Chapter 3 Network Noise and Intermodulation Distortion

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Contents

Introduction

Noise

Thermal Noise (Johnson Noise)

Current-source representation

Shot Noise

Flicker Noise

Avalanche Noise

Noise Models of IC Components

FET Transistor

Circuit Noise Calculations

Bipolar Transistor Noise Performance

Equivalent Input Noise and Minimum Detectable Signal

Equivalent Input Noise Generators

Bipolar Transistor Noise Generators

Field-Effect Transistor Noise Generators

Effect of Feedback on Noise Performance

Effect of Ideal Feedback

Practical Feedback

Amplifier Noise Model

Noise in Feedback Amplifiers

Amplifier Noise Model for Differential Inputs

Noise in Inverting Negative Feedback Circuits

Introduction

• 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

Noise

• 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

Thermal Noise (Johnson Noise)

• 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

Current-source representation

• 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

• 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

Flicker Noise

• 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
• K₁ 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

Avalanche Noise

• 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

Noise Models of IC Components

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 Transistor

• 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

Circuit Noise Calculations

• 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 v₁ and v₂ are statistically independent of each other arising from two separate resistors the average of the product v₁ v₂ 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 f₁, 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, i₀
• From 11.23a we have i₀ = g_mv_i
• From 11.23b we have g_mv_b + i_c = i₀
• 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_eq 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, i₀, are equated
• which gives
• Since i_b and i_c are independent generators, we obtain,

  where

• where b₀ 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 b₀ 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 f². Frequency f_b is estimated by equating the above equation to the midband noise and is 2q(I_B+[I_C/b₀²])
• For typical values of b₀ 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_iR_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 i₀ 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 i₀ 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^-24V²F

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.

Effect of Ideal Feedback

• 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.

Practical Feedback

• 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

Amplifier Noise Model

• 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_t²yields 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

Noise in Feedback Amplifiers

• 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_in and all the E's representing the noise voltages being injected at various critical points in the system. Blocks A₁ and A₂ represent amplifiers with voltage gains and b represents the feedback network. The output voltage V₀ is a function of all 5 inputs according to



• For comparison consider an open-loop system in which the feedback loop b is taken out V₀=A₁A₂'(V_in+E₁+E₂)+A₂'E₃+E₄
• To accomplish a meaningful comparison between the 2 cases we set and we find that V₀ 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_p and V_n 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
• An ideal differential amplifier occurs when we make the coefficients of V_in1 and V_in2 have identical magnitudes and opposite signs. This condition is satisfied by choosing the resistors such that R₁R₄=R₂R₃
• Thus the output becomes V₀=(R₂/R₁)(V_in2-V_in1)
• Thus the ideal difference mode voltage gain is R₂/R₁. To examine the noise behavior of the differential amplifier, first form a Thevenin equivalent circuit at the noninverting input as shown where R_p=R₃//R₄ and V'_in2=V_in2(R₄/(R₃+R₄))
• 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+R₂/R₁)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+R₂/R₁)².
• The negative input noise voltage ¡§flows through¡¨ the feedback resistor R₂ establishing a noise voltage directly at the output. Finally noise contribution due to R₂ 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 E²_no by (R₂/R₁)² to obtain



• where R₁²I_t2²=R₁²E_t2²/R₂²
• Note that two amplifier noise voltages plus are all increased at the input by (1+R₁/R₂)². Usually R₁<< R₂ for a typical high-gain amplifier applicationso the first 3 noise voltage sources essentially contribute
• directly to E_ni1² as does E²_t1. The noise current of the feedback resistor R₂ is multiplied by R₁². The I_n1 noise current ¡§flows through¡¨ R₁ creating a direct contribution to E_ni1². The I_n2 noise current ¡§flows through¡¨ R_pto produce a noise voltage and then is reflected to the inverting input by the same (1+R₂/R₁)² factor.
• When reflected to the noninverting input, we divide the noise equation by (1+R₂/R₁)²

  

• Here the two amplifier noise voltages as well as the noise voltage from R_p contribute directly to E²_ni2. The noise voltage in the feedback resistor is divided by the square of feedback factor. The noise in R₁ is slightly diminished but essentially unchanged when R₁<< R₂. The inverting noise flows through parallel combination of r₁ and r₂ then contributes directly to E²_ni1. The non-inverting current "flows through" R_p and contributes directly to E²_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 R₂.
• All noise source are now reflected to the V_in1 input, we obtain Fig. 3-22
• 
• where I_t2=E_t2/R₂
• 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