Circuit designers often need to know the relationship between the signal they are
working with and the noise generated by their circuit. Noise analysis is useful in
making sure that the expected signal is not buried by the noise that an electrical
circuit generates with its resistances and semiconductor devices. One measurement
that is useful is the total RMS noise voltage which can also be used to calculate the
signal to noise ratio of a circuit. Measuring the total RMS noise voltage is a simple
procedure in Micro-Cap. The simple resistor divider shown below will be used to
demonstrate how the theory matches the simulation results.
The noise voltage spectral density (thermal noise) of a resistance is calculated through
the well known equation:
Sqrt(4*k*T*R)
where k is Boltzmann's Constant, T is the temperature in Kelvin, and R is the
resistance. This produces a result in units of volts per root Hertz. To calculate the
total RMS noise voltage, the frequency bandwidth needs to be taken into account. The
spectral density needs to be integrated over the bandwidth of interest. Since the
noise voltage spectral density of a resistance is constant across all frequencies,
the integrated equation can be simplified as:
Sqrt(4*k*T*R*B)
where B is the bandwidth. For the resistor divider circuit, the theoretical noise
values are calculated using the following values:
k = 1.3807e-23
T = 300.15K (27C)
R = 500k (1Meg // 1Meg)
B = 19980 (20Hz to 20kHz)
The theoretical results for the resistor divider are:
Noise Voltage Spectral Density = 91.04nV/sqrt(Hz)
Total RMS Noise Voltage = 12.87uV
In Micro-Cap, the total RMS noise voltage of a circuit can be plotted using the
following expression.
Sqrt(SD(Onoise**2))
Onoise is a Micro-Cap operator which represents the noise voltage spectral density of
the circuit at the specified output node. The SD function will integrate the
expression with respect to frequency in AC analysis. An AC analysis is run on the
resistor divider circuit from 20Hz to 20kHz. The plot is displayed below.
Both the Onoise variable and the Sqrt(SD(Onoise**2)) have been plotted. Note that since
the SD function performs a running integral of the expression, only the last value in
the plot is important since the value at 20kHz will be the integrated value across the
entire simulated bandwidth. As can be seen in the cursor tables of the plot, the
simulated values are dead on with the theoretical values.
The resistor divider is a simplified example to show that the mathematics for the total
RMS noise voltage expression work in Micro-Cap. This same expression can be used with
any type of circuit that includes shot and flicker noise as well as thermal noise. To
demonstrate this, the basic audio amplifier (Ref 1) shown below will be
simulated.
This audio amplifier is simulated from 1Hz to 100kHz. Both the Onoise and the total
RMS noise voltage expressions are plotted. The results are shown below. The total RMS
noise voltage across the simulated bandwidth for this audio amplifier is 54.968uV as
shown in the cursor tables.
Thanks to Sigurd Ruschkowski for helping develop this technique to work with Micro-Cap.
