Contents

 

 

13.0 Signals and Noise

 

13.1 Noise Sources:

13.1.1 Combining Noise Voltages

13.1.2 Noise Figure

13.1.2.1 Friiss’ Formula

13.1.2.2 Noise Reduction

13.1.2.3 Noise Temperature

 

13.2 Review of RLC Networks

13.2.1 Series RLC Circuits

13.2.2 Parallel RLC Circuits

13.2.3 Self Resonance

 

13.3 Oscillators

13.3.1 Tuned Circuit Oscillators

13.3.2 Crystal Oscillator

 

Assignment Questions

 

For Further Research

 

 

 


13.0     Signals and Noise

 

13.1 Noise Sources:

Man made noise < 500 MHz

Hydro lines

Ignition systems

Fluorescent lights

Electric motors

Therefore deep space networks are placed out in the desert.

Atmospheric noise - lighting < 20 MHz

Solar noise - sun - 11 year sunspot cycle

Cosmic noise - 8 MHz to 1.5 GHz

Thermal or Johnson noise. Due to free electrons striking vibrating ions.

This equation applies to copper wire wound resistors, but is close enough to be used for all resistors.

Maximum power transfer occurs when the source and load impedance are equal.

 

Resistor Type

Cost

Noise

Carbon

Low

High

Metal film

Mid

Mid

Wire wound

High

Low

 

White noise - white noise has a constant spectral density over a specified range of frequencies. Johnson noise is an example of white noise.

Gaussian noise - Gaussian noise is completely random in nature however, the probability of any particular amplitude value follows the normal distribution curve. Johnson noise is Gaussian in nature.

Shot noise - bipolar transistors

Excess noise, flicker, 1/f, and pink noise< 1 KHz also

Inversely proportional to frequency

Directly proportional to temperature and dc current

Transit time noise - occurs when the electron transit time across a junction is the same period as the signal.

13.1.1 Combining Noise Voltages

The instantaneous value of two noise voltages is simply the sum of their individual values at the same instant.

This result is readily observable on an oscilloscope. However, it is not particularly helpful, since it does not result in a single stable numerical value such as one measured by a voltmeter.

If the two voltages are coherent [K = 1], then the total rms voltage value is the sum of the individual rms voltage values.

If the two signals are completely random with respect to each other [K = 0], such as Johnson noise sources, the total power is the sum of all of the individual powers:

A Johnson noise of power P = kTB, can be thought of as a noise voltage applied through a resistor.

A example of such a noise source may be a cable or transmission line. The amount of noise power transferred from the source to a load, such as an amplifier input, is a function of the source and load impedances.

If the load impedance is 0 Ω, no power is transferred to it since the voltage is zero. If the load has infinite input impedance, again no power is transferred to it since there is no current. Maximum power transfer occurs when the source and load impedance is equal.

The rms noise voltage at maximum power transfer is:

Observe what happens if the noise resistance is resolved into two components:

From this we observe that random noise resistance can be added directly, but random noise voltages add vectorially:

If the noise sources are not quite random, and there is some correlation between them [0 < K < 1], the combined result is not so easy to calculate:

13.1.2 Noise Figure

This parameter is specified in all high performance amplifiers and is measure of how much noise the amplifier itself contributes to the total noise. In a perfect amplifier or system, NF = 0 dB. This discussion does not take into account any noise reduction techniques such as filtering or dynamic emphasis.

Signal to noise ratio: It is either unitless or specified in dB. The S/N ratio may be specified anywhere within a system.

Noise Ratio: is a unitless quantity

Noise Figure:

 

13.1.2.1           Friiss’ Formula

It is informative to examine a cascade of amplifiers to see how noise builds up in a large system.

Therefore:   

Gain can be defined as:    

Therefore the output signal power is:   

and the noise ratio can be rewritten as:

The output noise power can now be written:  

From this we observe that the input noise is increased by the noise ratio and amplifier gain as it passes through the amplifier. A noiseless amplifier would have a noise ratio of 1 or noise figure of 0 dB. Consequently, the input noise would only be amplified by the gain.

The minimum noise that can enter any system is the Johnson Noise:

The minimum noise that can appear at the output of any amplifier is:

The output noise of a perfect amplifier would be:    

The difference between these two values is the noised added by the amplifier itself:

This is the added noise, as it appears at the output. The total noise coming out of the amplifier is then given by:

If a second amplifier were added in series, the total output noise would consist the first stage noise amplified by the second stage gain, plus the additional noise of the second amplifier:

If we divide both sides of this expression by the common term:

we obtain:

Recall:         

Then:

This process can be repeated for n stages, and the resulting equation is known as Friiss’ Formula:

It is more commonly written as:

From an examination of this equation, it is readily apparent that the overall system noise figure is largely determined by the noise figure of the first stage in a cascade.

Example

An amplifier has a noise figure of 8 dB and a power gain of 10 dB. It is followed by a mixer with a noise figure of 25 dB and a gain of -20 dB. The overall noise figure can be determined as follows:

Convert all numbers to ratios:

  G1 = 10 dB       10

  G2 = -20 dB      0.01

  F1 = 8 dB                   6.309

  F2 = 25 dB        316.22

Apply Friis’ formula:

Note that the overall noise figure is greater than that of the amplifier, but less than that of the mixer.

13.1.2.2           Noise Reduction

Inductive and capacitive reactances do not generate thermal noise because they cannot dissipate power. However, they can affect the noise spectrum.

The noise spectrum density is defined as:

but if it passes through a reactance:

13.1.2.3           Noise Temperature

In microwave applications, it is difficult to speak in terms of currents and voltages since the signals are more aptly described by field equations. Therefore, temperature is used to characterize noise. The total noise temperature is equal to the sum of all the individual noise temperatures.

13.2 Review of RLC Networks

13.2.1 Series RLC Circuits

The total impedance of the RLC circuit is given by:

When the magnitude of the inductance exceeds that of the capacitor, the impedance at a specific frequency may be graphically represented as:

By definition, the series RLC circuit is resonant when q = 0. This occurs when X = XL - XC = 0. The magnitude of the output voltage is determined by applying the voltage divider rule to the source and load resistances. The circuit impedance is its minimum value at resonance.

The frequency at which resonance occurs can be determined as follows:

The quality factor or Q is a measure of the amount of energy stored in a reactor. It can be defined as the ratio of voltage across the inductor to voltage across the resistor.

Since at resonance , then:

Typical values of Q in RLC circuits are: 10 < Q < 300.

Bandwidth

A sketch of the magnitude of the impedance as a function of frequency resembles:

The impedance is sometimes expressed in terms of Q and a new parameter we shall call y.

At resonance, the current flowing in the circuit is determined by R. The 3 dB point is defined when the current drops by 3 dB from its resonant value, at which point the modulus increases by . The magnitude of the impedance is therefore given by:

The 3 dB bandwidth can be determined by evaluating the impedance function at the 3 dB points:

At the upper cutoff frequency we obtain:

At the lower cutoff frequency we obtain:

The 3 dB bandwidth is the difference between these two frequencies:

13.2.2 Parallel RLC Circuits

Parallel circuits are easier to analyze in terms of admittance instead of impedance.

At resonance, the admittance is real and the imaginary components cancel. The resonant frequency can be determined as:

It is interesting to note that in a parallel resonant circuit, the resonant frequency is influenced by R, but in series resonant circuit, it is not.

Usually , and therefore: .

Since the admittance at resonance is purely conductive:

Recall that , therefore .. Since , the impedance at resonance is given by: .

The graph of  vs  is similar to that of  vs  for the series circuit. It therefore follows that .

The impedance of a parallel resonant circuit is high and is often referred to as the dynamic impedance where:

13.2.3 Self Resonance

A real coil has winding resistance and capacitance in addition to its inductance. As a result, it acts like a parallel tuned circuit. In some applications, it may be necessary to consider these secondary affects.

13.3 Oscillators

Many amplifiers and oscillators can be represented by:

The transfer function of this network can easily be determined as:

This expression is sometimes written:

As an example of this, we may consider an opamp.

If the open loop gain for this amplifier is: Ao = 104, the closed loop gain is given by:

The closed loop gain can be very closely approximated by:

Note that the feedback factor is 1/11 of the output signal.

With negative feedback the closed loop gain is less than of open loop gain, and the circuit is stable. With positive feedback the closed loop gain is greater than the open loop gain and the circuit is unstable.

For continuous oscillations:  and the total phase change around the loop must be .

If , oscillations build up until something limits the process. Creating an output with no input means that the closed loop gain is equal to infinity. This occurs when 1 + GH = 0. The loop gain at this point is GH = -1. This condition is known as the Barkhausen criteria for oscillation.

13.3.1 Tuned Circuit Oscillators

Note: The following analysis only works for high output impedance amplifiers such as transistors. In opamp circuits, the low output impedance effectively shorts out Z1.

Writing KVL in the output loop we obtain:

Writing KVL in the feedback loop we obtain:

From the input loop we obtain:

The loop gain can be determined as:

In order for oscillations to occur, the loop gain must be greater than 1, therefor an inversion is necessary in the circuit to change the sign of the gain. Such an inversion does take place between the base and collector of a transistor.

Recall that at resonance, a series RLC circuit has low impedance. Therefor  and the loop gain reduces to:

Oscillator Type

Z1

Z2

Z3

Hartley

L

L

C

Colpitts

C

C

L

Clapp

C

C

L + C

 

13.3.1.1           Hartley Oscillator

From our previous analysis, the loop gain at resonance is given by:

In order to guarantee oscillations, when the power is first turned on, the loop gain must be greater than 1.

Example

A worst-case design: the typical gain of a transistor is 35, but at the worst case operating temperature, it falls 30%. Determine the value of b to ensure oscillations with a 2:1 safety margin.

Solution

The lowest gain expected is 35 x.7 = 24.5

For a 2:1 safety margin, assume the gain is 24.5/2 = 12

Since Ab = 1 and A = 12, then b = 1/12

The operating frequency is determined by:

13.3.1.2           Colpitts Oscillator

This is the same as the Hartley oscillator, but with the inductors and capacitors reversed.

The feedback factor is given by: 

The oscillation frequency is given by:

13.3.1.3           Clapp Oscillator

This more stable variation of the Colpitts oscillator has a reduced tuning range.

The capacitors C1 and C2 must be much larger than the transistor junction capacitance. The oscillation frequency is given by:

13.3.2 Crystal Oscillator

Crystal Oscillators by MXCOM

 

The equivalent RLC circuit of a crystal is:

Applying an electric potential across the crystal causes it to deform. But since the crystal is a piezo-electric device, deforming it causes a slight electrical potential to develop across it. As a result, the crystal can act as an electromechanical resonator if the right excitation is applied.

The oscillation frequency is determined by the size and shape of the crystal. Crystals can be used to make very stable oscillators ranging from 15 KHz to 100 MHz.

Placing an inductor in series with the crystal lowers the series resonant frequency, and placing an inductor in parallel to the crystal increases the parallel resonant frequency.

13.3.2.1           Crystal Filter

 

Assignment Questions

 

Quick Quiz

1.  Friiss’ formula shows that noise figure is proportional to bandwidth. [True, False]

2.  Johnson noise is also called [pink, white, blue] noise.

3.  Admittance at resonance is purely conductive. [True, False]

Analytical Questions

1.  Determine the total noise power if two thermal noise sources of 12 µW and 16 µW are connected in series.

2.  A two stage RF amplifier has a 3 dB bandwidth of 150 KHz determined by an LC circuit at the input and operates at 27o C.

First stage:   PG  = 8 dB               NF = 2.4 dB

Second stage:         PG  = 40 dB              NF = 6.5 dB

Output Load = 300 Ω

     In testing the system, a 100 KΩ resistor is applied at the input. Determine:

a)  Input noise voltage

b)  Input noise power

c)  Output noise voltage

d)  Output noise power

e)  System noise figure

3.  Prove that under matched load conditions, the rms Johnson noise voltage is:         

Composition Questions

For Further Research

 

http://www.mth.msu.edu/~maccluer/Lna/noisetemp.html