Ae104a10_solutions_1

# Ae104a10_solutions_1 - Ae/APh 104a Homework Solution Set#1...

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Unformatted text preview: Ae/APh 104a Homework Solution Set #1 1/2 Microcalorimeter. (10 points) For small changes in temperature around an equilibrium given by ( T a ,T e ,T l ,V ), where ( T a ,T e ,T l ) are the temperatures of the absorber, the electron system and the lattice system, respectively, and V is the output voltage, the governing equations for the response at a given frequency, ω , can be written as iωC a Δ T a + G a Δ T a = W + G a Δ T e (1) iωC l Δ T l + ( G el ( T l ) + G )Δ T l = G el ( T e )Δ T e (2) iωC e Δ T e + ( G a + G el ( T e ) + G ETF )Δ T e = G a Δ T a + G el ( T l )Δ T l (3) Δ V V = αA tr Δ T e T e . (4) (i) By drawing an equivalent block diagram for each of equations (1-4), show that the microcalorimeter system can be represented by the block dia- gram shown in figure 2. The inputs and outputs of the block diagrams must be Δ T a , Δ T e , Δ T l or W only. (4 points) Answer: The block diagrams corresponding to equations (1) to (4) are repre- sented on Figure 1. By interconnecting the four block diagrams we obtain the block diagram associated with the microcalorimeter system shown on Figure 2. 1 G a + i ω C a ΔT a G a ΔT e W (a) 1 G el (T l ) + G + i ω C l ΔT l G el ΔT e (b) 1 G a + G el (T e ) + i ω C e ΔT e G a ΔT a G el ΔT l G ETF (c) V α A TR T e ΔV ΔT e (d) Figure 1: Block diagrams corresponding respectively to equations (1) to (4). 1 Figure 2: Block diagram for microcalorimeter. (ii) Find the responsivity of the circuit, S , i.e. determine the ratio between the change in output voltage to input power, where (2 points) S = Δ V W . Answer: We solve equations (1) to (3) for Δ T e and then use equation (4) to substitute Δ T e for Δ V , ( iωC e + G a + G el ( T e )+ G ETF )Δ T e = G a W + G a Δ T e iωC a + G a + G el ( T l ) G el ( T e )Δ T e iωC l + G el ( T l ) + G , which simplifies to iωC e + G a + G el ( T e ) + G ETF- G 2 a iωC a + G a- G el ( T l ) G el ( T e ) iωC l + G el ( T l ) + G Δ T e = G a W iωC a + G a , hence Δ T e W = G a iωC a + G a iωC e + G a + G el ( T e ) + G ETF- G 2 a iωC a + G a- G el ( T l ) G el ( T e ) iωC l + G el ( T l ) + G- 1 , and S = Δ V W = αA tr V T e Δ T e W = αA tr V G a ( iωC a + G a ) T e iωC e + G a + G el ( T e ) + G ETF- G 2 a iωC a + G a- G el ( T l ) G el ( T e ) iωC l + G el ( T l ) + G- 1 . 2 (iii) Where might noise be introduced to the system? (Assume that the noise will only have an interfering effect). Draw a modified block diagram which reflects the sources of experimental noise and suggest possible noise sources. (4 points) Answer: Noise might be introduced at the input due to ambiant radiation, blockage effects of the absorber fixture and reflections on the surface of the absorber. The effective inputthe absorber fixture and reflections on the surface of the absorber....
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## This note was uploaded on 01/05/2012 for the course AE 104a taught by Professor List during the Fall '09 term at Caltech.

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Ae104a10_solutions_1 - Ae/APh 104a Homework Solution Set#1...

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