has its miimum magitude ad vaishig phase (ie the drivig voltage ad the curret are i-phase while the charge o the capacitor lags the drivig voltage by 9) L Z R Z R I L L V V L R R Q L L (55) Next we observe that i the limit where the drivig frequecy is well below the atural frequecy (ad the goes to zero i the limit with V t V ) Z Z L Z ta I V cost Q V si t V (56) Physics 7 Lecture 5 6 Autum 8
which is the expected result for a D circuit The charge just builds up o the capacitor to match the applied D voltage I the opposite limit of a drivig frequecy well above the atural frequecy we have Z L Z L L L Z ta R V V I t cos t si t L L V V Q t si t cos t L L (57) The amplitudes of both the charge ad the curret vaish i the limit of very high frequecy The circuit simply caot respod to a drivig voltage that oscillates at a frequecy much larger tha its atural frequecy To see what sets the scale for how rapidly the impedace varies with frequecy we cosider the impedace i the form L R Z L L The we ask how far must vary from L (58) before Z Z R This is give by the equatio L R L Z varies by a factor L R 4 L L L R L L (59) Physics 7 Lecture 5 7 Autum 8
Z The width of the peak typically called a resoace i the quatity Z is determied by the dampig i the system This characteristic resoace shape will appear ofte i physics ad is illustrated i the ext figure correspodig to L RL ad the x-axis is the scaled frequecy L 8 6 4 ASIDE : Aother way to thik about the quatity Z is to cosider where the (complex) impedace vaishes Z R R i L (5) L 4L Thus the quatity Z has () simple poles at the complex frequecies i Eq (5) 9 95 5 L Z i L (5) Of course i physical applicatios with ad real we ca get close oly to the pole at We ca easily check that the expressio i Eq (5) vaishes i the limits ad ( Z see Eqs (56) ad (57)) ad goes to its maximum value for real frequecies R at L (see Eq (55)) The reader is ecouraged to perform this check (Note that the maximum value for real Physics 7 Lecture 5 8 Autum 8
frequecies is ot at Re due to the factor of i the umerator) The distace of the pole(s) from the real axis R L is just what sets the width of the peak i Eq (59) It is these poles that characterize the solutio to the homogeeous (udrive) equatio as we discuss below ASIDE : The (time) average power cosumed i the circuit correspodig to the frequecy ca also be easily calculated i terms of the complex curret ad voltage P dt I tv t V dt cos t t Z V Z Z cos cosz Re IV (5) which is very much like the D result except for the factor of ½ (from the time average of cos t ) ad the cosie of the phase differece betwee the I ad V If the curret ad drivig voltage are out of phase (as whe the R vaishes) eergy is oly stored i the circuit ad o power (eergy) is dissipated Now let s apply the rule of Feyma to simply write dow the form of the motio of a damped harmoic oscillator drive with the frequecy F t F cost hagig the ames i Eq (56) we have x F F i m i iho Fe e b (53) ta HO m b HO m HO m m ib k b m HO Physics 7 Lecture 5 9 Autum 8
Agai the maximum amplitude of motio arises for a drivig frequecy equal to the atural frequecy HO km F m x b k HO HO ta x HO HO (54) As we did i Eq (5) we ca thik of the amplitude i Eq (53) as havig simple poles i the complex frequecy plae ie we ca write x F F b m m i m HO HO HO b b b k b m 4m m m 4m HO i HO i (55) The correspodig time depedece for geeral is give by (see Eq (54)) x t F cos t HO b m HO m (56) For a geeral drivig force described by a sum over Fourier compoets Re i i o t F t Fe e F cos t complete form of the respose to the give drivig force called the particular solutio liear superpositio yields the F x t t cos (57) p HO b m HO m Physics 7 Lecture 5 Autum 8
Similar expressios with the ames chaged arise for the geeral drivig voltage i the RL circuit problem V I t t p cos Z Z V cos t L R L Z V Qp t si t Z (58) R L L Although Eqs (57) ad (58) appear very similar the astute studet will recogize that there is a subtle uderlyig issue arisig from our (covetioal) choice to defie the complex impedace to be real at the atural frequecy while the deomiator i Eq (53) is pure imagiary at the atural frequecy of the oscillator (this is why the expressio for Qt which is the closest aalog to xt is a sie fuctio istead of a cosie fuctio) We close this discussio of the use of complex variables to solve differetial equatios by cosiderig the homogeeous versio of Eq (5) (a similar aalysis works for Eq (53) with Vt ) This will allow us to write dow the complete solutio to such liear mechaics problems icludig both the particular ad homogeous (also called the complemetary) solutios (ad to fit ay iitial coditios) I particular cosider the equatio mx bx kx (59) t ad assume a Asatz of the familiar expoetial form xh xe with x a costat (ote that i geeral we are allowig both x ad to be complex) As above this leads to a algebraic equatio b b 4km b k b m b k i (53) m m m m Physics 7 Lecture 5 Autum 8
Note that whe we traslate to frequecies ad RL circuits these are just the positios of the poles of Z i Eq (5) i m L b R k R R i i L L L (53) For the usual case of small dampig where b km the secod term is truly imagiary ad the motio correspods to damped oscillatory motio with frequecy HO k m b m HO We ca express this behavior i a variety of forms H bt bt m i HOt i HOt m 3 cos HO x t e x e x e x e t bt m e x cos t x si t 4 HO 5 HO (53) We ca use the costats i each of the expressios ( x x or x 3 or x4 x 5) to fit the iitial coditios for x H ad x H Note that ulike the (ihomogeeous) drive case we are ot explicitly takig ay real parts here O the other had whe we fit to the (presumably real) iitial coditios we will fid a real xh t ie x 3 ad x4 x 5 will be real while x x will be appropriately complex ( x x x 4 ix x x5 ) I the opposite limit with b km called the over-damped case we have istead oly damped behavior ie the are pure real ad both egative ( b m k m b m ) H t t x t x e x e (533) I the special case of (so-called) critical dampig b km with b m we fid (the reader should cofirm that the followig expressio satisfies Eq (59)) H t x t e x x t (534) 6 7 Physics 7 Lecture 5 Autum 8
I all three cases (with b ) the homogeous solutio damps to zero asymptotically xh t Similar expressios arise for the udrive RL circuit as you may recall from Physics Fially for the geeral problem with a drivig force ad iitial coditios we sum (liear superpositio agai) the homogeeous solutios ad the particular solutio x t x t x t (535) H to obtai a solutio that (still) satisfies the ihomogeeous equatio Eq (5) that ca be matched to the iitial coditios usig the costats i xh t ad that is just the particular solutio due to the drivig force at asymptotic times (after the homogeous solutio has damped out) This is a extremely powerful result ad should be added to your kowledge base as quickly as possible! p Physics 7 Lecture 5 3 Autum 8