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
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
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
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
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.
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
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
ESE Lecture 4 The Tme onstant Vt V 0 0.368 V 0 0.05 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
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
ESE Lecture 4 Transent esponse of an Integratng rcuts V V V KVL: V V + V V t V O V V t t
ESE Lecture 4 Transent esponse of a Dfferentatng rcuts V V V KVL: V V + V V t V O V t V
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
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
- - 0 3 - - 0 3 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.0 0.5 0.707 0.0-3 0.00 ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο V o /V n 0-0 - -3 db 0 0 slope 0-0 db/dec -0-40 V 0 [db] 0-3 -3-60 0.00 ο 0.0 ο ο 0. ο 0 ο Frequency 00 ο 000 ο
-3 - - 0 3 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 -45-90 0.00 ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο
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
- - 0 3 - - 0 3 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 0 0.707 V n 0.0-3 0.00 ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο 0 Vo/V n 0-0 - 0 0 slope -0 db/dec or -6 db/oct -3 db pont at 0-0 -40 V 0 [db] 0-3 -3-60 0.00 ο 0.0 ο ο 0. ο Frequency 0 ο 00 ο 000 ο
-3 - - 0 3 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 0 45 0 0.00 ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο
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
- 0 3-3 - - 0 3 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 0 0.707 V n 0.0-3 - 0.00 ο.5 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο V /V n, V L /V n.0 0.5 V /V n V L /V n 0.0 0.00 ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο
-3 - - 0 3 - - 0 3 ESE Lecture 5 Ampltude and Phase esponse of the Ladder rcut Vn Vo 0 V o -90-80 0.00 ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο 0 0 0 V o /V n 0-0 -4 slope -40 db/dec -40-80 V 0 [db] 0-6 -3-0 0.00 ο 0.0 ο 0. ο ο 0 ο Frequency 00 ο 000 ο
-3 - - 0 3-0 3 ESE Lecture 5 Ampltude and Phase esponse of a Wen-Brdge rcut Vn Vo T Z Z + j + j + + OUT TOTAL j 00 50 V o 0-50 -00 0.00 ο 0.33 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο V /V o n 0.4 5 BandWdth V 0 0.707 V max 0.0-3 - 0.00 ο 0.0 ο 0. ο ο Frequency 0 ο 00 ο 000 ο