EAS327: Trial Problems

  1. Calculate the output voltage Vo from a half-bridge (ie. voltage divider) if the supply voltage V=6 volts, while R1=R2=1 KW. What is the current through R1 and R2?

  2. Calculate the output voltage Vo from a half-bridge if the supply voltage V=6 volts, while R1=R2=10 KW.What is the current through R1 and R2?

  3. If the output voltage Vo = 2 volts from a half-bridge with supply voltage V=6 volts, calculate R1 if given that R2=10 KW

  4. You are given a resistance thermometer whose R-T characteristic is:

    R(T) = R(To) [1 + a (T-To) + b (T-To)2 ]

    where R(To) = Ro = 10 KW at To= 20oC. Assume the coefficients a = 3.92E-3, and b =0.

    This resistance-thermomenter is placed as the element "R1" in a half-bridge with supply voltage V=12 volts, and with R2=10 KW. Calculate the output voltage Vo from this circuit if actual temperature T=[10, 15, 20, 25, 30] oC. Plot your calculation on graph paper. Is the voltage-temperature characteristic exactly linear?

  5. An RC lowpass filter is built using components R=5KW and C=1 microfarad. What is the power gain G of the filter at frequency f=f0=1/(2pRC)? At f=0? At f=infinity? At f=f0/2.0?

  6. At 20oC a certain thermistor has resistance R=20 KW and a sensitivity a=-4%/K. It is placed in a 1/2-bridge, as component R1, while R2=20KW. What is the bridge output voltage at T=25oC, if the bridge supply voltage is 5.0 volts?

  7. Suppose a resistance thermometer has resistance R=20 KW at 20oC, and has temperature coefficient +0.04 [1/K], ie.

    R(T) = 20 KW [ 1 + 0.04 (T-20) ]

    It is placed in a 1/2-bridge, as component R1, while R2=5KW. The bridge supply voltage is V=10 volts. What is the bridge output voltage at T=20oC? What is the bridge output voltage at T=20.05oC? can the output voltage be received by data-logger having a full-scale range (FSR) of +/- 5 millivolts?

    Suppose the output voltage is to be taken to a 12 bit logger having FSR = +/- 10.0 volts: what is the voltage resolution of the logger? Can this system discriminate a change in temperature of 0.05oC? Now suppose one added a second 1/2-bridge with R3=R4=5KW, to function as the "reference arm." That is, now place the sensor in a Wheatstone Bridge. What is the temperature resolution of this system, if the bridge error voltage is read by a 14 bit logger with FSR = +/- 5.0 millivolts?

  8. You are given a little stick with which to measure the geometry of your living room (this room happens to be rectangular): the length of this stick is your unit of measure for length. You find that the stick fits exactly NL times into what you call the "length," and NW times into the "width" of the room, where both NL and NW are integers. You record the following three numbers: NL, NW, NL/NW. Which of these is/are legitimately called a "dimensionless" number?

  9. Your partner doubts your skill with the measuring stick, and is particularly concerned to buy a carpet that fits nicely. S/he next morning repeats your measurements, obtaining answers NL', NW', NL'/NW'. However unknown to both of you, one of your children had broken the measuring stick precisely in half. What is the relationship between the two sets of measurements?

  10. A copper rod 1m long has its ends held at temperatures Ta=0oC and Tb=20oC. Neglecting any heat transport perpendicular to the axis of the rod, and assuming a linear variation of temperature with distance down the rod, calculate the heat flux density along the rod. (Go here for the conductivity of copper).

  11. A certain infra-red humidity sensor employs as a source of IR radiation a sphere of diameter 1.5 mm, controlled at a constant temperature of Ts=850 oC. Consider what terms might be important in the energy balance of such a device (eg. is a term that involves the heat capacity of the sphere itself important?).

    Assuming this sphere is exposed in still air at temperature Ta=25 oC, and that it radiates energy at a rate (per unit of its surface area) Q* [W m-2] = s Ts4 where s=5.67x10-8 [W m-2 K-4] is the Stefan-Boltzmann constant, calculate the power which must be provided to sustain its temperature. Hint: you will need to evaluate the Nusselt number for this sphere in free convection.

  12. Calculate the steady-state radiation error

    T - Ta = rH Q*/(ra cpa)

    of an unshielded cylindrical thermometer, having radius R=0.0005 m, and exposed in a breeze of windspeed U=1 m/s. Assume the net radiation Q*=700 W m-2. Pick plausible values for all other necessary variables/parameters.

  13. Calculate the steady-state radiation error

    T - Ta = rH Q*/(ra cpa)

    of the same cylindrical thermometer (radius R=0.0005 m) enclosed in a ventilated radiation shield. Assume the radiation shield has an interior surface temperature (as "seen" by the thermometer) Tenv=Ta+1oC, and that the ventilation speed is U=3 m/s.

    Assume the shield is 100% effective at rejection of shortwave energy, ie. that Q*=L*. Assume the net longwave radiation can be calculated as

    L* = s { Tenv4 - T4 }

  14. Suppose it happens that air temperature is T=25oC, and wet-bulb temperature is Tw=20oC. Suppose a wet-bulb thermometer, radiation-shielded, and exposed to a ventilating draft of speed u=4 m/s, is in equilibrium at the wet-bulb temperature (ie. it is performing ideally). Its wick is cylindrical, with diameter d=4 mm, and with length 2 cm. The water reservoir contains 5 cm3. Calculate the interval of time required to empty the reservoir.

  15. Suppose a pitot tube is aligned with a unidirectional airflow of velocity U and density r = 1 [kg m-3]. The pressure difference Dp between the stagnation and static ports is relayed to a manometer, which is filled with water. Calculate the "signal" h (height of the displaced column of water) if:
    • U=0.1 m/s
    • U=0.5 m/s
    • U = 1 m/s
    • U = 10 m/s
    • U = 100 m/s

  16. Now suppose the same pitot tube/manometer reports h = 15 mm, when aligned to a unidirectional flow of water. What is the water velocity?

  17. Suppose a cup anemometer, sitting in an airstream with speed s=5 m/s and density r = 1 [kg m-3], is restrained from rotating. Calculate the drag torque on the anemometer if: frontal area of the cups A=25 [cm2]; radius to centre of cups r = 10 [cm]; and drag coefficients cdf=1, cdb=0.5.

  18. Suppose x is uniformly distributed on the range -1 <= x <= 1. The the p.d.f. of x is:

    f(x) = 0.5, |x| <= 1

    = 0, |x| > 1

    Calculate E[x], E[x2], and the standard deviation sx of x. Find the probability that a random realisation (choice) of x lies in the range -0.6 <= x <= -0.5

  19. Suppose x can take on values, 0 <= x <= infinity, and that x has p.d.f.

    f(x) = a exp(-a x)

    • What are the units of a?
    • Prove the f(x) is normalised
    • Calculate E[x], E[x2], and the standard deviation sx of x
    • If the parameter a= 1, find the probability that a random realisation (choice) of x lies in the range 3 <= x <= 4.

  20. Calculate sample mean and standard deviation for the sample x={1, -1, 0, 2, 3}.

  21. Prove by substitution that the function bv = B [ 1 - cos (t/T)] satisfies the differential equation

    d2bv/dt2 = ( b - bv) / T2

    where b is a constant.


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    Last Modified: 4 April, 2005