Modelling a low-pass filter on LTSpice to filter an input square wave at 50kHz to obtain a sinusoidal output at 50Hz

Question

I'm trying to create a filter that filters the output of an inverter at 50kHz PWM (from reading the comments, I'm not sure if I should be calling this the switching freq?) to a sinusoidal one at 50Hz with a THD of less than 2% for the output voltage waveform (output voltage should be at 230Vrms). This circuit is supposed to model an inverted DC-AC output, with a filter attached to this output.

The schematics provided should model the inverter with an output of 50kHz PWM, although from reading the comments, I'm not sure if the voltage source allows this simulation?

I'm experimenting with a simple LCR one stage filter if possible. I understand that a low pass filter requires the shunt capacitor to short the high frequencies, but I'm not sure why the design recommended the use of a series inductor. In the schematics provided, R2 can be viewed as the parasitic resistance of the circuit, and R1 is the load resistor. I'm not sure how to calculate for the L and C values to be used to obtain the specified THD value.

I've tried running through a range of values, but I'm not sure which combination of L and C I should be using. The output waveform (a little messy because of the number of simulations I ran.)

The ideal values for L and C might not be in the range provided. Please advise.

Edit:

I've included the inverter circuit so that I can output a PWM waveform. I've modelled the switches to turn ON and OFF to output a triangular wave

Answer 1

Modelling a low-pass filter on LTSpice to filter an input square wave at 50kHz to obtain a sinusoidal output at 50Hz

There are three parts to such a problem statement

1. Model the inverter
2. Design the filter
3. Model the system

Model the inverter

In the OP a simple 50kHz pulse was used into an RLC "filter". This however isn't how an inverter would work to generate a fundamental of 50Hz (with a PWM of 50kHz).

What is needed is a 50Hz sinus waveform is modulated via a 50kHz carrier.

There are a number of way to perform this. The classic method is via generating the desired 50Hz waveform and using a comparator, compare this with a 50kHz triangular carrier to generate the desired modulation pattern in real time.

Another option is to pre-calculate the PWM patterns and store in a lookup table. Such a solution is very convenient for fixed-frequency, cheap inverters

With the PWM generation created, the inverter and a simple resistive load can be shown. A resistive load is prefered in such examples because an inductive load will smooth out the currents

nice switched voltage and current, but you can see an underlying harmonic. NOTE: an inductive load would have smoothed the currents but the voltage would still look chopped

Design the filter There are a number of methods associated with this. Typically they work by setting the resonant frequency of the filter at the logarithmic halfway be the fundamental and the switching. In this instance, around 1.5kHz

How you balance the L-C is based upon your design criteria. Working through one such method

L = 672uH

C = 16uF

targeting 15kW (10R per phase)

Model the system

With the pwm inverter model running and aspects of the output filter designed. They can be connected to confirm functionality

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LTspice.

To generate suitable pwm patterns, two voltage sources (pulse and sine) to create teh triangle carrier and the modulated signal are needed. A schmitt comparator can be used to generate the PWM signals to then feed to the inverter

Answer 2

As written : "a square wave at switching frequency of 50kHz" cannot be converted via any filter to another frequency.

what you are probably really asking is how to convert a PWM waveform, with a 50kHz frequency, and mark space ratio modulated at 50Hz, into a 50Hz sinewave.

Now there will be two contributions to THD in the output waveform:

1. Errors in the modulation imposed on the mark space ratio
2. Harmonic content arising from the 50kHz PWM ratio ( the thousandth harmonic of the fundamental, and its multiples).

I don't know how you want to apportion your error budget between these two, so I'm going to arbitrarily choose 1% harmonic content for the latter.

So you have an outline specification for a low pass filter, comprising:

1. Minimum (ideally zero) attenuation at the 50Hz fundamental frequency
2. Better than 40dB attenuation across your stopband starting at 50 kHz.

So pick a cutoff frequency much higher than 50 Hz, but sufficiently lower than 50 kHz that you can easily achieve 40 dB stopband attenuation. You are allowed an L/C filter (treating the load as a termination resistor) so a second order filter giving 40dB/decade attenuation, therefore the cutoff frequency must be below (ideally well below) 5 khz.

Note that the filter must meet your goals across the range of possible load impedances (RL in your schematic) defined by the output power you require.

This is a simple filter design following well established practice, so have at it.

You can probably substantially improve that 1% figure while maintaining low loss in the filter's parasitic components.

Answer 3

What you're asking for can't be done.

A linear filter like the one you drew with a sinusoidal input always gives an output at the same frequency as the input. It can attenuate the signal differently depending on the frequency, but it does not change the frequency.

To produce a 50 Hz output from a 50 kHz reference, you need a frequency divider or a phase-locked loop circuit, which are substantially more complicated than your RLC filter, and also require active elements of some kind.

Edit

In comments you wrote,

from my understanding, a square wave is made up of many sinusoidal harmonics - attenuating the higher order harmonics will not result in a sinusoidal 50Hz waveform?

The harmonics of a square wave are at odd multiples of the fundamental. So a 50 kHz square wave has harmonics at 150 kHz, 250 kHz, 350 kHz, etc. If the duty cycle is not exactly 50%, there will also be components at the even harmonics: 100 kHz, 200 kHz, ... There is no frequency content below the fundamental.

You could conceivably start with a 50 Hz square wave and filter out the harmonic at 49.95 kHz or 50.05 kHz (Of course picking out just one of these would require a very high order filter). But you can't start with 50 kHz and select out a harmonic at 50 Hz, because 50 Hz is not a harmonic of 50 kHz.