- Noise is one of the most important factors affecting the operations of IC circuits. This is because noise represents the smallest signal the circuit can process.
- The principle noise sources are Johnson noise generated in resistors due to random motion of carriers; shot noise arising from the discreteness of charge quanta; mixer noise arising from non-ideal properties of mixers; undesired cross coupling of signals between two sections of the receiver; flicker noise due to defects in the semiconductor; and power supply noise.
- Except for Johnson noise and shot noise the other noise sources can be improved or eliminated through proper design
- The ¡§Noise Figure¡¨ measures the noise generated in a network, together with the ¡§dynamic range¡¨ are used to quantify the receiver performance

- All signals are contaminated with noise
- The noisiness of a signal is specified by the signal-to-noise ratio defined as
- The last definition will be adopted.
- Noise of human origin is usually the dominant factor in receiver noise. This can usually be eliminated through proper design, layout, and shielding. Random noise cannot be eliminated. It sets the theoretical lower limit on receiver noise
- The mean square noise voltage are referred to as the noise power
- The noise power is normally frequency-dependent and is usually expressed as a power spectral density function. The total noise power is

- Discovered by J.B. Johnson and is therefore commonly known as Johnson noise
- The rms value of thermal noise voltage E
_{n}is given by - Since the noise voltage squared is proportional to Df. This implies that if the interval is infinite, the noise power contributed by the resistor is also infinite.
- In reality the above equation must be modified above 100 MHz, but is sufficiently accurate for low frequency
- R(f) is the real part of the impedance Z(f) looking into the two terminals
between which E
_{n}is measured. - If a resistor is connected to a frequency-dependent network as shown in Fig. 3-1
- then the total noise due to R is
- Where G(f) is the magnitude squared of the frequency-dependent transfer
function between the input and the output voltages
- Since R(f) depends on frequency
- The integral of G(f) is known as the noise bandwidth B
_{n}of the system. - If 2 resistors are connected in series, it is the voltage squared, not the noise voltages, which are added
- Example 3.1
- The impedance of the parallel combination of a resistor R and capacitor C is given by
- The real part is given by
- We can calculate E02 using
- Since

- So far we used voltage sources in series with a noiseless resistor to represent thermal noise. Norton's theorem shows that the voltage noise source can also be represented by a current generator in parallel with a noiseless resistor as shown in Fig. 3-2

- Shot noise is due to discreteness of the electronic charge arriving at the anode giving rise to pulses of current. The current noise power spectrum is

- This type of noise is found in all semiconductor devices under the application of a current bias
- The mean squared current fluctuation over a frequency range Df is
- Metal films show no or very small flicker noise, thus they should be used for CKT design if low 1/f noise is desired
- K
_{1}may vary by over orders of magnitude because flicker noise is caused by various unknown mechanisms such crystal imperfections, contamination. - Although flicker noise appears to be dominant at low-frequencies, it may still affect rf applications of the communication circuits through the nonlinear properties of the oscillators and mixers which mixes the noise to the carrier frequencies

- This is caused by Zener or avalanche breakdown in a pn junction
- Electrons and holes in the depletion region of a reversed-biased junction acquire enough energy
- Since additional electrons and holes are generated in the collision process a random series of large noise spikes will be generated.
- The most common situation is when Zener diodes are used in the circuit and should therefore be avoided in low noise circuits.
- The magnitude of the noise is hard to predict due to its dependence on the materials

**I. Junction Diode**

- The equivalent circuit for a junction diode the equivalent circuit is shown in Fig. 3-3
- R
_{s}is a physical resistor due to resistivity of the silicon, it exhibits thermal noise. - The current noise source is due to shot noise and flicker noise. Thus

**II. Bipolar Transistors**

- In a bipolar transistor in the forward-active region, minority carriers
entering the collector-base depletion region are being accelerated to the
collector. The time of arrival is a random process process, thus I
_{C}shows full shot noise. - The base current I
_{B}is due to recombination in the base and base-emitter depletion regions and also due to carrier injection from the base into the emitter. - Thus I
_{B}also shows full shot noise characteristics. The recombination process in the region also contribute to burst noise and flicker noise. - Transistor base resistor is a physical resistor and thus has thermal noise
- Collector r
_{c}also shows thermal noise, but since this is in series with the high-impedance collector node, this noise is usually neglected. r_{p}and r_{b}are fictitious resistors used for modeling and therefore do not contribute to thermal noise - The equivalent circuit model for a BJT transistor is shown in Fig. 3-4

- FET shows full shot noise for the leakage current at the gate as well as thermal noise and flicker noise in the channel region.
- Very often in JFETs the dominant type of noise is burst noise instead and in MOSFETs the dominant type of noise is flicker noise

- The device equivalent circuits can be used for calculation of noise performance. Consider a current noise source
- if the rms current noise is represented by i, Within a small bandwidth, Df, the effect of the noise current can be calculated by substituting by a sinusoidal generator and performing circuit analysis in the usual fashion. When the circuit response to the sinusoid is calculated, the mean-squared value of the output sinusoid gives the mean squared value of the output noise in bandwidth Df.
- In this way network noise calculations reduce to familiar sinusoidal circuit analysis calculations.
- When multiple noise sources exists which is the case in most practical situations, each noise source is represented by a separate sinusoidal generator, and the output contribution of each source is calculated separately.
- The total output noise in bandwidth Df is calculated as a mean-squared value by adding the individual mean-squared contributions from each output sinusoid.
- For example if we have 2 resistors in series the total voltage is
- Since the noise sources v
_{1}and v_{2}are statistically independent of each other arising from two separate resistors the average of the product v_{1}v_{2}will be zero - Analogous results is true for independent current noise sources placed in parallel. The spectra are summed together.

**Bipolar
Transistor Noise Performance**

- Consider the noise performance of the simple transistor stage as shown in Fig. 3-5
- The total output noise can be calculated by considering each noise source in turn and performing the calculation as if each noise source were a sinusoid with rms value equal to that of the noise source being considered.
- Consider the noise generator v
_{s}due to R_{s}where Z is the parallel combination of r

_{p}and C_{p}. The output noise voltage due to v_{s} - Similarly it can shown that the output noise voltages by v
_{b}and i_{b}are - Noise at the output due to is
- The total output noise is
- Substituting for Z we have
- The output noise power spectral density has a frequency-dependent part,
which arises because the gain stage begins to fall above frequency f
_{1}, and noise due to which appears amplified in the output, also begins to fall. The constant term is due to noise generators . Note that this noise contribution would also be frequency dependent if the effect of Cƒ_{m}had not been neglected. The noise voltage spectral density is shown in the Fig. 3-6

**Equivalent
Input Noise and Minimum Detectable Signal**

- The significance of the noise performance of a circuit is the limitation it places on the smallest input signal. For this reason the noise performance is usually expressed in terms of an equivalent input noise signal, which gives the same output noise as the circuit under consideration.
- Such representation allows one to compare directly with incoming signals and the effect of the noise on those signals is easily determined.
- Thus the circuit previously studied can be represented by Fig. 3-7
- where is an input noise voltage generator that produces the same output noise as all of the original noise generators. All other source of noise are considered removed. Thus
- The above equation rises at high frequencies due to variation of |Z| with frequencies. This is due to the fact that as the gain of the device falls with frequency, output noise generators have larger effects when referred back to the input.
- Example: Calculate the total input noise voltage, , for the circuit of the following circuit from 0 to 1 MHz
- Fig. 3-8
- Using the above equation for equivalent input noise
- On the other hand we can use for the calculation of
- If A
_{v}is the low-frequency gain - using the data
- The examples shows that from 0 to 1 MHz the noise appears to come from a 3.78 mV rms noise-voltage source in series with the input. This can be used to estimate the smallest signal that the circuit can effectively amplify, sometimes called the minimum detectable signal (MDS). If a sine wave of magnitude 3.78 mV were applied to this circuit, and the output in a 1-MHz bandwidth examined on an oscilloscope, the sine wave would be barely detectable

**Equivalent
Input Noise Generators **

- Using the equivalent input noise voltage an expression for equivalent input noise generator dependent on the source resistance can be determined.
- To extend this to a more general and more useful representation using 2 equivalent input noise generators. The situation is shown in Fig. 3-9
- Here the two-port network containing noise generators is represented by the same network with internal noise sources removed and with a noise voltage and current generator connected to the input. It can be shown that this representation is valid for any source impedance, provided that the correlation of between the two noise generators is considered.
- The 2 noise sources are correlated because they are both dependent on the same set of original noise sources.
- However, correlation my significantly complicate the calculation. If the correlation is large, it may be simpler to go to the original circuit.
- The need for both voltage and current equivalent input noise generators to represent the noise performance of the circuit for any source resistance can be appreciated as follows. Consider the extreme cases of source resistance RS=¥, cannot produce output noise and represents the noise performance of the original noisy network
- The values of the equivalent input generators are readily determined. This is done by first short circuiting the input of both circuits and equating the output noise in each case to calculate . The value of is found by open circuiting the input of each circuit and equating the output noise in each case

**Bipolar
Transistor Noise Generators**

- The equivalent input noise generators for BJT can be calculated from the equivalent circuit of Fig. 3-10
- The 2 circuits are equivalent and should give the same output noise for any source impedance
- The value of can
be calculated by short circuiting the input of each circuit and equating the
output noise, i
_{0} - From 11.23a we have i
_{0}= g_{m}v_{i} - From 11.23b we have g
_{m}v_{b}+ i_{c}= i_{0} - Here we use rms noise quantities and make no attempts to preserve the signs
of the noise quantities as the noise generators are all independent and have
random phase. We also assume that . r
_{b}<< r_{p} - Thus we have
- Since r
_{b}is small the effect of is neglected - Using the fact that v
_{b}and i_{c}are independent, we obtain - Using previous definition of
- The equivalent noise-voltage spectral density thus appears to come from
a resistor R
_{e}q such that - This is known as the ¡§equivalent input noise resistance¡¨
- Here rb is a physical resistor in series with the input, whereas 1/2g
_{m}represents the effect of collector shot noise referred back to the input - The above equations allows one to compare the relative significance of noise
from r
_{b}and I_{C}in contributing to . - Good noise performance requires the minimization of R
_{eq}. This can be accomplished by designing the transistor to have a low r_{b}, and running the devices at a large collector bias current to reduce 1/2g_{m}. - To calculate the equivalent input noise current, the inputs of both circuits
are open circuited and the output noise currents, i
_{0}, are equated - which gives
- Since i
_{b}and i_{c}are independent generators, we obtain,where

- where b
_{0}is the low-frequency, small signal current gain. Substituting for gives - where . The
last term is due to collector current noise referred to the input. At low
frequencies this becomes and
is negligible compared with I
_{B}for typical b_{0}values. The equivalent input noise current spectral density appears to come from a current source I_{eq}and - I
_{eq}is minimized by utilizing low bias currents in the transistor, and using high b transistors. It should be noted that low current requirement to reduce contradicts that for reducing - Spectral density for is
frequency dependent both at low and high frequency regime due to flicker noise
and collector current noise referred to the input respectively.
^{fb}and^{fa}are defined as in the Fig. 3-11 - Using the definition
- The collect current noise is
- at high frequencies, which increases as f
^{2}. Frequency f_{b}is estimated by equating the above equation to the midband noise and is 2q(I_{B}+[I_{C}/b_{0}^{2}]) - For typical values of b
_{0}it is approximately 2qI_{B}We obtain - The large signal current gain is
- Therefore
- Once the input noise generators have been calculated, the transistor noise performance with any source impedance is readily calculated.
- Consider the circuit in Fig. 3-12
- with a source resistance R
_{S}. The noise performance of this circuit can be represented by the total equivalent noise voltage in series with the input of the circuit as shown. - Neglecting noise in R
_{l}and equating the total noise voltage at the base of the transistor v_{iN}=v_{s}+v_{i}+i_{i}R_{S} - If correlation between v
_{i}and i_{i}is neglected this equation gives - Thus the expression for total equivalent noise voltage is
- Using the data from previous example and neglecting 1/f noise we calculate
the total input noise voltage for the circuit in a bandwidth 0 to 1 MHz. The
total input noise in a 1 MHz bandwidth is

**Field-Effect
Transistor Noise Generators**

- The equivalent noise generators for a field-effect transistor can be calculated from the equivalent circuit in Fig. 3-13
- Figure (a) is made equivalent to figure (b).
- The output noise in each case is calculated with a short-circuit load and Cgd is neglected.
- If the input of each circuit in the figure is short circuited and the resulting
output noise currents i
_{0}are equated we obtain from shorting fig. a that - Figure (a) is made equivalent to figure (b).
- The output noise in each case is calculated with a short-circuit load and
C
_{gd}is neglected. - If the input of each circuit in the figure is short circuited and the resulting
output noise currents i
_{0}are equated we obtain from shorting fig. a that - From Fig. b we have
- Thus
- Substituting the expression for into the equation for total noise we have
- The equivalent input noise resistance R
_{eq}is defined as - where in which K' = K/4kT
- The input noise-voltage generator contains a flicker noise component which may extend into the Mega Hertz region. The magnitude of flicker noise depends on the details of the processing procedure, biasing and the area of the device.
- Flicker noise generally increase as 1/A this is because larger devices contains more defects at the Si-SiO2 interface. An averaging effect occurs that reduces the overall noise.
- Flicker noise varies inversely with the gate capacitance because trapping and detrapping lead to variation of the threshold voltage which is inversely proportional to the gate capacitance. The equivalent input-referred voltage noise can often be written as
- Typical value for K
_{f}is 3 x 10^{-24}V^{2}F

**Effect of
Feedback on Noise Performance**

- The representation of circuit noise performance with two equivalent input noise generators is extremely useful in the consideration of the effect of feedback on noise performance.

- The series-shunt feedback amplifier is shown where the feedback network is ideal in the signal feedback to the input is a pure voltage source and the feedback network is unilateral. Noise in the basic amplifier is represented by input noise generators
- The noise performance of the overall circuit is represented by equivalent input generators
- Fig. 3-14
- The value of can be found by short circuiting the input of each circuit and equating the output signal. However, since the output of the feedback network has a zero impedance, the current generators in each circuit are then short circuited and the two circuits are then identical if
- If the input terminals are open circuited, both voltage generators have a floating terminal and thus no effect on the circuit, for equal outputs, it is necessary that
- Thus for the case of ideal feedback, the equivalent input noise generators can be moved unchanged outside the feedback loop and the feedback has no effect on the circuit noise performance.
- Since the feedback reduces circuit gain and the output noise is reduced by the feedback, but desired signals are reduced by the same amount and the signal-to-noise ratio will be unchanged.

- Series-shunt feedback circuit is typically realized using a resistive divider
consisting of R
_{E}and R_{F}as shown in Fig. 3-15 - If the noise of the basic amplifier is represented by equivalent input noise
generators and
, and the thermal
noise generators in R
_{E}and R_{F}are included in b as shown above. To calculate consider the inputs of the circuits of b and c short circuited, and equate the output noise - where R = R
_{F}// R_{E}. Assuming that all noise sources are independent we have where - Thus in a practical situation the equivalent input noise voltage of the overall amplifier contains the input noise of the basic amplifier plus two other terms. The second term is usually negligible, but the third represents the thermal noise in R and is often significant.
- The equivalent input noise current, , is calculated by open circuiting both inputs and equating output noise. For the case of shunt feedback at the input as shown, opening circuiting the inputs of b and c and equating the output noise we have
- Fig. 3-16
- Thus the equivalent input noise current with shunt feedback applied consists of the input noise current of the basic amplifier together with a term representing thermal noise in the feedback resistor. The second term is usually negligible. If the inputs of the circuits of b and c are short circuited ant the output noise equated it follows that

- As in the case before, amplifier noise represented by a zero impedance voltage generator in series with the input port and an infinite impedance current generator in parallel in the input and by a complex complex correlation coefficient C.
- The equivalent model is shown in the next page where noise sources E
_{n}, E_{t}and I_{n}are used. Here E_{t}is the noise generator for the signal source. Again we determine the equivalent input noise, E_{ni}, to represent all 3 sources. The levels of signal voltage and noise voltage that reach Z_{in}in the circuit are multiplied by the noiseless gain A_{v} - Fig. 3-17
- The system gain is defined by
- For signal voltage, linear voltage and current division principles can be applied. However, for the evaluation of noise, we must sum each contribution in mean square values. The total noise at the output port is
- The noise at the input to the amplifier is
- The total output noise above divided by K
_{t}^{2}yields the expression for equivalent input noise - is independent of the amplifier¡¦s gain and its input impedance. This makes the most useful index for comparing the noise characteristics of various amplifiers and devices. If the individual noise sources are correlated an additional term must be added to the above expression

- Feedback is an important technique to alter gains, impedance levels, frequency response and reduce distortion. When negative feedback is properly applied the critical performance indexes are improved by a factor 1+Ab. However, with noise it was shown that feedback does not affect the equivalent input noise, but the added feedback resistive elements themselves will add noise to the system.
- To examine how noise is affected by feedback we consider the block diagram Fig. 3-18
- The desired input voltage
*V*and all the E's representing the noise voltages being injected at various critical points in the system. Blocks_{in}*A*and_{1}*A*represent amplifiers with voltage gains and_{2}*b*represents the feedback network. The output voltage*V*is a function of all 5 inputs according to_{0} - For comparison consider an open-loop system in which the feedback loop
b is taken out V
_{0}=A_{1}A_{2}'(V_{in}+E_{1}+E_{2})+A_{2}'E_{3}+E_{4} - To accomplish a meaningful comparison between the 2 cases we set and we
find that V
_{0}for the open loop case is

- Thus feedback does not give any improvement for any noise source introduced at the input to either amplifier regardless of whether this noise source exists before or after the summer. Noise injected at the amplifier¡¦s output is attenuated in the feedback amplifier.
- In fact, if the feedback consists of resistive elements will actually increase the output noise level due to added thermal noise from the feedback resistors.

**Amplifier Noise
Model for Differential Inputs **

- Since operational amplifiers are configured with differential inputs. Users can configure the feedback network an input signal so as to produce a noninverting amplifier, an inverting amplifier or a true differential amplifier. Therefore all op amp model having equivalent noise sources must be able to handle all of these different configurations.
- The basic amplifier noise model is expanded as in Fig. 3-19
- Noise sources E
_{n1}and I_{n1}are noise contributions from the amplifier reflected to the inverting input terminal referenced to ground. I_{n2}and E_{n2}are that reflected to the non-inverting terminal. Consider the typical amplifier circuit shown in Fig. 3-20 - Voltages
*V*and_{p}*V*are the voltages at the respective positive and negative inputs to the amplifier referenced to ground. The output voltage for an ideal op amp is_{n} - An ideal differential amplifier occurs when we make the coefficients of
*V*and_{in1}*V*have identical magnitudes and opposite signs. This condition is satisfied by choosing the resistors such that R_{in2}_{1}R_{4}=R_{2}R_{3} - Thus the output becomes V
_{0}=(R_{2}/R_{1})(V_{in2}-V_{in1}) - Thus the ideal difference mode voltage gain is R
_{2}/R_{1}. To examine the noise behavior of the differential amplifier, first form a Thevenin equivalent circuit at the noninverting input as shown where R_{p}=R_{3}//R_{4}and V'_{in2}=V_{in2}(R_{4}/(R_{3}+R_{4})) - Next insert noise voltage and current sources for the op amp and Thevenin equivalent noise sources for the resistors as shown in Fig. 3-21
- Here 7 signal source have arbitrary polarities as shown. Here we assume
the op-amp has a finite open loop voltage gain A but is ideal otherwise. The
four defining equations for this circuit are
- The four equations give
- As A®¥ we obtain
- Previously for clarity, we substituted voltage and current signal sources for corresponding noise sources. The gain to the output will be the same for both signal sources and noise sources from the same circuit position
- The result is
- The equation shows that each noise source contributes to the total squared
output noise. Both equivalent input noise voltages and the noise from R
_{p}are reflected to the output by the square of the noninverting voltage gain,(1+R_{2}/R_{1})^{2}. - The positive input noise current ¡§flows through¡¨ R
_{p}establishing a noise voltage which, in turn, is reflected to the output by the same gain factor (1+R_{2}/R_{1})^{2}. - The negative input noise voltage ¡§flows through¡¨ the feedback resistor
R
_{2}establishing a noise voltage directly at the output. Finally noise contribution due to R_{2}appears directly at the output. - To determine E
_{ni}we first decide which terminal will be the reference. This is critical since the K_{t}¡¦s are different for the inverting and non-inverting inputs - First reflect E2no to the inverting input by dividing E
^{2}_{no}by (R_{2}/R_{1})^{2}to obtain - where R
_{1}^{2}I_{t2}^{2}=R_{1}^{2}E_{t2}^{2}/R_{2}^{2} - Note that two amplifier noise voltages plus are all increased at the input
by (1+R
_{1}/R_{2})^{2}. Usually R_{1}<< R_{2}for a typical high-gain amplifier applicationso the first 3 noise voltage sources essentially contribute - directly to
*E*as does_{ni1}^{2}*E*. The noise current of the feedback resistor R^{2}_{t1}_{2}is multiplied by*R*. The_{1}^{2}*I*noise current ¡§flows through¡¨_{n1}*R*creating a direct contribution to_{1}*E*. The_{ni1}^{2}*I*noise current ¡§flows through¡¨ R_{n2}to produce a noise voltage and then is reflected to the inverting input by the same (1+R_{p}_{2}/R_{1})^{2}factor. - When reflected to the noninverting input, we divide the noise equation by
(1+R
_{2}/R_{1})^{2} - Here the two amplifier noise voltages as well as the noise voltage from
R
_{p}contribute directly to*E*. The noise voltage in the feedback resistor is divided by the square of feedback factor. The noise in R^{2}_{ni2}_{1}is slightly diminished but essentially unchanged when R_{1}<< R_{2}. The inverting noise flows through parallel combination of r_{1}and r_{2}then contributes directly to*E*. The non-inverting current "flows through" R^{2}_{ni1}_{p}and contributes directly to*E*^{2}_{ni2}

**Noise in Inverting
Negative Feedback Circuits**

- The inverting amplifier configuration with resistive negative feedback is
the most widely used stage configuration. The input offset voltage due to
bias current will be canceled by making R
_{p}a single resistor equal to the parallel combination of R_{s}and R_{2}. - All noise source are now reflected to the V
_{in1}input, we obtain Fig. 3-22 - where I
_{t2}=E_{t2}/R_{2} - An op amp specification sheet normally provide En and In which are defined as
- We can now define a new equivalent amplifier noise voltage
- and