Linear Electrical and Acoustic Systems Some Basic Concepts

Transcription

1 Linear Electrical and Acoustic Systems Some Basic Concepts

2 Learning Objectives Two port systems transfer matrices impedance matrices reciprocity 1-D compressional waves in a solid equation of motion/constitutive equation acoustic transfer matri of a layer Single Input-Single Output systems LTI systems impulse response, transfer functions convolution/ deconvolution Wiener filter

3 Two Port Systems R Consider our simple system again: V i C If we take off the voltage source we are left with: R C This is an eample of a two port system: I 1 I 2 V 1 V 2

4 I 1 I 2 V 1 V 2 [ T ] Transfer matri: V1 T11 T12 V2 I T T I Alternately, we can epress this two port system in terms of an impedance matri: I 1 I' 2 - I 2 V1 Z11 Z12 I1 [ ] V V 1 Z 2 V Z Z I

5 (1) I 1 I 2 (1) State (1): A (1) [ T ] V 1 V 2 (1) B (2) I 1 I 2 (2) State (2): C (2) [ T ] V 1 V 2 (2) D Reciprocity ( 1) ( 2) ( 2) ( 1) ( 1) ( 2) ( 2) ( 1) V I V I V I V I

6 For reciprocal systems, the impedance matri is symmetric, i.e. and the determinant of the transfer matri is equal to one Z Z [ T] T11T22 T12T21 det 1 V1 Z11 Z12 I1 V Z Z I V1 T11 T12 V2 I T T I I 1 I' 2 I 1 I 2 [ ] V 1 Z V 2 V 1 V 2 [ T ]

7 Relationship between transfer matri components and impedance matri components (reciprocal system) T T Z, T ( 2 Z ) 11Z22 Z Z12 Z12 1, T Z Z12 Z12 Note: T T 12 21

8 1-D plane compressional wave in an elastic solid σ σ dz d dy σ σ + d d dydz dydz ddydz σ + σ ρ σ Constitutive equation E ( 1 ν ) σ 1 ν 1 2ν ( )( ) ρc 2 P 2 u ρ t u + u 2 c P σ 2 t u u 2 stress displacement compressional (P) wave speed E ( 1 ν ) ( 1+ ν )( 1 2 ) ν ρ

9 2 2 u 1 u cp t displacement u A ikp i t B ikp i t velocity v v i A ikp i t i B ikp i t ω i u stress σ ωρ P P ω ωρ P P ω A waves in a solid layer ( ) ep[ ω ] + ep[ ω ] ( ) ω ep[ ω ] ω ep[ ω ] ( ) i c Aep[ ik i t] i c Bep[ ik i t] 0 l B l compressive force F σ S ( ) ( ) a ( kl P ) iz0 ( kl P ) a ( ) ( ) ( ) () F 0 cos sin F l v 0 isin kpl / Z0 cos kpl v l Z acoustic impedance of the layer a 0 ρc S P

10 Equivalent transfer matrices I 1 I 2 V 1 [T 1 ] [T 2 ] [T N ] V 2 I 1 I 2 V 1 [T g ] V 2 [ 1][ 2] [ N] [ ] [ ] [ ] Tg T T T det Tg det T1 det T2 det TN 1

11 When we specify termination conditions at both ports of a two port system, we end up with a system where single inputs and outputs are related: (terminated with voltage source) R I 0 (open circiut termination) V i C V 0 V i V 0

12 R V i C V 0 i ( ) 0 ( ) ( ) dv0 () t () V t V t i t R i t C dt so dv ( t ) V 0 0( t ) V i ( t ) + dt RC RC Note: V 0 (t) is defined here in terms of V i (t) only implicitly as the solution of this differential equation, i.e. ( ) ( ) V0 t L Vi t L linear operator

13 An important class of these single input-output systems is a linear time-shift invariant (LTI) system i(t) L o(t) if then () ( ) () () o1 t L i1 t o2 t L i2 t ( ) 11( ) + 2 2( ) al i() t al i () t ot Lai t ai t linearity if ot () Lit ( ) then ot t Lit t ( ) ( ) 0 0 time-shift invariance

14 Impulse response of LTI systems and the convolution integral delta δ(t) function L g(t) impulse response function i(t) o(t) L + () ( ) ( ) ot iτ gt τ dτ + ( ) ( ) g τ i t τ dτ convolution of g and i

15 i(t) i(τ) ( τ ) τδ( t τ) i Δ Δτ τ ( ) ( τ ) τ ( τ) Δot i Δ gt t linearity and time shift invariance ( ) ( τ ) Δτ ( τ) ot i gt + ( ) ( ) i τ g t τ dτ linearity (additivity)

16 If + ( ) ep( ω ) I i t i t dt + ( ) ep( ω ) O o t i t dt + ( ) ep( ω ) G g t i t dt + and () ( ) ( ) ot iτ g t τ dτ O ω G ω I ω then ( ) ( ) ( )

17 + () ( ) ( ) ot iτ gt τ dτ + ( ) ( ) g τ i t τ dτ O ω G ω I ω ( ) ( ) ( ) convolution in the frequency domain is just (comple-valued) multiplication

18 I(ω) G 1 (ω) G 2 (ω) G N (ω) O(ω) O ω G ω G ω G ω I ω ( ) ( ) ( ) ( ) ( ) 1 2 N The frequency components of the impulse response function of an LTI system are also called the transfer function, t(ω), for the system since this function "transfers" the inputs to the outputs: I(ω) t(ω) O(ω) t ( ω ) O I ( ω ) ( ω )

19 input voltage V i (ω) Pulser cabling Transducer (transmitter) F t (ω) output force flaw F B (ω) flaw signal V R (ω) Receiver cabling Transducer (receiving) force on receiver output voltage V R ( ω ) Vi( ω ) V F F R B t F F V B t i t t t V R A G i

20 Deconvolution deconvolution in the frequency domain is just (comple-valued) division but it must be done with care G ω ( ) O I ω ( ) ω ( ) Generally, a Wiener filter is used in ultrasonics applications to desensitize the division process to noise G I * O I + ε 2 ma I { } 2 2 small "noise" constant ( ) * comple conjugate 2 * I I I

21 It is easier to see the Wiener filter if we rewrite the deconvolution in the form G O ( ) 2 I ( ) + 2 ma ( ) { } I ω I ω ε I ω O I W 2 2 where W I I { } + ε 2 ma I Wiener filter

22 MATLAB eample showing effects of choice of ε >> f linspace(0, 10, 200); >> I f.*(f<5) +(10-f).*(f>5); >> plot(f, I) >> e.01; >> W I.^2./(I.^2 + e^2*ma(i.^2)); >> plot(f, W) >> hold on >> e.1; >> W I.^2./(I.^2 + e^2*ma(i.^2)); >> plot(f, W,'--') >> label(' frequency, f') >> ylabel(' W') >> hold off W I Wε ε frequency, f f