Taming the Chaos: Correlated and Uncorrelated Sources in LTspice .NOISE Simulations: Part 3 of 3

Noise analysis of a circuit requires tracking several things simultaneously: each noise source, how those noise sources are shaped by the circuit, and how to combine them at the output.  LTspice excels at this analysis; manual calculations can be error-prone.

Noise Analysis when there are Multiple Noise Sources

I’ll use an example circuit to demonstrate these considerations. Figure 1 shows a circuit that converts a single-ended input signal to a differential signal, including an RC filter at the output to band-limit noise.  This is one of many circuit options for interfacing a single-ended signal to a differential ADC input.  

Figure 1: Single-Ended to Differential Circuit Using Op Amps

This simplified LTspice schematic uses ideal, noiseless op amps, and an ideal signal source.  We’ll explore each resistor as a noise source, and how their noise combines at the output.

Noise Simulation of the Full Circuit

Let’s run a .NOISE simulation in LTspice.  To follow along, download Noise_Examples.zip, and unzip the contents.

Open Single-ended to Differential Example - Complete.asc and examine the .NOISE directive:

.noise V(out+,out-) V1 dec 100 1 1Meg

The output-referred noise is defined as the voltage across out+ and out- nodes.

Simulate by clicking the green play button (or ALT+R).  Click the Pick Visible Traces button (or View -> Visible Traces from the menu) and select v(onoise) to plot the output referred noise. Click on the waveform name to display the cursor for measurements, and CTRL+Click to calculate integrated RMS noise (review Part 1 for these basic topics).  Figure 2 shows the noise density of v(onoise) low frequencies, and Figure 3 shows the integrated RMS noise.

Figure 2: Plotting v(onoise) of Differential Output (out+, out-)

Figure 3: Integrating v(onoise) to Calculate Total RMS Noise

LTspice makes simulation easy, but do we understand the results?  Could we verify simulation accuracy with hand calculations?

Analyzing the Positive Output Path

Let’s examine noise sources observable at out+ only. R1, R2, and R3 all appear as noise sources here.  Open and run Single-ended to Differential Example - Positive Output.asc, and plot v(onoise). Click on R1, R2, and R3 to plot each element’s noise contribution.

Select View -> SPICE Output Log to view .MEAS directive results (review Part 2 for .MEAS directives in .NOISE simulations).  See Figure 4 for waveform and SPICE Output Log results.

Figure 4: Individual Noise Contributions at out+

U1, R1, and R2 form an inverting amplifier; R3 and C1 form a low-pass filter.  In this example, R1, R2, and R3 each contribute the same noise density at the output. 

This partial circuit shows uncorrelated noise sources that combine using root sum of squares (RSS).

\sqrt{V_n(R1)^2 + V_n(R2)^2 + V_n(R3)^2} = \sqrt{(4.07nV/\sqrt[]{Hz})^2+(4.07nV/\sqrt[]{Hz})^2+(4.07nV/\sqrt[]{Hz})^2}=7.05nV/\sqrt[]{Hz}

This manual calculation matches the simulated total noise density at lower frequencies shown in Figure 4.

Analyzing the Negative Output Path

Next, examine noise sources at out-. R4 and R5 combine with U2 to form an inverting amplifier, and R6 combines with C2 to form a low-pass filter.  Additionally, noise from R1 and R2 passes through that noninverting amplifier and appears at out-.

Open and run Single-ended to Differential Example – Negative Output.asc, click out- to plot v(onoise).  Click on R1, R2, R4, R5, and R6 to plot each element’s noise contribution.  R3 does not contribute noise at out-.

View the SPICE Output Log for .MEAS directives (Figure 5).

Figure 5: Individual Noise Contributions at out-

Each noise source is uncorrelated, combining via RSS.

\sqrt{V_n(R1)^2 + V_n(R2)^2 + V_n(R4)^2 + V_n(R5)^2 + V_n(R6)^2} = \sqrt{5*(4.07nV/\sqrt[]{Hz})^2}=9.10nV/\sqrt[]{Hz}

Again, our hand calculations match the simulated total output noise density shown in Figure 5.

Differential Output Analysis

Now examine out+ and out- combined as a differential output.  Return to Single-ended to Differential Example - Complete.asc, and examine the SPICE Output Log (Figure 6).

Figure 6: Individual Noise Contributions at (out+,out-)

These results resemble the single-ended output simulations, except R1 and R2 noise contributions have doubled. This makes intuitive sense, because R1 and R2 noise appear at both out+ and out-.  The key insight: R1 and R2 at each output is correlated, combining differently than uncorrelated noise sources.

The R1 noise at out- is an inverted version of the R1 noise at out+.  Therefore, the differential noise across out+ and out- is twice the noise source, and LTspice simulations validate this.

Uncorrelated noise adds using RSS, whereas correlated noise combines (or cancels, depending on correlation relationship) with simple addition.  A tabular summary of all three simulations is shown below.

Conclusion

The moral of the story?  You can’t assume that the total output noise density from out+ (7.05nV/rtHz) combines with the total output noise density from out- (9.10nV/rtHz) in a purely uncorrelated (RSS) manner.  Extra caution is needed with your calculations; and when in doubt, simulate in LTspice!

 Read all the blogs in A Pinch of LTspice series.