In the previous appends, we looked at using Shooting Newton Periodic Steady-State analysis to analyze analog circuits. In this append, we will look at using Harmonic Balance Periodic Steady-State, HBPSS, to analyze analog circuits. HBPSS is widely used for RF and microwave circuit design. However, designers often do not realize that it can also be useful for analog circuit design, in particular, when they would like to analyze distortion. As an example, we will simulate the Total Harmonic Distortion, THD, of an amplifier. We will compare and contrast using transient analysis with the Fourier transform and using HBPSS to analyze distortion. The test circuit is a simple Audio Amplifier for headphones built from an LM386 op-amp, shown in Figure 1.
Figure 1: LM386 Audio Amplifier
Typically, transient analysis with the Fourier transform is used to simulate the THD of an audio amplifier. The challenge with using transient analysis is to optimize the transient analysis simulation configuration for accurate Fourier analysis. Fourier analysis requires that the circuit has reached sinusoidal steady-state, that is, we need to measure the response after the start-up transient of the system has completed settling. Achieving sinusoidal steady-state can require settling for many periods in audio designs because of the large time constants due to the large off-chip capacitors for dc blocking. Of course performing Fourier analysis can alter the spectrum of the amplifier unless designers are careful with their simulation and Fourier analysis setup.
To illustrate the limitations of Fourier analysis and the benefit of steady-state analysis for this application, the several simulations were run. In each case the THD was calculated for one period of the fundamental frequency, in this case 1kHz. Four transient simulations were performed with different amounts of delay allowed to settle the start-up transients of the circuit before performing the Fourier analysis. The delay times were: 0 periods of the fundamental frequency, 1 period of the fundamental frequency, 3 periods of the fundamental frequency, and 10 periods of the fundamental frequency. The THD for each simulation condition is shown is Table I. In this case, the simulation is performed using the Spectre's conservative error preset. The conversion from the time domain to the frequency domain was performed using the ViVA Waveform Calculator FFT function and the Spectre Fourier Integral.
Table 1: THD Results for Various Simulation Conditions
Some observations about the simulation results,
- As expected, the simulated THD is sensitive to the delay time. The longer the delay time the closer the amplifier is to sinusoidal state and the more accurate the Fourier analysis.
- After about 10 periods, the amplifier has reached sinusoidal steady-state and the results for the Fourier Integral and FFT are consistent with HB PSS analysis.
- In this case, the HBPSS analysis was performed based on the dc operating point of the circuit, transient-assisted harmonic balance analysis was not required.
For this simple example, the simulation time using harmonic balance PSS analysis is >5x faster than using transient analysis with the Fourier Transform. As circuit become larger and especially for post-layout simulations, we would expect to see that the difference in the transient analysis time and the dc operating point calculation become larger and HBPSS becomes even more effective. Reducing simulation enables designers to analyze THD across process variations, with corner and Monte Carlo analysis, or to optimize THD.
One question maybe why didn't we use Shooting Newton for the periodic steady-state analysis? The short answer is that Shooting Newton is not required in this case. Harmonic Balance analysis provides the steady state solution in terms of finite Fourier series and is very effective for simulating distortion. If time domain waveforms were more non-linear, for example, when simulating a Switched Capacitor circuit or a DC-to-DC Converter then Shooting Newton would be appropriate.
To help illustrate the need to settle the initial start-up transient, I have plotted the non-periodicity, on of the outputs of the Spectre's Fourier Integral analysis, as a function of settling time, see Figure 2. The non-periodicity measures the difference between the initial value and final value. When the response is in sinusoidal steady-state the non-periodicity will be 0.
Figure 2: Effect of Settling Time on Periodicity
This approach, using harmonic balance analysis for periodic steady-state analysis to supplement transient analysis with the FFT, can be applied whenever you need to measure the distortion of a linear amplifier. In the next append, we will look at extending this approach to using PSS for distortion analysis of non-linear circuits, for example.
Hope you found this append useful, please let me know!