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(d) C = 60 F and R = 25 (e) L = 60 H and R = 25 42 Suppose two pure sine waves have no dc components, have the same frequency, and have the same peak-to-peak voltages, but they cancel each other out when combined What is the phase difference between the waves (a) 45 (b) 90 (c) 180 (d) 270 (e) 360 43 Suppose a series RC circuit has a resistance of 50 and a capacitive reactance of 37 What is the phase angle (a) 37 (b) 53 (c) 37 (d) 53 (e) It cannot be calculated from the data given 44 Suppose a 200- resistor is in series with a coil and capacitor, such that XL = 200 and XC = 100 What is the complex impedance (a) 200 j100 (b) 200 j200 (c) 200 + j100 (d) 200 + j200 (e) Impossible to determine from the data given

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The error state vector is x = [ n, e, , RL , RR ] , where each error term is de ned as x = x x In this section, we will also use the following variables to decrease the complexity of the resulting equations: uL = RL L , uR = RR R , and u = ( L + uR )/2 u

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45 The characteristic impedance of a transmission line (a) is negative imaginary (b) is positive imaginary (c) depends on the frequency (d) depends on the construction of the line (e) depends on the length of the line 46 Suppose the period of a pure sine wave is 2 10 8s What is the frequency (a) 2 108 Hz (b) 20 MHz (c) 50 kHz (d) 50 MHz (e) 500 MHz 47 Suppose a series RC circuit has a resistance of 600 and a capacitance of 220 pF What is the phase angle (a) 20 (b) 20 (c) 70 (d) 70 (e) Not determinable from the data given 48 A capacitor with a negative temperature coefficient (a) works less well as the temperature increases (b) works better as the temperature increases (c) heats up as its value is made larger (d) cools down as its value is made larger (e) exhibits increasing capacitance as the temperature drops 49 Suppose three coils are connected in parallel Each has an inductance of 300 H There is no mutual inductance What is the net inductance (a) 100 H (b) 300 H (c) 900 H (d) 173 H (e) 173 H 50 Suppose a coil has 100 of inductive reactance at 300 MHz What is its inductance (a) 0531 H (b) 188 mH (c) 531 H (d) 188 H (e) It can t be found from the data given

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Each of these quantities is computable as an average over any time increment based on the encoder readings Forming the di erence of the Taylor series expansion of eqn (99) and eqn (915) results in the error state equations n = 1 uL RL + uR RR u sin( ) + n cos( ) 2 RL RR 1 uL RL + uR RR + u cos( ) + e sin( ) 2 RL RR uR 1 uL RL RR + L RL RR (919) (920) (921)

e = =

have acquired a dominating role in electronics The term semiconductor arises from the ability of these materials to conduct some of the time, but not all the time The conductivity can be controlled to produce effects such as amplification, rectification, oscillation, signal mixing, and switching

The error models for RR and RL are given by eqns (912 913) The error state equations can be put into be in the standard form of eqn (457) with L R 0 0 sin( ) u cos( ) u cos( ) u 2RL 2RR L R 0 0 u cos( ) u sin( ) u sin( ) 2RL 2RR 1 uL 1 uR (922) F(t) = 0 0 L 0 L RL RR 0 R 0 0 0 0 0 0 0 R and = I5 where = [ n , e , , L , R ] 2 2 2 Let 1 , 2 , and 3 denote the PSD s of n , e , and , respectively These process driving noise terms are included in the model due to the fact that the assumptions will not be perfectly satis ed For example, the wheels may slip or the surface may not be perfectly planar The wheel slip is a function of wheel speed or acceleration Similarly, the e ect of a non-planar surface on the computed position is a function of the speed Therefore, the 2 2 2 quantities 1 , 2 , and 3 and hence the Q matrix are sometimes de ned to be increasing functions of the speed u and/or the acceleration (ie as indicated by the change u)

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