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h| | e kBT 1
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L dk 2 (10.13)
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e kBT 1 The current generated in the circuit by the fluctuating voltage is simply I= V 2R (10.14)
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so that the average power dissipated (emitted) by the resistor is Pemitted = The time average is V 2 = lim 1 T T V 2 4R (10.15)
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T dt V(t)V * (t)
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(10.16)
In Eq. (10.16) and the expressions that follow, the voltage function V(t) is considered at frequency , Eq. (10.9). Then, the power spectrum the distribution of power, or square voltage, over many frequencies is defined as follows: S( ) = where R( ) = lim 1
T T
d R( )e i T , T dt V(t)V * (t )
(10.17)
(10.18)
is the autocorrelation function of the function V(t) over the period [ T, T]. Then, integrating S(w) over all frequencies,
d S( ) = Tlim T d d T dt V(t)V * (t ) e i T
= lim
T
2 T 2 T
d T dt V(t)V * (t ) ( )
= lim
T dt V(t)V * (t)
= 2 V 2
(10.19)
Equating the power dissipated by the resistor, Eq. (10.15) and that absorbed due to thermal radiation Eq. (10.13), it leads to 1 V 2 = 4R 2
d
h| | e kBT 1
h| |
(10.20)
By the relationship in Eq. (10.19), 1 1 4R 2 so that S( ) = 4R h| | e kBT 1
h| |
d S( ) = 2 d
h| |
h| | e kBT 1
(10.21)
(10.22)
Ten
Note: Equating the integrands of two definite integrals cannot be justified mathematically. A strict derivation would then appeal to the balance of emitted and absorbed power over an arbitrarily small frequency interval, rather than simply over the full range; this is the argument Nyquist constructs in Fig. 10.9. There, he considers the following circuit thought experiment : Imagine that over all frequencies, the exchange of power between the two resistors (at equal temperature) is equal, but over a small range of frequencies, the resistor on the left radiates more power than it absorbs from that on the right. Then, introduce a non-dissipative circuit element between the two that interferes more with the transfer of energy in the frequency range of interest than in any other range, so that now more total power flows from the right resistor to the left than in the other direction. But since the resistors are initially at the same temperature, to let the system evolve would be to raise the temperature of a hotter thermal body using the heat of a colder body, by simply introducing a passive circuit element. So, there must be a detailed balance of power over each tiny range of frequency, in accordance with the second law of thermodynamics.) Then, in the classical limit, h kBT , S( ) SV ( ) = 4RkBT . (10.23)
Equation (10.23) shows the Nyquist relation between the voltage power spectrum and temperature. Accordingly, the current power spectrum is closely related to the following equation, SI ( ) = 4k T 1 S ( ) = B R R2 V (10.24)
Since the actual resistor will be situated in a circuit with an amplifier designed to enhance the minuscule voltage fluctuations, and since the transmission line will exhibit capacitive effects, a realistic configuration of the circuit diagram would therefore, appear as in Fig. 10.9. The parallel combination of current source i and resistor R is the Norton equivalent of R and a small fluctuating voltage source,
Vamp
C i R Vin
Iamp
FIGURE 10.9
Enhancing small voltage uctuations.
S p e c t R x N I R Te c h n o l o g y
illustrated in Fig. 10.9. The net input voltage is then denoted by Vin, and the additional current and voltage associated with the operation of the amplifier itself are labeled Iamp and Vamp. The input voltage power spectrum takes the simple form SV = SI [ (R| C)]2 + SI, amp [ (R| C)]2 | |
(10.25)
where SI is the current power spectrum due to the resistor as illustrated in Eq. (10.24), SI, amp is the current power spectrum due to the amplifier, and [ (R| C)]2 is the squared real part of the parallel resistor capacitor | combination. The voltage power spectrum output generated through the amplifier can be represented by: SV out = (SV + SV , amp ) G 2
(10.26)
where SV, amp is the voltage power spectrum of the amplifier itself, and G is its gain. Eqs. (10.23) through (10.26) relate the measured quantity SV, out. Another important feature of Nyquist s model is that Johnson noise is independent of the material chosen for the resistor. This feature is exploited in accurate thermometry, allowing precise temperatures measurements without worrying about the particular type of sensor material having a contaminating effect. Additionally, Johnson noise found many applications in classical information cryptography.
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