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1.7: Exercises

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    1. Consider a photon at \(1\text{ GHz}\).
      1. What is the energy of the photon in joules?
      2. Is this more or less than the random kinetic energy of an electron at room temperature?
    2. Consider a photon at various frequencies.
      1. What is the photon’s energy at \(1\text{ GHz}\) in terms of electron-volts?
      2. What is the photon’s energy at \(10\text{ GHz}\) in terms of electron-volts?
      3. What is the photon’s energy at \(100\text{ GHz}\) in terms of electron-volts?
      4. What is the photon’s energy at \(1\text{ THz}\) in terms of electron-volts?
    3. Consider a photon at \(1\text{ THz}\).
      1. What is the energy of the photon in terms of electron-volts?
      2. What is the energy of the photon in joules?
      3. Is this more or less than the random kinetic energy of an electron at room temperature (\(300\text{ K}\))?
      4. Discuss if it is necessary to consider quantum effects of the \(1\text{ THz}\) photon at room temperature.
    4. Consider a photon at \(10\text{ GHz}\).
      1. What is the energy of the photon in terms of electron-volts?
      2. What is the energy of the photon in joules?
      3. What is the random kinetic energy of an electron at room temperature (\(300\text{ K}\))?
      4. Calculate the temperature, in kelvins, at which the random kinetic energy of an electron is equal to the energy you calculated in (a).
    5. A \(10\text{ GHz}\) transmitter transmits a \(1\text{ W}\) signal. How many photons are transmitted?
    6. A receiver receives a \(1\text{ pW}\) signal at \(60\text{ GHz}\). How many photons per second are received?
    7. At what frequency is the photon energy equal to the thermal energy of an electron at \(300\text{ K}\)?
    8. What is the frequency at which the energy of a photon is equal to the thermal energy of an electron at \(77\text{ K}\)?
    9. What is the wavelength in free space of a signal at \(4.5\text{ GHz}\)?
    10. Consider a monopole antenna that is a quarter of a wavelength long. How long is the antenna if it operates at \(3\text{ kHz}\)?
    11. Consider a monopole antenna that is a quarter of a wavelength long. How long is the antenna if it operates at \(500\text{ MHz}\)?
    12. Consider a monopole antenna that is a quarter of a wavelength long. How long is the antenna if it operates at \(2\text{ GHz}\)?
    13. A dipole antenna is half of a wavelength long. How long is the antenna at \(2\text{ GHz}\)?
    14. A dipole antenna is half of a wavelength long. How long is the antenna at \(1\text{ THz}\)?
    15. Write your family name in Morse code (see Table 1.3.3).
    16. A transmitter transmits an FM signal with a bandwidth of \(100\text{ kHz}\) and the signal is received by a receiver at a distance \(r\) from the transmitter. When \(r = 1\text{ km}\) the signal power received by the receiver is \(100\text{ nW}\). When the receiver moves further away from the transmitter the power received drops off as \(1/r^{2}\). What is \(r\) in kilometers when the received power is \(100\text{ pW}\). [Parallels Example 1.6.1]
    17. A transmitter transmits an AM signal with a bandwidth of \(20\text{ kHz}\) and the signal is received by a receiver at a distance \(r\) from the transmitter. When \(r = 10\text{ km}\) the signal power received is \(10\text{ nW}\). When the receiver moves further away from the transmitter the power received drops off as \(1/r^{2}\). What is \(r\) in kilometers when the received power is equal to the received noise power of \(1\text{ pW}\)? [Parallels Example 1.6.1]
    18. In a legacy, i.e. 0G, broadcast radio system a transmitter broadcasts an AM signal and the signal can be successfully received if the AM signal is \(20\text{ dB}\) higher than the \(10\text{ fW}\) noise power received. The received signal power when the transmitter and receiver are separated by \(r = 1\text{ km}\) is \(100\text{ nW}\). The received signal power falls off as \(1/r^{2}\) as the receiver moves further away.
      1. What is the radius of the broadcast circle in which the broadcast signal is successfully received?
      2. At what distance does the power of the broadcast signal match the noise power?
      3. If two transmitters both transmit similar AM signals at the same frequency, how far should the transmitters be separated so that the interference received is \(10\text{ dB}\) below the noise level?
    19. In a legacy radio system a transmitter broadcasts an FM signal and for noise-free reception the FM signal must be \(30\text{ dB}\) higher than the received noise power of \(10\text{ fW}\). When the transmitter and receiver are separated by \(r = 1\text{ km}\) the signal power received is \(100\text{ nW}\). The received signal power falls off as \(1/r^{3}\) with greater separation.
      1. What is the radius of the circle in which the broadcast signal is successfully received?
      2. At what distance does the power of the broadcast signal match the noise power?
      3. If two transmitters both transmit similar FM signals at the same frequency and power. One transmitter transmits the desired signal while the second transmits an interfering signal. How far should the transmitters be separated so that the interference received is \(10\text{ dB}\) below the noise level?
    20. A transmitter broadcasts a signal to a receiver that is a distance \(d\) away. The noise power received is \(1\text{ pW}\) and when \(d = 5\text{ km}\) the signal power received is \(100\text{ nW}\). What is the radius of the noise threshold circle where the noise and signal powers are equal, when the received signal power falls off as:
      1. \(1/d^{2}\)? [Parallels Example 1.6.1]
      2. \(1/d^{2.5}\)? [Parallels Example 1.6.2]
    21. A signal is transmitted to a receiver that is a distance \(r\) away. The noise power received is \(100\text{ fW}\) and when \(r\) is \(1\text{ km}\) the received signal power is \(500\text{ nW}\). What is \(r\) when the noise and signal powers are equal when the received signal power falls off as:
      1. \(1/d^{2}\)? [Parallels Example 1.6.1]
      2. \(1/d^{3}\)? [Parallels Example 1.6.2]
    22. The logarithm to base \(2\) of a number \(x\) is \(0.38\) (i.e., \(\log_{2}(x)=0.38\)). What is \(x\)?
    23. The natural logarithm of a number \(x\) is \(2.5\) (i.e., \(\ln (x)=2.5\)). What is \(x\)?
    24. The logarithm to base \(2\) of a number \(x\) is \(3\) (i.e., \(\log_{2}(x)=3\)). What is \(\log_{2}(\sqrt[2]{x}\))?
    25. What is \(\log_{3}(10)\)?
    26. What is \(\log_{4.5}(2)\)?
    27. Without using a calculator evaluate \(\log \{[\log_{3} (3x) − \log_{3} (x)]\}\).
    28. A \(50\:\Omega\) resistor has a sinusoidal voltage across it with a peak voltage of \(0.1\text{ V}\). The RF voltage is \(0.1 \cos(\omega t)\), where \(\omega\) is the radian frequency of the signal and \(t\) is time.
      1. What is the power dissipated in the resistor in watts?
      2. What is the power dissipated in the resistor in \(\text{dBm}\)?
    29. The power of an RF signal is \(10\text{ mW}\). What is the power of the signal in \(\text{dBm}\)?
    30. The power of an RF signal is \(40\text{ dBm}\). What is the power of the signal in watts?
    31. An amplifier has a power gain of \(2100\).
      1. What is the power gain in decibels?
      2. If the input power is \(−5\text{ dBm}\), what is the output power in \(\text{dBm}\)? [Parallels Example 1.6.3]
    32. An amplifier has a power gain of \(6\). What is the power gain in decibels? [Parallels Example 1.6.3]
    33. A filter has a loss factor of \(100\). [Parallels Example 1.6.3]
      1. What is the loss in decibels?
      2. What is the gain in decibels?
    34. An amplifier has a power gain of \(1000\). What is the power gain in \(\text{dB}\)? [Parallels Example 1.6.3]
    35. An amplifier has a gain of \(14\text{ dB}\). The input to the amplifier is a \(1\text{ mW}\) signal, what is the output power in \(\text{dBm}\)?
    36. An RF transmitter consists of an amplifier with a gain of \(20\text{ dB}\), a filter with a loss of \(3\text{ dB}\) and then that is then followed by a lossless transmit antenna. If the power input to the amplifier is \(1\text{ mW}\), what is the total power radiated by the antenna in \(\text{dBm}\)? [Parallels Example 1.6.5]
    37. The final stage of an RF transmitter consists of an amplifier with a gain of \(30\text{ dB}\) and a filter with a loss of \(2\text{ dB}\) that is then followed by a transmit antenna that looses half of the RF power as heat. [Parallels Example 1.6.5]
      1. If the power input to the amplifier is \(10\text{ mW}\), what is the total power radiated by the antenna in \(\text{dBm}\)?
      2. What is the radiated power in watts?
    38. A \(5\text{ mW}\) RF signal is applied to an amplifier that increases the power of the RF signal by a factor of \(200\). The amplifier is followed by a filter that losses half of the power as heat.
      1. What is the output power of the filter in watts?
      2. What is the output power of the filter in \(\text{dBW}\)?
    39. The power of an RF signal at the output of a receive amplifier is \(1\:\mu\text{W}\) and the noise power at the output is \(1\text{ nW}\). What is the output signal-to-noise ratio in \(\text{dB}\)?
    40. The power of a received signal is \(1\text{ pW}\) and the received noise power is \(200\text{ fW}\). In addition the level of the interfering signal is \(100\text{ fW}\). What is the signal-to-noise ratio in \(\text{dB}\)? Treat interference as if it is an additional noise signal.age gain of \(1\) has an input impedance of \(100\:\Omega\), a zero output impedance, and drives a \(5\:\Omega\) load. What is the power gain of the amplifier?
    41. A transmitter transmits an FM signal with a bandwidth of \(100\text{ kHz}\) and the signal power received by a receiver is \(100\text{ nW}\). In the same bandwidth as that of the signal the receiver receives \(100\text{ pW}\) of noise power. In decibels, what is the ratio of the signal power to the noise power, i.e. the signal-to-noise ratio (SNR), received?
    42. An amplifier with a voltage gain of \(20\) has an input resistance of \(100\:\Omega\) and an output resistance of \(50\:\Omega\). What is the power gain of the amplifier in decibels? [Parallels Example 1.6.6]
    43. An amplifier with a voltage gain of \(1\) has an input resistance of \(100\:\Omega\) and an output resistance of \(5\:\Omega\). What is the power gain of the amplifier in decibels? Explain why there is a power gain of more than \(1\) even though the voltage gain is \(1\). [Parallels Example 1.6.6]
    44. An amplifier with a volt
    45. An amplifier has a power gain of \(1900\).
      1. What is the power gain in decibels?
      2. If the input power is \(−8\text{ dBm}\), what is the output power in \(\text{dBm}\)? [Parallels Example 1.6.3]
    46. An amplifier has a power gain of \(20\).
      1. What is the power gain in decibels?
      2. If the input power is \(−23\text{ dBm}\), what is the output power in \(\text{dBm}\)? [Parallels Example 1.6.3]
    47. An amplifier has a voltage gain of \(10\) and a current gain of \(100\).
      1. What is the power gain as a number?
      2. What is the power gain in decibels?
      3. If the input power is \(−30\text{ dBm}\), what is the output power in \(\text{dBm}\)?
      4. What is the output power in \(\text{mW}\)?
    48. An amplifier with \(50\:\Omega\) input impedance and \(50\:\Omega\) load impedance has a voltage gain of \(100\). What is the (power) gain in decibels?
    49. An attenuator reduces the power level of a signal by \(75\%\). What is the (power) gain of the attenuator in decibels?

    1.10.1 Exercises By Section

    \(†\)challenging, \(‡\)very challenging

    \(§1.2 1, 2†, 3†, 4†, 5, 6, 7†, 8†\)

    \(§1.3 9, 10, 11, 12, 13, 14, 15\)

    \(§1.5 16, 17, 18‡, 19†, 20†, 21†\)

    \(§1.6 22, 23, 24, 25, 26, 27, 28, 29, 30 31†, 32, 33, 34, 35, 36† , 37†, 38† 39, 40, 41, 42, 43, 44, 45, 46, 47 48, 49, 50\)

    1.10.2 Answers to Selected Exercises

    1. (d) \(41.36\text{ meV}\)
    1. (b) \(662.6\text{ fJ}\)
    1. \(3.25\text{ cm}\)
    1. \(2.096\)
    1. \(10\text{ dBm}\)
    2. \(10\text{ W}\)
    1. \(7.782\text{ dB}\)
    1. \(1.301\)
    2. \(50.12\text{ mW}\)
    3. (b) \(3.162\text{ W}\)
    1. (b) \(-6\text{ dB}\)

    This page titled 1.7: Exercises is shared under a CC BY-NC license and was authored, remixed, and/or curated by Michael Steer.

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