Phasor Diagram of an RC Circuit V R

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ÚΑἰσχύλος Ταρσούλη
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1 ESE Lecture 3 Phasor Dagram of an rcut VtV m snt V t V o t urrent s a reference n seres crcut KVL: V m V + V V ϕ I m V V m

2 ESE Lecture 3 Phasor Dagram of an L rcut VtV m snt V t V t L V o t KVL: V m V + V L V L ϕ V m V I m

3 ESE Lecture 3 Phasor Dagram of a Seres L rcut V V L L V V KVL: V m V + V L + V V L V m V L- V I m V Voltages across capactor and nductor compensate each other

4 ESE Lecture 4 Integratng rcut V V t V o t V V + V or V V + V Assume hgh frequency / << then V >> V V V V V Vdt V O t Idt Vdt 0 0 t O Accurate ntegratng can be obtaned at hgh frequency that leads to a low output sgnal

5 ESE Lecture 4 Dfferentatng rcut V V t V o t V V + V or V V + V Assume low frequency / >> then V << V V V V V V t Idt I d dt V 0 V V I V d dt V O O d dt V An accurate result can be achved at low frequency.

6 ESE Lecture 4 Transent Processes n Passve rcuts rcut wthout a Source The crcut response s due only to the energy stored n the capactor t0 The capactor s precharged to the voltage V 0 Applyng KL: dv/dt +V/ 0 Ths s the st order dfferental equaton

7 ESE Lecture 4 Soluton of the Dfferental Equaton dv/dt + V/ 0 dv/v -dt/ ln V -t/ +a V A e -t/, Ae +a e s the base of the natural logarthms, e.78 ontnuty requres the ntal condton: At t 0 V0 V 0 Vt V 0 e -t/ The response governed by the elements themselves wthout external force s called a natural response

8 ESE Lecture 4 The Tme onstant Vt V V V 0 τ tme 3τ The tme constant τ s the rate at whch the natural response decays to zero Tme τ s requred to decay by a factor of /e0.368 After the tme perod of 3τ the transent process s consdered to be completed

9 ESE Lecture 4 Determnaton of the Tme onstant V 0 V V V/e τ t From the soluton of the equaton By defnton: as the tme when the sgnal decreases by /e Vt+τ/Vt /e The tme t can be chosen arbtrarly From the slope of the lne at t0 Dfferentatng the soluton dv/dt -{V 0 /τ } e t/τ V t -{V 0 /τ }t +V 0 V τ 0

10 ESE Lecture 4 Transent esponse of an Integratng rcuts V V V KVL: V V + V V t V O V V t t

11 ESE Lecture 4 Transent esponse of a Dfferentatng rcuts V V V KVL: V V + V V t V O V t V

12 ESE Lecture 5 Transfer Functon V t V o t V t V m cost V o t V om cost+ϕ V om ϕ V o t e { V o e jt }, V o V om e jϕ T V o V Ampltude response T V om /V m Phase response T ϕ

13 ESE Lecture 5 The Transfer Functon of an Integratng rcut V t V o t j j j V V T o + + τ j T + τ

14 ESE Lecture 5 Ampltude esponse of the Integratng rcut The magntude of the transfer functon s T + τ The Integratng crcut s a LOW-PASS flter V o /V n ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο V o /V n db 0 0 slope 0-0 db/dec V 0 [db] ο 0.0 ο ο 0. ο 0 ο Frequency 00 ο 000 ο

15 ESE Lecture 5 Phase esponse of the Integratng rcut The angle of the transfer functon s T Im arctan e [ T ] [ T ] jτ T arctan τ + jτ jτ 0 V o ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο

16 ESE Lecture 5 The Transfer Functon of a Dfferentatng rcut V t V o t o X I I V V T j j j X T + + τ τ τ τ τ τ j j j T τ

17 ESE Lecture 5 Ampltude response of the Dfferentatng rcut Ampltude response of the crcut s the magntude of T T τ τ + + j τ τ + τ The Dfferentatng crcut s a HIGH-PASS flter.0 Vo/V n 0.5 V V n ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο 0 Vo/V n slope -0 db/dec or -6 db/oct -3 db pont at V 0 [db] ο 0.0 ο ο 0. ο Frequency 0 ο 00 ο 000 ο

18 ESE Lecture 5 Phase response of the Dfferentatng rcut Phase response of the crcut s the phase angle of T τ τ + + τ T j T arctan τ 90 V ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο

19 ESE Lecture 5 Frequency esponse of a Seres L rcut V V L L V V KVL: V V + V L + V V L V L- V V I m V Voltages across capactor and nductor compensate each other

20 ESE Lecture 5 Ampltude esponse of a Seres L rcut V 0 / V IN jl + j + + L.0 V o /V n 0.5 V V n ο ο 0. ο ο Frequency 0 ο 00 ο 000 ο V /V n, V L /V n V /V n V L /V n ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο

21 ESE Lecture 5 Ampltude and Phase esponse of the Ladder rcut Vn Vo 0 V o ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο V o /V n slope -40 db/dec V 0 [db] ο 0.0 ο 0. ο ο 0 ο Frequency 00 ο 000 ο

22 ESE Lecture 5 Ampltude and Phase esponse of a Wen-Brdge rcut Vn Vo T Z Z + j + j + + OUT TOTAL j V o ο ο 0. ο ο Frequency 0 ο 00 ο 000 ο V /V o n BandWdth V V max ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο

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