It is a well-known relationship that noise voltage increases with higher resistor values according to the Johnson noise equation,
where is the rms voltage noise, is Boltzmann’s constant, is temperature in Kelvin, is the resistance, and is the bandwidth.
This leads many engineers to the conclusion that resistor values should be reduced in order to reduce the noise. Although this is often true, it cannot be assumed because there are specific examples where larger resistors improve the noise performance.
As an example, in most cases, current is measured by passing it through a resistor and measuring the resulting voltage. The voltage developed is proportional to the resistor value according to Ohm’s law,
but as shown above, the Johnson noise of the resistor is proportional to the square root of the resistor value. Because of this relationship, a 3 dB improvement in the signal-to-noise ratio can be achieved each time the resistor value is doubled.
公式推导
我们以测电流信号为例：
1. 信号电压：
2. 电阻的热噪声电压：
3. 信噪比（SNR）表达式：
因此：
结论：在测量电流时，增加电阻值会提升 SNR（直到电压过大或功耗过高为止）。
Python 仿真与可视化
我们来模拟不同电阻值 对 SNR 的影响：
import numpy as np import matplotlib.pyplot as plt
# 常数 k = 1.38e-23 # Boltzmann 常数 (J/K) T = 300 # 温度 (K) B = 1e3 # 带宽 1kHz I = 1e-9 # 电流信号 1nA
# 绘图 plt.figure() plt.semilogx(R_vals, SNR_dB) plt.xlabel("Resistor R (Ohm)") plt.ylabel("SNR (dB)") plt.title("SNR vs Resistor Value for Current Measurement") plt.grid(True, which="both") plt.show()
The Noise Spectral Density of All Noise Sources Can Be Added Up and the Bandwidth Can Be Taken into Account at the End of the Calculation
It can save time to combine the noise spectral density (nV/√Hz) of multiple noise sources (voltage noise sources are combined as the root sum of squares) rather than computing the rms noise of each noise source separately, but this simplification is only applicable if the bandwidth seen by each noise source is the same.
It becomes a dangerous trap if the bandwidths seen by each of the noise sources are different.
Figure 1 shows the implications in an oversampled system. It would appear from the noise spectral density that the gain amplifier will dominate the total noise of the system, but once the bandwidth is taken into account, the rms noise contributed by each stage is very similar.
It Is Important to Include Every Noise Source in Hand Calculations
It may be tempting to consider every noise source in a design, but a designer’s time is valuable and this can be very time consuming in large designs. Comprehensive noise calculations are best left to simulation software. But how does a designer simplify the hand noise calculations needed during the design process?
Ignore minor noise sources below a certain threshold. If a noise source is 1/5 value of the dominant noise source, it contributes less than 2% to the total noise and can reasonably be ignored.
Designers argue about where to draw the threshold, but common rules of thumb are：
相对幅度比例
对总噪声的影响
1/3
+5%
1/5
+2%
1/10
+0.5%
数学推导
设：
主噪声源：
次要噪声源：，其中
总噪声为平方和：
相对误差（相比于仅使用 ）：
我们可以画图观察不同 对误差的影响。
我们绘制如下内容：
横轴为次要噪声源占主源的比例
纵轴为相对误差
并标出：
1/3 → 5%
1/5 → 2%
1/10 → 0.5%
import numpy as np import matplotlib.pyplot as plt
# 绘图 plt.figure(figsize=(8, 5)) plt.plot(alpha, relative_error * 100, label="Relative Increase (%)") plt.xlabel("e_minor / e_dominant") plt.ylabel("Relative Total Noise Increase (%)") plt.title("Effect of Adding Minor Noise Sources") plt.grid(True)
# 标出典型点 for t, label in zip(thresholds, threshold_labels): y = (np.sqrt(1 + t**2) - 1) * 100 plt.plot(t, y, 'ro') plt.text(t + 0.01, y + 0.2, f"{label} at {t:.2f}", color='red')
The noise equivalent bandwidth (NEB) is a useful simplification for noise calculations. Some noise from beyond the bandwidth of the circuit is able to get into the circuit because the gain above the cutoff frequency is not zero. The NEB is the cutoff frequency of a calculated ideal brick wall filter that would let in the same amount of noise that the actual circuit does. The NEB is larger than the –3 dB bandwidth and it has been calculated for common filter types and orders, for example, it is 1.57× larger than the –3 dB bandwidth for a 1-pole, low-pass filter or, in equation form,
However, there seems to be consistent confusion about where to put that multiplication factor in the noise equation. Remember that the NEB is an adjustment for the bandwidth, not the noise, so it goes under the square root, as follows:
推导
我们从基础出发推导：
一、理想白噪声系统中的噪声积分：
若某系统的噪声谱密度 恒定（白噪声），则其总均方噪声为：
其中 是系统的频率响应（单位增益低通时为一阶低通响应）：
二、单位增益一阶低通滤波器的增益平方：
则：
令 ，则 ：
对比“理想矩形滤波器”：
若我们假设理想滤波器带宽为 ，其增益为 1，在 内积分：
则为了等效两个积分：
即：
Python 仿真
我们来模拟一阶低通滤波器的频率响应，并计算其噪声等效带宽（NEB）与实际积分面积。
import numpy as np import matplotlib.pyplot as plt from scipy.integrate import simps
