11.11: Exercises

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An amplifier consists of three cascaded stages with the following characteristics.

	Stage 1	Stage 2	Stage 3
Gain (\(\text{dB}\))	\(-3\)	\(15\)	\(5\)
NF (\(\text{dB}\))	\(3\)	\(2\)	\(2\)

Table \(\PageIndex{1}\)

What is the overall gain of the amplifier?
What is the overall noise figure of the amplifier?

What is the available noise power of a \(50\:\Omega\) resistor in a \(10\text{ MHz}\) bandwidth. The resistor is at standard temperature.
A \(50\:\Omega\) resistor a \(20\:\Omega\) resistor are in shunt. If both resistors have a temperature of \(300\text{ K}\), what is the total available noise power spectral density of the shunt resistors?
A \(2\text{ GHz}\) amplifier in a \(50\:\Omega\) system has a bandwidth of \(10\text{ MHz}\), a gain of \(40\text{ dB}\), and a noise figure of \(3\text{ dB}\). The amplifier is driven by a circuit with a Thevenin equivalent resistance of \(50\:\Omega\) held at \(290\text{ K}\) (standard temperature). What is the available noise power at the output of the amplifier?
A \(30\text{ dB}\) attenuator is terminated at Port \(2\) in a matched resistor and both are at \(290\text{ K}\). What is the noise temperature at Port \(1\) of the attenuator?
A receive amplifier with a gain of \(30\text{ dB}\), a noise figure of \(2\text{ dB}\), and bandwidth of \(5\text{ MHz}\) is connected to an antenna which has a noise temperature of \(20\text{ K}\). [Parallels Example 11.5.2]
1. What is the available noise power presented to the input of the amplifier in the \(5\text{ MHz}\) bandwidth (recall that the antenna noise temperature is \(20\text{ K}\)?
2. If instead the input of the amplifier is connected to a resistor held at standard temperature, what is the available noise power presented to the input of the amplifier in the \(5\text{ MHz}\) bandwidth?
3. What is the noise factor of the amplifier?
4. What is the excess noise power of the amplifier referred to the its output?
5. What is the effective noise temperature of the amplifier when the amplifier is connected to the antenna with a noise temperature of \(20\text{ K}\). That is, what is the effective noise temperature of the resistor in the Thevenin equivalent circuit of the amplifier output?
A receive amplifier has a bandwidth of \(5\text{ MHz}\), a \(1\text{ dB}\) noise figure, a linear gain of \(20\text{ dB}\). The minimum acceptable SNR is \(10\text{ dB}\).
1. What is the output noise power in \(\text{dBm}\)?
2. What is the minimum detectable output signal in \(\text{dBm}\)?
3. What is the minimum detectable input signal in \(\text{dBm}\)?
The system shown below is a receiver with bandpass filters, amplifiers, and a mixer. [Parallels Example 11.5.2]

Figure \(\PageIndex{1}\)

What is the total gain of the system?
What is the noise factor of the first filter?
What is the system noise factor?
What is the system noise figure?

An amplifier consists of three cascaded stages with the following characteristics:

	Stage 1	Stage 2	Stage 3
Gain	\(10\text{ dB}\)	\(15\text{ dB}\)	\(30\text{ dB}\)
NF	\(0.8\text{ dB}\)	\(2\text{ dB}\)	\(2\text{ dB}\)

Table \(\PageIndex{2}\)
What is the noise figure (NF) and gain of the cascade amplifier?

The first stage of a two-stage amplifier has a linear gain of \(40\text{ dB}\) and a noise figure if \(3\text{ dB}\). The second stage has a gain of \(10\text{ dB}\) and a noise figure of \(5\text{ dB}\).
1. What is the overall gain of the amplifier?
2. What is the overall noise figure of the amplifier?
A subsystem consists of a matched filter with an insertion loss of \(2\text{ dB}\) then an amplifier with a gain of \(20\text{ dB}\) and a noise figure, NF, of \(3\text{ dB}\).
1. What is the overall gain of the subsystem?
2. What is NF of the filter?
3. What is NF of the subsystem?
A subsystem consists of a matched amplifier with a gain of \(20\text{ dB}\) and a noise figure of \(2\text{ dB}\), followed by a \(2\text{ dB}\) attenuator, and then another amplifier with a gain of \(10\text{ dB}\) and NF of \(3\text{ dB}\).
1. What is the overall gain of the subsystem?
2. What is NF of the attenuator?
3. What is NF of the subsystem?
A connector used in a \(50\:\Omega\) system introduces a series resistance of \(0.5\:\Omega\). What is the insertion loss of the connector?
A microwave switch is used in a \(75\:\Omega\) system and has a \(5\:\Omega\) on resistance. The reactive parasitics of the switch are negligible.
1. What is the insertion loss of the switch in the on state?
2. What is the return loss of the switch in the on state?
A microwave switch is used in a \(50\:\Omega\) system and has a \(5\:\Omega\) on resistance. The reactive parasitics of the switch are negligible.
1. What is the insertion loss of the switch in the on state?
2. If the available power of the source is \(50\text{ W}\), what is the power dissipated by the switch?
A microwave switch is used at \(1\text{ GHz}\) in a \(50\:\Omega\) system and it has a \(2\:\Omega\) on resistance and a \(2\text{ k}\Omega\) off resistance. The reactive parasitics of the switch are negligible.
1. What is the insertion loss of the switch?
2. What is the isolation of the switch (i.e., what is the insertion loss of the switch when it is in the off state)?
The RF front end of a communications unit consists of a switch, then an amplifier, and then a mixer. The switch has a loss of \(0.5\text{ dB}\), the amplifier has a gain of \(20\text{ dB}\), and the mixer has a conversion gain of \(3\text{ dB}\). What is the overall gain of the cascade?
A three-port circulator has the \(S\) parameters
\[\left[\begin{array}{ccc}{0}&{0}&{0.5}\\{20.5}&{0}&{0}\\{0}&{0.5}&{0}\end{array}\right]\nonumber \]
Port \(3\) is terminated in a matched load creating a two-port network.
1. Find the \(S\) parameters of the two-port.
2. What is the return loss in \(\text{dB}\) at Port \(1\) if Port \(2\) is terminated in a matched load?
3. What is the insertion loss in \(\text{dB}\) for a signal applied at Port \(1\) and leaving at Port \(2\) with matched source and load impedances?
4. What is the insertion loss in \(\text{dB}\) for a signal applied at Port \(2\) and leaving at Port \(1\) with matched source and load impedances?
Two isolators are used in cascade. Each isolator has an isolation of \(20\text{ dB}\). The isolators are matched so that their input and output reflection coefficients are zero. Determine the isolation of the cascaded isolator system?
A three-port circulator in a \(50\:\Omega\) system has the \(S\) parameters
\[\left[\begin{array}{ccc}{0.1}&{0.01}&{0.5}\\{0.5}&{0.1}&{0.01}\\{0.01}&{0.5}&{0.1}\end{array}\right]\nonumber \]
If port \(3\) is terminated in a matched load to create a two-port network
1. Find the \(S\) parameters of the two-port.
2. What is the return loss in \(\text{dB}\) at Port \(1\) if Port \(2\) is terminated in \(50\:\Omega\)?
3. What is the insertion loss in \(\text{dB}\) for a signal applied at Port \(2\) and leaving at Port \(1\) with \(50\:\Omega\) source and load impedances?
4. What is the insertion loss in \(\text{dB}\) for a signal applied at Port \(1\) and leaving at Port \(2\) with \(50\:\Omega\) source and load impedances?
5. What is is the name of this network?
A mixer in a receiver has a conversion loss of \(16\text{ dB}\). If the applied RF signal has an available power of \(100\:\mu\text{W}\), what is the available power of the IF at the output of the mixer?
The RF signal applied to the input of a mixer has a power of \(1\text{ nW}\) and the output of the mixer at the IF has a power level of \(100\text{ pW}\). What is the conversion loss of the mixer in decibels?
A mixer in a receiver has a conversion gain of \(10\text{ dB}\). If the applied RF signal has a power of \(100\:\mu\text{W}\), what is the available power of the IF at the output of the mixer?
A mixer in a receiver has a conversion loss of \(6\text{ dB}\). If the applied RF signal has a power of \(1\:\mu\text{W}\), what is the available power of the IF at the output of the mixer?
The phase noise of an oscillator was measured as \(−125\text{ dBc/Hz}\) at \(100\text{ kHz}\) offset. What is the normalized phase noise at \(1\text{ MHz}\) offset, assuming that the phase noise power varies as the square of the inverse of frequency?
The phase noise of an oscillator was measured as \(−125\text{ dBc/Hz}\) at \(100\text{ kHz}\) offset. What is the normalized phase noise at \(1\text{ MHz}\) offset, assuming that the phase noise power varies inversely with frequency offset?

11.14.1 Exercises by Section

\(†\)challenging,

\(§11.5 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\)

\(§11.7 13, 14†, 15†, 16†, 17†\)

\(§11.8 18, 19, 20\)

\(§11.9 21, 22, 23, 24\)

\(§11.10 25†, 26\)

11.14.2 Answers to Selected Exercises

(d) \(5.17\text{ dB}\)

(b) \(3\text{ dB}\)

(b) \(60\text{ dB}\)

(b) \(29.8\text{ dB}\)
(a) \(0.424\text{ dB}\)

\(22.5\text{ dB}\)

\(40\text{ dB}\)

\(-26\text{ dBm}\)

\(0\text{ dBm}\)