It is useful to get the phase and magnitude of response of a circuit as a

function of frequency. A DDS unit is used to synthesize a cosine wave,

which is then played through a SPI attached DAC.

The output of the DAC is attached to the circuit under test. For

starts, I am just testing on an RC lowpass with a nominal cutoff

frequency of 10000 radians/sec (1590 Hz) and a resistance of 10 kohm.

The voltage across the capacitor feeds back to the AN4 analog input (pin

6). An ISR does the DDS sine synthesis, records the voltage, and

computes the quadrature values by multiplying the analog input voltage with the DDS value (I-signal) and a sine wave in quadrature

with the DDS value (Q-signal). The quadrature signals, which have zero

average frequency (product of two equal frequencies) are lowpass

filtered in the ISR, then the main thread computes the phase and

magnitude. **Test code****.**

-- A modification of the low pass algorithm allows for much better

averaging by adding together up to 200,000 of the quadrature signals,

representing an average of the measurements

for up to one second. More points means accuracy at lower

frequency. Averaging 50,000 points works down to about 100 Hz, 200,000

works down to lower than 5 Hz. There is a systematic phase error which

can be calibrated out. The SPI transaction is ends in the ISR following

the ISR where it started. That implies a 5 microsecond delay in the DAC

update, and thus a frequency-dependent phase-shift. Also, as the

frequency increases, the sine approximation gets worse using DDS. The

phase correction is linear in frequency and corresponds to 5

degrees/KHz. **Code**.

The first two plots below compare the theoratical curves (in Matlab)

with the data generated by the program, shown as '+' symbols. Modifying

the frequency-dependent phase correction from f_main*0.005 to f_main*0.0048 makes the phase fit better in the third image. Addition of a *scan* command (**code**) produces a table of frequencies, phases, and amplitudes to feed to matlab.

-- Adding a lowpass filter with a time constant of 3 microseconds to

the sinewave output (200 KHz sample rate implies 5 microsecond sample

time), *and recording both the input and output *of the

circuit-under-test (CUT) drops the error to below 0.25 degrees/KHz. The

phase of the CUT is then output phase minus input phase, and the gain is

the ratio of output/input amplitudes. This extends the range of correct

phase above 10 KHz, as shown in the 4th plot below. **Code**