EE692 Applied EM- FDTD Method One-Dimensional Transmission Lines Notes- Lecture 4

Transcription

1 ee692_fdtd_d_tras_lie_lecture4.doc Page of 6 EE692 Applied EM FDTD Method OeDimesioal Trasmissio ies Notes ecture 4 FDTD Modelig of Voltage ources ad Termiatios with Parallel/eries R oads As the fial step toward modelig trasmissio lie circuits (see Figure ) this lecture will preset the FDTD update equatios ecessary to implemet voltage sources ad trasmissio lie termiatios with parallel or series R loads i the oedimesioal (D) FDTD trasmissio lie model []. This is a extesio of the work to implemet sigle lumped circuit elemets (e.g. capacitors iductors ad resistors) [2] ad parallel or series R loads [34] placed i parallel or series with the trasmissio lie preseted i prior lectures. The work o sigle lumped elemets ca be adapted i a similar fashio. ome results will be show to demostrate the accuracy ad validity of the update equatios by compariso with aalytic results. V t ( ) Voltage ource Parallel/eries R oad p ossless trasmissio lie possibly with R loads i parallel or series Parallel/eries R oad Z Z Parallel/eries Z v v R oad v p p Termiatig oad Parallel/eries R oad Figure Trasmissio lie circuit. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

2 ee692_fdtd_d_tras_lie_lecture4.doc Page 2 of 6 Voltage source with series R load i series Figure 2 shows the circuit represetatio icremetal ad discretized icremetal models for a voltage source with series R elemets i series. I Figure 2 l is the iductace per uit legth ad c is the capacitace per uit legth ad R ad are the lumped elemet resistor iductor ad capacitor respectively. R v t ( ) Z vp (a) v ( z t) izt () v R (b) I.5 ( k.5) I V (.5 ( k.5.5) ) k V ( k ) R V ( k ) (c) Figure 2 (a) ircuit represetatio (b) icremetal ad (c) discretized icremetal models of a voltage source with lumped elemet series R load i series. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

3 ee692_fdtd_d_tras_lie_lecture4.doc Page 3 of 6 Applyig Kirchoff s urret aw (K) to the top right ode of Figure 2b yields vz ( zt ) izt () iz ( zt ) c z =. Applyig Kirchoff s Voltage aw (KV) clockwise aroud the outside loop of Figure 2b yields izt () izt () v( z z t) v( z t) R i( z t) v l z = where v () zt = izt () t v ( zt) t. t These equatios whe discretized (see R loads lecture material) rearraged ad simplified yield the update equatios t.5 V ( k.5) = V ( k.5) I ( k.5) v t I ( k.5) = B I ( k.5) B V ( k ) V ( k ).5.5 p 2 Z v t B V ( k.5) p 2 Z ad vp t.5.5 V ( k ) = V ( k ) Z I ( k.5) I ( k.5) where ad vp t R Z 2 t B = vp t R Z 2 t B 2 = vp t R Z 2 t. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

4 ee692_fdtd_d_tras_lie_lecture4.doc Page 4 of 6 l Agai defie the characteristic impedace Z = phase velocity c ourat stability factor v t p =. This implies z Z l = ad v p v p = ad lc c =. Note there vz is a auxiliary equatio to update the voltage across the series lumped capacitor V. Further the curret update equatio has several modificatios (whe compared to that for a D lossless trasmissio lie) that accout for the series R load i series a additioal term for V the lefthad voltage discretized source voltage p has bee replaced with the V ( k ) ad there are coefficiets that accout for the series lumpedelemet iductor ad resistor i series. The update equatio for the ( k) voltage is uchaged from that for a D lossless trasmissio lie. The series R load i series with the voltage source ad the lossless D trasmissio lie is located at z= ( k.5) z V spatial grid the voltage source is placed at locatio k = ad z = i the FDTD grid. I the FDTD z= k z. I most istaces are selected (source at the begiig of the trasmissio lie). This allows the update equatios to begi with the curret ad voltage odes at spatial idices of k = which is well suited to most programmig laguages. Further the voltage source is a hard source i.e. V ( k ) = v ( t) that is ot effected by surroudig currets ad voltages. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

5 ee692_fdtd_d_tras_lie_lecture4.doc Page 5 of 6 Voltage source with parallel R load i series Figure 3 shows the circuit represetatio icremetal ad discretized icremetal models for a voltage source with parallel R elemets i series. R v ( t) Z vp v ( z t) izt () i P R (a) v P V ( k ) R (b) I.5 ( k.5) I.5 ( k.5).5 I P ( k.5) V ( k ) V P ( k.5) (c) Figure 3 (a) ircuit represetatio (b) icremetal ad (c) discretized icremetal models of a voltage source with lumped elemet parallel R load i series. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

6 ee692_fdtd_d_tras_lie_lecture4.doc Page 6 of 6 Applyig K to the top right ode of Figure 3b yields vz ( zt ) izt () iz ( zt ) c z = Applyig K to the top left ode of Figure 3b yields vp vp izt () ip = R where t ip () zt = vp () zt t ip ( zt). t Applyig KV clockwise aroud the outside loop of Figure 3b yields izt () vz ( zt ) v( zt ) vp l z =. These equatios whe discretized (see R loads lecture material) rearraged ad simplified yield the update equatios.5.5 t IP ( k.5) = IP ( k.5) VP ( k. 5). 2 (.5) t R VP k = VP ( k.5) t 2R.5.5 I ( k.5) IP ( k.5) t 2R ad v t I ( k.5) = I ( k.5) V ( k ) V ( k ).5.5 p Z vp t VP ( k.5) Z vp t.5.5 V ( k ) = V ( k ) Z I ( k.5) I ( k.5). EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

7 ee692_fdtd_d_tras_lie_lecture4.doc Page 7 of 6 There are two auxiliary equatios to update the voltage across the lumped elemet capacitor v P ad curret through the lumped elemet iductor i P. Also the curret update equatio has two modificatios (whe compared to that for a D lossless trasmissio lie) that accout for the parallel R load i series a additioal term for V P ad the lefthad voltage voltage V ( k) has bee replaced with the discretized source V ( k ). However the update equatio for the voltage is uchaged from that for a D lossless trasmissio lie. The parallel R load i series with the voltage source ad the lossless D trasmissio lie is located at z= ( k.5) z spatial grid the voltage source is placed at locatio k = ad z = i the FDTD grid. I the FDTD z= k z. I most istaces are selected (source at the begiig of the trasmissio lie). This allows the update equatios to begi with the curret ad voltage odes at spatial idices of k = which is well suited to most programmig laguages. Further the voltage source is a hard source i.e. V ( k ) = v ( t) that is ot effected by surroudig currets ad voltages. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

8 ee692_fdtd_d_tras_lie_lecture4.doc Page 8 of 6 eries R trasmissio lie termiatio Figure 4 shows the circuit represetatio icremetal ad discretized icremetal models for a lossless D trasmissio lie termiated with series R elemets. R Z vp izt () (a) R i P vzt () v P I.5 ( k.5) (b) I.5 ( k.5) = R I P.5 ( k k ) D = V () k V P ( k D = k) V ( k = k) D (c) Figure 4 (a) ircuit represetatio (b) icremetal ad (c) discretized icremetal models of a termiatig lumpedelemet series R load. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

9 ee692_fdtd_d_tras_lie_lecture4.doc Page 9 of 6 Applyig K to the top right ode of Figure 4b (o curret at k.5) yields vz ( zt ) izt () c z ip =. Applyig KV clockwise aroud the outside loop of Figure 4b yields izt () vz ( zt ) vzt ( ) l z =. Applyig KV couterclockwise aroud the righthad loop of the circuit of Figure 4b yields ip ip R vp v( z z t) = where t vp( z zt ) = ip ( z zt ) t vp( z zt ). t These equatios whe discretized (see R loads lecture material) rearraged ad simplified yield the update equatios t.5 VP ( k ) = VP ( k ) IP ( k ) R IP ( k ) t = IP ( k ) R V ( k ) VP ( k ) R t 2 t 2 ad v t I ( k.5) = I ( k.5) V ( k ) V ( k) Z.5.5 p vp t.5 vp t V ( k ) = V ( k ) Zc I ( k.5) Zc I ( k c.5 P ). Note that the update equatios cotai two itermediate auxiliary variables oe for the curret i P through the termiatig series R load ad oe for the voltage across the lumped capacitor v P. The voltage update equatio has a additioal term (whe compared to that for a D lossless trasmissio lie) that accouts for the termiatig series R load ad is mius the righthad curret term. Also the termiatig series R load is located at z= ( k ) z= k z i the FDTD grid. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28 D

10 ee692_fdtd_d_tras_lie_lecture4.doc Page of 6 Parallel R trasmissio lie termiatio Figure 5 shows the circuit represetatio icremetal ad discretized icremetal models for a lossless D trasmissio lie with a termiatig parallel lumped elemet R load. Z vp R vzt () izt () (a) i P R.5 I ( k.5) (b).5 I (.5) k =.5 IP () k V () k R V ( kd= k) (c) Figure 5 (a) ircuit represetatio (b) icremetal ad (c) discretized icremetal models of a termiatig lumped elemet parallel R load. Applyig Kirchoff s urret aw (K) to the top right ode of Figure 5b yields vz ( zt ) vz ( zt ) vz ( zt ) izt () c z ip = R where i P is defied by the itegral equatio EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

11 ee692_fdtd_d_tras_lie_lecture4.doc Page of 6 t ip( z z t) = v( z z t) t ip( z z t ). t Applyig Kirchoff s Voltage aw (KV) clockwise aroud the outside loop of Figure 5b yields izt () vz ( zt ) vzt ( ) l z =. These equatios whe discretized (see R loads lecture material) rearraged ad simplified yield the update equatios.5.5 vp t I ( k.5) = I ( k.5) V ( k ) V ( k) Z z.5.5 t IP ( k ) = IP ( k ) V ( k ) ad vp t.5 vp t.5 V ( k ) = AV ( k ) A2Z I ( k.5) A2Z I ( k ) P where ad vp t Z z 2R t A = vp t Z 2R t A 2 = vp t Z 2R t Note that the update equatio for the curret is uchaged from that for a D lossless trasmissio lie. However there is ow a itermediate auxiliary equatio to update the curret through the parallel iductor. The voltage update equatio (whe compared to that for a D lossless trasmissio lie) is sigificatly modifieda additioal term to accout for the curret through coefficiets that accout for the parallel resistor R ad capacitor load i parallel ad is missig a term for the curret ode beyod the termiatig load. Also the termiatig parallel R load is located at z= ( k ) z= k z i the FDTD grid. D. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

12 ee692_fdtd_d_tras_lie_lecture4.doc Page 2 of 6 Verificatio of FDTD Update Equatios for voltage sources ad termiatig R oads To demostrate the validity of the derived FDTD update equatios voltage sources ad termiatig R loads a series of simple trasmissio lie circuits icorporatig all of the types of voltage sources ad termiatig R loads were modeled. The source voltage ad voltage at the iput of each trasmissio lie circuit were examied (see Figures 6 ). The time axes were ormalized (i.e. t/t) by the oeway trasit time of the trasmissio lie T = /c. The FDTD results are compared with umerically derived results. To obtai the umerically derived results frequecydomai lossless trasmissio lie theory was used to obtai the iput impedace lookig ito the trasmissio lie. The a simple equivalet circuit model (i.e. voltage divisio) the spectrum of the applied voltage ad a iverse Fourier trasform were used to fid the iput voltage. For these examples a timedelayed uitamplitude Gaussiapulse voltage source was used 2 2 p.5( t τ ) / τ d v () t = e where τ p is the characteristic time ad τ d is a time delay. The R elemet values were arbitrarily selected so that the cotributio from each elemet type was evidet. I the FDTD models the trasmissio lie (Z = 5 Ω v p = m/s =.4 m) z =.5 mm τ p = ps ad =.5. The time delay τ d was selected to place the peak of v () t at t/t =. Figures 6 9 show examples of series R voltage sources ad series R termiatig loads. As show there is excellet agreemet betwee the aalytic ad FDTD results. Figures show examples of parallel R voltage sources ad parallel R termiatig loads. As show there is good agreemet betwee the aalytic ad FDTD results. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

13 ee692_fdtd_d_tras_lie_lecture4.doc Page 3 of 6 Refereces [] J.G. Maloey K.. hlager ad G.. mith ``A imple FDTD Model for Trasiet Excitatio of Ateas by Trasmissio ies'' IEEE Tras. Ateas Propagatio vol. 42 o. 2 pp Feb [2] T. P. Motoya ad G.. mith Modelig Trasmissio ie ircuit Elemets i the FDTD Method Microwave ad Optical Techology etters vol. 2 o. 7 pp.54 April [3] T. P. Motoya ad W. R. cott Jr. Modelig Parallel ad eries R oads i a D FDTD Trasmissio ie UN/URI Natioal Radio ciece Meetig alt ake ity UT p. 3 July [4] T. P. Motoya Improved D FDTD Modelig of Parallel ad eries R oads i a ossless Trasmissio ie 26 IEEE AP Iteratioal ymp. Albuquerque NM pp July EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28

14 ee692_fdtd_d_tras_lie_lecture4.doc Page 4 of 6.8 V ( t) FDTD Aalytic V( z= t/ T) (V) v ( t) V( t) Figure 6 ource voltage v (t) ad iput voltage V( t) for a trasmissio lie circuit cosistig of a matched voltage source a lossless trasmissio lie sectio ad a mismatched resistive termiatig load. t /T 6.8 V ( t) FDTD Aalytic V( z= t/ T) (V) pf v ( t) V( t) Figure 7 ource voltage v (t) ad iput voltage V( t) for a trasmissio lie circuit cosistig of a series R voltage source a lossless trasmissio lie sectio ad a matched resistive termiatig load. t/ T EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28 6

15 ee692_fdtd_d_tras_lie_lecture4.doc Page 5 of 6.8 V ( t) FDTD Aalytic V ( z = t / T ) (V) v ( t) V( t) H 2 pf t / T Figure 8 ource voltage v (t) ad iput voltage V( t) for a trasmissio lie circuit cosistig of a matched voltage source a lossless trasmissio lie sectio ad a series R termiatig load. 6.8 V ( t) FDTD Aalytic V ( z = t / T ) (V) H.5 pf v ( t) V( t) t / T Figure 9 ource voltage v (t) ad iput voltage V( t) for a trasmissio lie circuit cosistig of a series R voltage source a lossless trasmissio lie sectio ad a mismatched resistive termiatig load. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28 6

16 ee692_fdtd_d_tras_lie_lecture4.doc Page 6 of 6.8 V ( t) FDTD Aalytic V ( z = t / T ) (V) H.5 pf v ( t) V( t) t / T Figure ource voltage v (t) ad iput voltage V( t) for a trasmissio lie circuit cosistig of a parallel R voltage source a lossless trasmissio lie sectio ad a matched resistive termiatig load. 6.8 V ( t) FDTD Aalytic V ( z = t / T ) (V) v ( t) V( t) H pf t / T Figure ource voltage v (t) ad iput voltage V( t) for a trasmissio lie circuit cosistig of a matched voltage source a lossless trasmissio lie sectio ad a parallel R termiatig load. EE 692 Applied EM FDTD Method Dr. Thomas P. Motoya 9/27/28 6