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Introduction

This document is an appendix to the submitted manuscript titled: Magnetic flux distor-tion in two-phase liquid metal flow. We have provided in this appendix, the difference emf time signals recorded in the Lockin amplifier and corresponding FFT spectra, at various values of U , ω and α. Even though each experiment was repeated six times, we only show the first two measurements in each case. All the time plots and FFT spectrum densities presented here, have been used in the submitted manuscript.

Notations

α Void volume fraction

ω Pulsation of AC current in primary coil of ECFM

U Velocity of ECFM

∆V Difference emf between secondary coils S1 and S2 I Current in primary coil

∆V_|| Component of difference emf in-phase with I

∆V⊥ Component of difference emf quadrature (π/2 out of phase) with I I|| Component of measured current in primary coil circuit, in-phase with I ||∆V ||2 _{Squared norm of difference emf}

Contents

Chapter 1

t (s) 0 1 2 3 4 " V || (V) -2 -1 0 t (s) 0 1 2 3 4 " V ? (V) -2 -1 0 t (s) 0 1 2 3 4 || " V|| 2 (V 2 ) #10-6 0.5 1 1.5 2 t (s) 0 1 2 3 4 I || 0.44 0.45 0.46 0.47

Figure 1.1: ∆V and I vs t at U = 0 m s−1, ω = 1571 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-7 2 2.5 3 3.5 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-7 8.2 8.3 8.4 8.5 8.6 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 2.5 3 3.5 4

Figure 1.2: FFT spectral density of ∆V vs ωM at U = 0 m s−1, ω = 1571 rad s−1 and

t (s) 0 1 2 3 4 " V || (V) #10-4 -4 -3 -2 -1 0 t (s) 0 1 2 3 4 " V ? (V) #10-4 -5 -4 -3 -2 -1 t (s) 0 1 2 3 4 || " V|| 2 (V 2 ) #10-7 0 1 2 3 t (s) 0 1 2 3 4 I || 0.438 0.4385 0.439 0.4395 0.44

Figure 1.3: ∆V and I vs t at U = 0 m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-7 0.8 1 1.2 1.4 1.6 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-7 4.35 4.4 4.45 4.5 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -10 1.6 1.7 1.8 1.9

Figure 1.4: FFT spectral density of ∆V vs ωM at U = 0 m s−1, ω = 3142 rad s−1 and

t (s) 0 1 2 3 4 " V || (V) -3 -2.5 -2 t (s) 0 1 2 3 4 " V ? (V) -4 -3.5 -3 t (s) 0 1 2 3 4 || " V|| 2 (V 2 ) #10-7 1 1.5 2 2.5 t (s) 0 1 2 3 4 I || 0.4238 0.424 0.4242 0.4244

Figure 1.5: ∆V and I vs t at U = 0 m s−1, ω = 4712 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-7 1.4 1.6 1.8 2 2.2 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-7 0 0.5 1 1.5 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -11 5 5.5 6 6.5

Figure 1.6: FFT spectral density of ∆V vs ωM at U = 0 m s−1, ω = 4712 rad s−1 and

t (s) 0 1 2 3 4 " V || (V) #10-4 -2.8 -2.6 -2.4 -2.2 t (s) 0 1 2 3 4 " V ? (V) #10-4 -4.4 -4.2 -4 -3.8 t (s) 0 1 2 3 4 || " V|| 2 (V 2 ) #10-7 2 2.2 2.4 2.6 t (s) 0 1 2 3 4 I || 0.4093 0.4093 0.4094 0.4094 0.4095

Figure 1.7: ∆V and I vs t at U = 0 m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-8 2.6 2.8 3 3.2 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-8 4 5 6 7 8 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -11 2 3 4 5

Figure 1.8: FFT spectral density of ∆V vs ωM at U = 0 m s−1, ω = 6283 rad s−1 and

t (s) 0 1 2 3 4 " V || (V) -2.8 -2.7 -2.6 t (s) 0 1 2 3 4 " V ? (V) -4.8 -4.7 -4.6 -4.5 t (s) 0 1 2 3 4 || " V|| 2 (V 2 ) #10-7 2.6 2.7 2.8 2.9 3 t (s) 0 1 2 3 4 I || 0.3948 0.3948 0.3949 0.3949

Figure 1.9: ∆V and I vs t at U = 0 m s−1, ω = 7854 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-8 4 4.5 5 5.5 6 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-8 2 4 6 8 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -11 4 5 6 7 8

Figure 1.10: FFT spectral density of ∆V vs ωM at U = 0 m s−1, ω = 7854 rad s−1 and

t (s) 0 1 2 3 4 " V || (V) #10-4 -2.9 -2.8 -2.7 -2.6 t (s) 0 1 2 3 4 " V ? (V) #10-4 -5.3 -5.2 -5.1 -5 t (s) 0 1 2 3 4 || " V|| 2 (V 2 ) #10-7 3.2 3.3 3.4 3.5 t (s) 0 1 2 3 4 I || 0.3803 0.3803 0.3803 0.3803 0.3803

Figure 1.11: ∆V and I vs t at U = 0 m s−1, ω = 9425 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-7 0.5 1 1.5 2 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-7 1.4 1.5 1.6 1.7 1.8 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -10 1.8 1.9 2 2.1 2.2

Figure 1.12: FFT spectral density of ∆V vs ωM at U = 0 m s−1, ω = 9425 rad s−1 and

t (s) 0 1 2 3 4 " V || (V) -3 -2.9 -2.8 t (s) 0 1 2 3 4 " V ? (V) -5.65 -5.6 -5.55 -5.5 t (s) 0 1 2 3 4 || " V|| 2 (V 2 ) #10-7 3.7 3.8 3.9 4 t (s) 0 1 2 3 4 I || 0.3660 0.366 0.3660 0.3660 0.3661

Figure 1.13: ∆V and I vs t at U = 0 m s−1, ω = 10 996 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-7 1 1.5 2 2.5 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-7 2.9 3 3.1 3.2 3.3 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -10 2.5 2.6 2.7 2.8

Figure 1.14: FFT spectral density of ∆V vs ωM at U = 0 m s−1, ω = 10 996 rad s−1 and

t (s) 0 1 2 3 4 " V || (V) #10-4 -3 -2.95 -2.9 -2.85 -2.8 t (s) 0 1 2 3 4 " V ? (V) #10-4 -6.05 -6 -5.95 -5.9 -5.85 t (s) 0 1 2 3 4 || " V|| 2 (V 2 ) #10-7 4.2 4.3 4.4 4.5 t (s) 0 1 2 3 4 I || 0.3521 0.3521 0.3522 0.3522 0.3522

Figure 1.15: ∆V and I vs t at U = 0 m s−1, ω = 12 566 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-7 2.558 2.559 2.56 2.561 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-7 2.4 2.5 2.6 2.7 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -10 1 1.5 2 2.5

Figure 1.16: FFT spectral density of ∆V vs ωM at U = 0 m s−1, ω = 12 566 rad s−1 and

t (s) 7 8 9 10 11 12 " V || (V) -2.5 -2 -1.5 t (s) 7 8 9 10 11 12 " V ? (V) -2.2 -2 -1.8 -1.6 t (s) 7 8 9 10 11 12 || " V|| 2 (V 2 ) #10-8 4 6 8 10 t (s) 7 8 9 10 11 12 I || 0.4536 0.4536 0.4537 0.4537 0.4538

Figure 1.17: ∆V and I vs t at U = 0.1 m s−1, ω = 1571 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 0.5 1 1.5 2 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 0.5 1 1.5

Figure 1.18: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 1571 rad s−1 and

t (s) 2 4 6 8 10 " V || (V) #10-4 -4 -3 -2 -1 0 t (s) 2 4 6 8 10 " V ? (V) #10-4 -6 -4 -2 0 t (s) 2 4 6 8 10 || " V|| 2 (V 2 ) #10-7 0 1 2 3 t (s) 2 4 6 8 10 I || 0.437 0.438 0.439 0.44

Figure 1.19: ∆V and I vs t at U = 0.1 m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 0.5 1 1.5 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 0.5 1 1.5 2

Figure 1.20: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 3142 rad s−1 and

t (s) 2 4 6 8 10 " V || (V) -3 -2.5 -2 t (s) 2 4 6 8 10 " V ? (V) -3.6 -3.4 -3.2 t (s) 2 4 6 8 10 || " V|| 2 (V 2 ) #10-7 1.55 1.6 1.65 1.7 t (s) 2 4 6 8 10 I || 0.4240 0.4240 0.424 0.4240 0.4240

Figure 1.21: ∆V and I vs t at U = 0.1 m s−1, ω = 4712 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 0.5 1 1.5 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3

Figure 1.22: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 4712 rad s−1 and

t (s) 3 4 5 6 7 8 " V || (V) #10-4 -3 -2.5 -2 t (s) 3 4 5 6 7 8 " V ? (V) #10-4 -4.5 -4 -3.5 t (s) 3 4 5 6 7 8 || " V|| 2 (V 2 ) #10-7 1.8 2 2.2 2.4 t (s) 3 4 5 6 7 8 I || 0.4091 0.4092 0.4093 0.4094

Figure 1.23: ∆V and I vs t at U = 0.1 m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 1 2 3 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3

Figure 1.24: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 6283 rad s−1 and

t (s) 5 6 7 8 9 10 " V || (V) -3 -2.8 -2.6 t (s) 5 6 7 8 9 10 " V ? (V) -4.8 -4.6 -4.4 t (s) 5 6 7 8 9 10 || " V|| 2 (V 2 ) #10-7 2.5 2.6 2.7 2.8 2.9 t (s) 5 6 7 8 9 10 I || 0.3946 0.3946 0.3947 0.3947 0.3947

Figure 1.25: ∆V and I vs t at U = 0.1 m s−1, ω = 7854 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 1 2 3 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3 4

Figure 1.26: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 7854 rad s−1 and

t (s) 5 6 7 8 9 10 " V || (V) #10-4 -3 -2.8 -2.6 -2.4 t (s) 5 6 7 8 9 10 " V ? (V) #10-4 -5.4 -5.2 -5 -4.8 -4.6 t (s) 5 6 7 8 9 10 || " V|| 2 (V 2 ) #10-7 3 3.1 3.2 3.3 3.4 t (s) 5 6 7 8 9 10 I || 0.3801 0.3802 0.3802 0.3802

Figure 1.27: ∆V and I vs t at U = 0.1 m s−1, ω = 9425 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 1 2 3 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 1 2 3 4 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 2 4 6

Figure 1.28: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 9425 rad s−1 and

t (s) 5 6 7 8 9 10 " V || (V) -3.2 -3 -2.8 t (s) 5 6 7 8 9 10 " V ? (V) -5.8 -5.6 -5.4 -5.2 t (s) 5 6 7 8 9 10 || " V|| 2 (V 2 ) #10-7 3.4 3.6 3.8 4 t (s) 5 6 7 8 9 10 I || 0.3659 0.3659 0.366 0.3660 0.3661

Figure 1.29: ∆V and I vs t at U = 0.1 m s−1, ω = 10 996 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 1 2 3 4 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 2 4 6

Figure 1.30: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 10 996 rad s−1

t (s) 4 6 8 10 " V || (V) #10-4 -3.2 -3 -2.8 -2.6 t (s) 4 6 8 10 " V ? (V) #10-4 -6 -5.8 -5.6 -5.4 t (s) 4 6 8 10 || " V|| 2 (V 2 ) #10-7 3.8 4 4.2 4.4 t (s) 4 6 8 10 I || 0.3520 0.3521 0.3521 0.3522 0.3522

Figure 1.31: ∆V and I vs t at U = 0.1 m s−1, ω = 12 566 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 1 2 3 4 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 2 4 6

Figure 1.32: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 12 566 rad s−1

t (s) 5 6 7 8 9 " V || (V) -2 -1 0 1 t (s) 5 6 7 8 9 " V ? (V) -2 -1 0 1 t (s) 5 6 7 8 9 || " V|| 2 (V 2 ) #10-6 0 1 2 3 4 t (s) 5 6 7 8 9 I || 0.44 0.45 0.46 0.47

Figure 1.33: ∆V and I vs t at U = 1 m s−1, ω = 1571 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-5 0 2 4 6 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-5 0 1 2 3 4 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -8 0 2 4 6

Figure 1.34: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 1571 rad s−1 and

t (s) 2 4 6 8 10 " V || (V) #10-4 -10 -5 0 5 t (s) 2 4 6 8 10 " V ? (V) #10-4 -6 -4 -2 0 2 t (s) 2 4 6 8 10 || " V|| 2 (V 2 ) #10-7 0 2 4 6 t (s) 2 4 6 8 10 I || 0.4375 0.438 0.4385 0.439 0.4395

Figure 1.35: ∆V and I vs t at U = 1 m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-5 0 1 2 3 4 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-5 0 0.5 1 1.5 2 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -8 0 1 2 3

Figure 1.36: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 3142 rad s−1 and

t (s) 4.5 5 5.5 6 6.5 7 " V || (V) -6 -4 -2 0 t (s) 4.5 5 5.5 6 6.5 7 " V ? (V) -6 -4 -2 t (s) 4.5 5 5.5 6 6.5 7 || " V|| 2 (V 2 ) #10-7 1 2 3 4 t (s) 4.5 5 5.5 6 6.5 7 I || 0.4236 0.4238 0.424 0.4242

Figure 1.37: ∆V and I vs t at U = 1 m s−1, ω = 4712 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-5 0 1 2 3 4 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-5 0 1 2 3 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -8 0 0.5 1 1.5 2

Figure 1.38: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 4712 rad s−1 and

t (s) 4 5 6 7 " V || (V) #10-4 -6 -4 -2 0 t (s) 4 5 6 7 " V ? (V) #10-4 -6 -5 -4 -3 -2 t (s) 4 5 6 7 || " V|| 2 (V 2 ) #10-7 1.5 2 2.5 3 t (s) 4 5 6 7 I || 0.4092 0.4092 0.4093 0.4093 0.4094

Figure 1.39: ∆V and I vs t at U = 1 m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-5 0 2 4 6 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-5 0 1 2 3 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -8 0 0.5 1 1.5

Figure 1.40: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 6283 rad s−1 and

t (s) 4 5 6 7 8 " V || (V) -6 -4 -2 t (s) 4 5 6 7 8 " V ? (V) -6 -5 -4 t (s) 4 5 6 7 8 || " V|| 2 (V 2 ) #10-7 2 2.5 3 3.5 t (s) 4 5 6 7 8 I || 0.3945 0.3946 0.3946 0.3947 0.3947

Figure 1.41: ∆V and I vs t at U = 1 m s−1, ω = 7854 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-5 0 1 2 3 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-5 0 1 2 3 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 2 4 6 8

Figure 1.42: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 7854 rad s−1 and

t (s) 4.5 5 5.5 6 6.5 7 " V || (V) #10-4 -5 -4 -3 -2 -1 t (s) 4.5 5 5.5 6 6.5 7 " V ? (V) #10-4 -7 -6 -5 -4 -3 t (s) 4.5 5 5.5 6 6.5 7 || " V|| 2 (V 2 ) #10-7 3 3.5 4 t (s) 4.5 5 5.5 6 6.5 7 I || 0.3801 0.3802 0.3802 0.3802 0.3803

Figure 1.43: ∆V and I vs t at U = 1 m s−1, ω = 9425 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-5 0 1 2 3 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-5 0 1 2 3 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 2 4 6

Figure 1.44: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 9425 rad s−1 and

t (s) 3.5 4 4.5 5 5.5 6 " V || (V) -5 -4 -3 -2 t (s) 3.5 4 4.5 5 5.5 6 " V ? (V) -7 -6 -5 t (s) 3.5 4 4.5 5 5.5 6 || " V|| 2 (V 2 ) #10-7 3 3.5 4 4.5 t (s) 3.5 4 4.5 5 5.5 6 I || 0.3659 0.366 0.3661 0.3662

Figure 1.45: ∆V and I vs t at U = 1 m s−1, ω = 10 996 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-5 0 1 2 3 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-5 0 1 2 3 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 2 4 6

Figure 1.46: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 10 996 rad s−1 and

t (s) 4 5 6 7 " V || (V) #10-4 -5 -4 -3 -2 -1 t (s) 4 5 6 7 " V ? (V) #10-4 -7 -6 -5 -4 t (s) 4 5 6 7 || " V|| 2 (V 2 ) #10-7 3.5 4 4.5 5 t (s) 4 5 6 7 I || 0.3521 0.3522 0.3523 0.3524

Figure 1.47: ∆V and I vs t at U = 1 m s−1, ω = 12 566 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-5 0 1 2 3 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-5 0 1 2 3 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 2 4 6

Figure 1.48: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 12 566 rad s−1 and

t (s) 0 100 200 300 400 " V || (V) -4 -3 -2 -1 t (s) 0 100 200 300 400 " V ? (V) -6 -4 -2 t (s) 0 100 200 300 400 || " V|| 2 (V 2 ) #10-7 0 1 2 3 t (s) 0 100 200 300 400 I || 0.438 0.4385 0.439 0.4395 0.44

Figure 1.49: ∆V and I vs t at U = 10−3m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-6 0 0.5 1 1.5 2 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 0.5 1 1.5 2

Figure 1.50: FFT spectral density of ∆V vs ωM at U = 10−3m s−1, ω = 3142 rad s−1

t (s) 0 100 200 300 400 " V || (V) #10-4 -2.8 -2.6 -2.4 -2.2 t (s) 0 100 200 300 400 " V ? (V) #10-4 -4.4 -4.2 -4 -3.8 -3.6 t (s) 0 100 200 300 400 || " V|| 2 (V 2 ) #10-7 1.8 2 2.2 2.4 2.6 t (s) 0 100 200 300 400 I || 0.4093 0.4094 0.4095 0.4096 0.4097

Figure 1.51: ∆V and I vs t at U = 10−3m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V || (V) #10-6 0 1 2 3 !_M (rad/s) 0 0.2 0.4 0.6 0.8 1 " V ? (V) #10-6 0 1 2 3 4 ! M (rad/s) 0 0.2 0.4 0.6 0.8 1 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3 4

Figure 1.52: FFT spectral density of ∆V vs ωM at U = 10−3m s−1, ω = 6283 rad s−1

t (s) 0 50 100 150 " V || (V) -4 -3 -2 -1 t (s) 0 50 100 150 " V ? (V) -6 -4 -2 t (s) 0 50 100 150 || " V|| 2 (V 2 ) #10-7 0 1 2 3 t (s) 0 50 100 150 I || 0.437 0.438 0.439 0.44

Figure 1.53: ∆V and I vs t at U = 3 × 10−3m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 1 2 3 " V || (V) #10-7 0 2 4 6 8 !_M (rad/s) 0 1 2 3 " V ? (V) #10-6 0 0.5 1 ! M (rad/s) 0 1 2 3 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -10 0 2 4 6 8

Figure 1.54: FFT spectral density of ∆V vs ωM at U = 3 × 10−3m s−1, ω = 3142 rad s−1

t (s) 20 40 60 80 100 120 " V || (V) #10-4 -2.8 -2.6 -2.4 -2.2 t (s) 20 40 60 80 100 120 " V ? (V) #10-4 -4.2 -4 -3.8 -3.6 t (s) 20 40 60 80 100 120 || " V|| 2 (V 2 ) #10-7 1.8 2 2.2 2.4 t (s) 20 40 60 80 100 120 I || 0.4093 0.4094 0.4095 0.4096

Figure 1.55: ∆V and I vs t at U = 3 × 10−3m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 1 2 3 " V || (V) #10-6 0 1 2 3 !_M (rad/s) 0 1 2 3 " V ? (V) #10-6 0 1 2 3 4 ! M (rad/s) 0 1 2 3 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3 4

Figure 1.56: FFT spectral density of ∆V vs ωM at U = 3 × 10−3m s−1, ω = 6283 rad s−1

t (s) 10 15 20 25 30 35 " V || (V) -4 -3 -2 -1 t (s) 10 15 20 25 30 35 " V ? (V) -6 -4 -2 t (s) 10 15 20 25 30 35 || " V|| 2 (V 2 ) #10-7 0 1 2 3 t (s) 10 15 20 25 30 35 I || 0.437 0.438 0.439 0.44

Figure 1.57: ∆V and I vs t at U = 10−2m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 2 4 6 8 10 " V || (V) #10-6 0 0.5 1 1.5 2 !_M (rad/s) 0 2 4 6 8 10 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 2 4 6 8 10 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 0.5 1 1.5 2

Figure 1.58: FFT spectral density of ∆V vs ωM at U = 10−2m s−1, ω = 3142 rad s−1

t (s) 10 20 30 40 " V || (V) #10-4 -2.8 -2.6 -2.4 -2.2 t (s) 10 20 30 40 " V ? (V) #10-4 -4.2 -4 -3.8 -3.6 t (s) 10 20 30 40 || " V|| 2 (V 2 ) #10-7 1.8 2 2.2 2.4 t (s) 10 20 30 40 I || 0.4091 0.4092 0.4093 0.4094

Figure 1.59: ∆V and I vs t at U = 10−2m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 2 4 6 8 10 " V || (V) #10-6 0 0.5 1 1.5 !_M (rad/s) 0 2 4 6 8 10 " V ? (V) #10-6 0 0.5 1 1.5 ! M (rad/s) 0 2 4 6 8 10 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 0.5 1 1.5 2

Figure 1.60: FFT spectral density of ∆V vs ωM at U = 10−2m s−1, ω = 6283 rad s−1

t (s) 5 10 15 20 " V || (V) -4 -3 -2 -1 t (s) 5 10 15 20 " V ? (V) -6 -4 -2 t (s) 5 10 15 20 || " V|| 2 (V 2 ) #10-7 0 1 2 3 t (s) 5 10 15 20 I || 0.437 0.438 0.439 0.44

Figure 1.61: ∆V and I vs t at U = 3 × 10−2m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 10 20 30 " V || (V) #10-6 0 0.5 1 !_M (rad/s) 0 10 20 30 " V ? (V) #10-6 0 0.5 1 ! M (rad/s) 0 10 20 30 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -10 0 2 4 6 8

Figure 1.62: FFT spectral density of ∆V vs ωM at U = 3 × 10−2m s−1, ω = 3142 rad s−1

t (s) 0 5 10 15 20 " V || (V) #10-4 -2.8 -2.6 -2.4 -2.2 t (s) 0 5 10 15 20 " V ? (V) #10-4 -4.4 -4.2 -4 -3.8 -3.6 t (s) 0 5 10 15 20 || " V|| 2 (V 2 ) #10-7 1.8 2 2.2 2.4 t (s) 0 5 10 15 20 I || 0.4093 0.4093 0.4094 0.4094 0.4095

Figure 1.63: ∆V and I vs t at U = 3 × 10−2m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 10 20 30 " V || (V) #10-6 0 1 2 3 !_M (rad/s) 0 10 20 30 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 10 20 30 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3 4

Figure 1.64: FFT spectral density of ∆V vs ωM at U = 3 × 10−2m s−1, ω = 6283 rad s−1

t (s) 2 4 6 8 10 " V || (V) -4 -3 -2 -1 t (s) 2 4 6 8 10 " V ? (V) -6 -4 -2 t (s) 2 4 6 8 10 || " V|| 2 (V 2 ) #10-7 0 1 2 3 t (s) 2 4 6 8 10 I || 0.437 0.438 0.439 0.44

Figure 1.65: ∆V and I vs t at U = 0.1 m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 0.5 1 1.5 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 0.5 1 1.5 2

Figure 1.66: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 3142 rad s−1 and

t (s) 3 4 5 6 7 8 " V || (V) #10-4 -3 -2.5 -2 t (s) 3 4 5 6 7 8 " V ? (V) #10-4 -4.5 -4 -3.5 t (s) 3 4 5 6 7 8 || " V|| 2 (V 2 ) #10-7 1.8 2 2.2 2.4 t (s) 3 4 5 6 7 8 I || 0.4091 0.4092 0.4093 0.4094

Figure 1.67: ∆V and I vs t at U = 0.1 m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 20 40 60 80 100 " V || (V) #10-6 0 0.5 1 1.5 2 !_M (rad/s) 0 20 40 60 80 100 " V ? (V) #10-6 0 2 4 6 ! M (rad/s) 0 20 40 60 80 100 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3 4

Figure 1.68: FFT spectral density of ∆V vs ωM at U = 0.1 m s−1, ω = 6283 rad s−1 and

t (s) 4 5 6 7 8 " V || (V) -6 -4 -2 0 t (s) 4 5 6 7 8 " V ? (V) -6 -4 -2 t (s) 4 5 6 7 8 || " V|| 2 (V 2 ) #10-7 0 1 2 3 t (s) 4 5 6 7 8 I || 0.4375 0.438 0.4385 0.439 0.4395

Figure 1.69: ∆V and I vs t at U = 0.25 m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 50 100 150 200 250 " V || (V) #10-6 0 0.5 1 1.5 2 !_M (rad/s) 0 50 100 150 200 250 " V ? (V) #10-6 0 0.5 1 1.5 2 ! M (rad/s) 0 50 100 150 200 250 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 0.5 1 1.5

Figure 1.70: FFT spectral density of ∆V vs ωM at U = 0.25 m s−1, ω = 3142 rad s−1

t (s) 3 4 5 6 7 " V || (V) #10-4 -3.5 -3 -2.5 -2 -1.5 t (s) 3 4 5 6 7 " V ? (V) #10-4 -4.5 -4 -3.5 -3 t (s) 3 4 5 6 7 || " V|| 2 (V 2 ) #10-7 1.8 2 2.2 2.4 2.6 t (s) 3 4 5 6 7 I || 0.4092 0.4092 0.4093 0.4093 0.4094

Figure 1.71: ∆V and I vs t at U = 0.25 m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 50 100 150 200 250 " V || (V) #10-6 0 0.5 1 1.5 !_M (rad/s) 0 50 100 150 200 250 " V ? (V) #10-6 0 0.5 1 1.5 2 ! M (rad/s) 0 50 100 150 200 250 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 0.5 1 1.5 2

Figure 1.72: FFT spectral density of ∆V vs ωM at U = 0.25 m s−1, ω = 6283 rad s−1

t (s) 4 5 6 7 " V || (V) -4 -3 -2 t (s) 4 5 6 7 " V ? (V) -5 -4.5 -4 -3.5 t (s) 4 5 6 7 || " V|| 2 (V 2 ) #10-7 1.8 2 2.2 2.4 2.6 t (s) 4 5 6 7 I || 0.4092 0.4092 0.4093 0.4093 0.4094

Figure 1.73: ∆V and I vs t at U = 0.5 m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 100 200 300 400 500 " V || (V) #10-6 0 1 2 3 !_M (rad/s) 0 100 200 300 400 500 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 100 200 300 400 500 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3

Figure 1.74: FFT spectral density of ∆V vs ωM at U = 0.5 m s−1, ω = 6283 rad s−1 and

t (s) 4 5 6 7 " V || (V) #10-4 -6 -4 -2 0 2 t (s) 4 5 6 7 " V ? (V) #10-4 -6 -4 -2 0 2 t (s) 4 5 6 7 || " V|| 2 (V 2 ) #10-7 0 1 2 3 4 t (s) 4 5 6 7 I || 0.4375 0.438 0.4385 0.439 0.4395

Figure 1.75: ∆V and I vs t at U = 0.75 m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 " V || (V) #10-6 0 1 2 3 !_M (rad/s) 0 200 400 600 800 " V ? (V) #10-6 0 2 4 6 ! M (rad/s) 0 200 400 600 800 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3

Figure 1.76: FFT spectral density of ∆V vs ωM at U = 0.75 m s−1, ω = 3142 rad s−1

t (s) 3 4 5 6 7 8 " V || (V) -6 -4 -2 t (s) 3 4 5 6 7 8 " V ? (V) -6 -5 -4 -3 t (s) 3 4 5 6 7 8 || " V|| 2 (V 2 ) #10-7 1.5 2 2.5 3 t (s) 3 4 5 6 7 8 I || 0.4091 0.4092 0.4092 0.4093 0.4093

Figure 1.77: ∆V and I vs t at U = 0.75 m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 " V || (V) #10-6 0 1 2 3 !_M (rad/s) 0 200 400 600 800 " V ? (V) #10-6 0 1 2 3 4 ! M (rad/s) 0 200 400 600 800 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3 4

Figure 1.78: FFT spectral density of ∆V vs ωM at U = 0.75 m s−1, ω = 6283 rad s−1

t (s) 2 4 6 8 10 " V || (V) #10-4 -10 -5 0 5 t (s) 2 4 6 8 10 " V ? (V) #10-4 -6 -4 -2 0 2 t (s) 2 4 6 8 10 || " V|| 2 (V 2 ) #10-7 0 2 4 6 t (s) 2 4 6 8 10 I || 0.4375 0.438 0.4385 0.439 0.4395

Figure 1.79: ∆V and I vs t at U = 1 m s−1, ω = 3142 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-6 0 2 4 6 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-6 0 2 4 6 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 2 4 6

Figure 1.80: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 3142 rad s−1 and

t (s) 4 5 6 7 " V || (V) -6 -4 -2 t (s) 4 5 6 7 " V ? (V) -6 -5 -4 -3 t (s) 4 5 6 7 || " V|| 2 (V 2 ) #10-7 1.5 2 2.5 3 t (s) 4 5 6 7 I || 0.4092 0.4092 0.4093 0.4093 0.4094

Figure 1.81: ∆V and I vs t at U = 1 m s−1, ω = 6283 rad s−1 and α = 0 %.

!_M (rad/s) 0 200 400 600 800 1000 " V || (V) #10-6 0 2 4 6 !_M (rad/s) 0 200 400 600 800 1000 " V ? (V) #10-6 0 1 2 3 ! M (rad/s) 0 200 400 600 800 1000 || " V|| 2 -h || " V|| 2 i (V 2 ) #10 -9 0 1 2 3 4

Figure 1.82: FFT spectral density of ∆V vs ωM at U = 1 m s−1, ω = 6283 rad s−1 and

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