| Linear analog electronic filters |
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The bridged-T delay equaliser is an electrical all-pass filter circuit utilising bridged-T topology whose purpose is to insert an (ideally) constant delay at all frequencies in the signal path. It is a class of image filter.
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Passive RC low pass filter tutorial!
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Transcription
In this tutorial I'm going to talk about passive low-pass RC filter circuits. So at a basic level... what are filters? Filters are used to change the frequency content of signals. You can experiment with this in software by playing with the graphical equalizer settings in your MP3 player. Here is a normal unfiltered sound clip... Now if you decrease the low frequencies and allow the higher frequencies to pass through unchanged, you create a high pass filter. It sounds like this... You can hear that there is much less bass and only the higher frequencies remain untouched. Now if you decrease the high frequencies and allow the low frequencies to pass through unchanged, you get a low pass filter. And that sounds like this... You can hear all the bass but the high frequency sounds are much quieter. Now I'm going to show you how to do this in hardware and you'll be able to do this with any signal, whether it be audio, video signals, radio-frequency signals or whatever. Let's start with RC low pass filters because they are easy. An RC filter is just a filter made out of a resistor and a capacitor. The original signal goes in and the filtered signal comes out. The reason why this works is that the voltage on a capacitor cannot instantaneously change. When you have a resistor that slows the charging of the capacitor, your output voltage might not be able to follow the sudden changes in the input voltage. As a result higher frequencies get filtered out. I'm going to build a low pass filter with a 10 nanofarad capacitor and a 10 kilohm resistor and show you how it affects different frequency signals. There's no particular reason why I'm using these values it's just for the sake of an example and I'll do more examples later. So this is very important equation. This is the equation that is used to determine the cut-off frequency of the filter. If I plug in my chosen values of 10 kilohms and 10 nanofarads I get a cut-off frequency of 1592Hz. And I'm going to round that up to 1600Hz. Okay so this low pass filter has a cut-off frequency of 1600Hz, but what does that mean? This means that at frequencies below 1600Hz the signal passes through unchanged. If I feed in sine waves of different frequencies, you can see that from 45Hz to 200Hz the output is the same as the input: ten volts peak to peak. Now at frequencies approaching 1600Hz, the amplitude is slowly dropping on the output. By the time we reach the 1600Hz cut-off frequency we've got a reduction of about thirty percent. The cut-off frequency is the point where the filtering effect really starts being effective. And another thing you'll notice is that the two sine waves no longer line up. There has been a shift in phase. That's because the filter is introducing a small delay in the signal. Most the time you won't care but it is something you should be aware of. Now let's increase the frequency way beyond the cut-off point. At 15kHz there is a massive reduction in amplitude. And as we get into higher frequencies like 50kHz and beyond, there's almost nothing left of the signal. Now the oscilloscope example was just a visual introduction to the concept of a low pass filter and how it affects voltages in your circuits. Normally what engineers do is describe filter characteristics with a graph called a bode plot. Earlier I mentioned that the cut-off frequency is the point at which the filter starts being effective. Well you should know that in the real world you will never get a filter like this where as soon as you hit the cut-off frequency all the higher frequencies get reduced to zero. Here is a more realistic bode plot. In realistic filters what happens is that for low frequencies you get this flat response where the amplitude doesn't change. Then as you slowly approach the cut-off frequency, things start to decay a little. At the exact cut-off frequency the amplitude is reduced by three decibels. (Or even simpler than that, at the cut-off frequency the amplitude is reduced by twenty nine percent.) Now a little bit after you've hit the cut-off point, the filter keeps reducing the amplitude by twenty decibels per decade. That means that for every time the frequency increases by a factor of ten, the amplitude decreases by a factor of ten. Bode plots like this are a great way to predict how your filter will perform at a wide range of frequencies. In another video I will show you how to make them. (Search for "LTSpice tutorial") For now let's get some practice creating some more low pass filters. Let's say I want a low pass filter that goes before subwoofer's amplifier to make sure that the woofer is only outputting really low bass frequencies. Let's say I want a cut off frequency of about 150Hz. Here's the equation again to calculate the cut-off frequency. Now I can play around with random values of R and C to get the cutoff value of 150Hz... But from experience that I know that I should choose R, and then let the equation determine the value of C. I'm going to choose a 1 kilohm resistor because I don't want to overload the audio source. In this case it's an MP3 player. So here I've rearranged the equation to give me a way of calculating the capacitance needed for the filter based on a chosen resistance value. And it turns out that the value I need is 1.06 microfarads. Let's round that down to 1 microfarad. So here's the 150Hz low-pass filter as a schematic, and here it is on the breadboard. Now I'm going to play two sound clips. The first one has no filtering. The second one has been filtered by our low pass filter and amplified a little. You can hear that the bass is clearly there, but most of the rest of the music is missing. Okay that was an audio example but I want to show you how low-pass filters can be used with much more than just sine waves. Let's say you've got a microcontroller that is outputting a pulse width modulated square wave. Now let's say you want to convert that to a smooth continuous voltage that is an average of the square wave's voltage. Essentially we want to filter out all high frequencies and be left with D.C. So let's take our handy cutoff equation and make the cut-off a very low value of 1Hz. I'm going to choose a resistor value of 100 ohms so that this filter can still power something useful like an LED. You should always think about what resistor value will work best with your source signal and your chosen load. So with R = 100 ohms, and reusing the same equation, we get a capacitance of 1592 microfarads. Some extra capacitance won't hurt in this case because we want as much filtering as possible so let's round that up to 2000 microfarads. Now let's see how this heavy filtering affects a square wave. Okay here I have a 54Hz PWM signal and average voltage is 4.24 volts. At the output of the filter it looks like we've got almost pure D.C. with an average voltage of 4.16 volts. That's pretty close! Now I've lowered the duty cycle of the signal and on the input the average voltage is 1.68 volts. After the filter it looks like we're very close to clean D.C. with an average voltage of 1.44 volts. Again that's pretty close and as you have probably figured out, real world filters aren't 100% efficient so you'll always lose a little energy as heat. Now you might be wondering... can you put filters back to back in series for an even heavier filtering effect? The answer is yes! However since each filter puts a load on the previous filter, the math behind calculating the cut-off frequencies gets really complicated. That's the point where I bust out a circuit simulator like LTSpice and let it do the hard work for me. I'll make a tutorial on filter simulation with LTSpice in another video. (Search "LTSpice tutorial") In the meantime check out my video on passive RC high-pass filters. The good news is they are almost exactly the same as low-pass filters!
Applications
The network is used when it is required that two or more signals are matched to each other on some form of timing criterion. Delay is added to all other signals so that the total delay is matched to the signal which already has the longest delay. In television broadcasting, for instance, it is desirable that the timing of the television waveform synchronisation pulses from different sources are aligned as they reach studio control rooms or network switching centres. This ensures that cuts between sources do not result in disruption at the receivers. Another application occurs when stereophonic sound is connected by landline, for instance from an outside broadcast to the studio centre. It is important that delay is equalised between the two stereo channels as a difference will destroy the stereo image. When the landlines are long and the two channels arrive by substantially different routes it can require many filter sections to fully equalise the delay.
Operation
The operation is best explained in terms of the phase shift the network introduces. At low frequencies L is low impedance and C' is high impedance and consequently the signal passes through the network with no shift in phase. As the frequency increases, the phase shift gradually increases, until at some frequency, ω0, the shunt branch of the circuit, L'C', goes in to resonance and causes the centre-tap of L to be short-circuited to ground. Transformer action between the two halves of L, which had been steadily becoming more significant as the frequency increased, now becomes dominant. The winding of the coil is such that the secondary winding produces an inverted voltage to the primary. That is, at resonance the phase shift is now 180°. As the frequency continues to increase, the phase delay also continues to increase and the input and output start to come back into phase as a whole cycle delay is approached. At high frequencies L and L' approach open-circuit and C approaches short-circuit and the phase delay tends to level out at 360°.
The relationship between phase shift (φ) and time delay (TD) with angular frequency (ω) is given by the simple relation,
It is required that TD is constant at all frequencies over the band of operation. φ must, therefore, be kept linearly proportional to ω. With a suitable choice of parameters, the network phase shift can be made linear up to about 180° phase shift.
The network is terminated in a characteristic impedance (not shown in the circuit diagram), ideally a resistance R, which is the input impedance to the successive circuit or transmission line.
Design
The four component values of the network provide four degrees of freedom in the design. It is required from image theory (see Zobel network) that the L/C branch and the L'/C' branch are the dual of each other (ignoring the transformer action) which provides two parameters for calculating component values. These are
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Equivalently, every transmission pole, sp in the s-domain left half-plane must have a matching zero, sz in the right half-plane such that sp=−sz.[1] A third parameter is set by choosing a resonant frequency, this is set to (at least) the maximum frequency the network is required to operate at.
There is one remaining degree of freedom that the designer can use to maximally linearise the phase/frequency response. This parameter is usually stated as the L/C ratio. As stated above, it is not practical to linearise the phase response above 180°, i.e. half a cycle, so once a maximum frequency of operation, fm is chosen, this sets the maximum delay that can be designed in to the circuit and is given by,
For broadcast sound purposes, 15 kHz is often chosen as the maximum usable frequency on landlines. A delay equaliser designed to this specification can, therefore, insert a delay of 33μs. In reality, the differential delay that might be required to equalise may be many hundreds of microseconds. A chain of many sections in tandem will be required. For television purposes, a maximum frequency of 6 MHz might be chosen, which corresponds to a delay of 83ns. Again, many sections may be required to fully equalise. In general, much greater attention is paid to the routing and exact length of television cables because many more equaliser sections are required to remove the same delay difference as compared to audio.
Superconductor planar implementation
Losses in the circuit cause the maximum delay to be reduced, a problem that can be ameliorated with the use of high-temperature superconductors. Such a circuit has been realised as a lumped-element planar implementation in thin-film using microstrip technology. The traces are the superconductor yttrium barium copper oxide and the substrate is lanthanum aluminate. The circuit is for use in the microwave band and has a centre frequency of approximately 2.8 GHz and achieves a peak group delay of 0.7 ns. The device operates at a temperature of 77 K. The layout of the components corresponds to the layout shown in the circuit diagram at the head of this article, except that the relative positions of L' and C' have been interchanged so that C' can be implemented as a capacitance to ground. One plate of this capacitor is the ground plane and it thus has a much simpler pattern (a simple rectangle) than the pattern of C which needs to be a series capacitor in the main transmission line.[2]
See also
References
Cited references
- H. J. Chaloupka, S. Kolesov, "Design of lumped-element 2D RF devices", H. Weinstock, Martin Nisenoff (eds), Microwave Superconductivity, Springer, 2012 ISBN 9401004501.
General references
- Jay C. Adrick, "Analog television transmitters", in, Edmund A. Williams (editor-in-chief), National Association of Broadcasters Engineering Handbook, 10th edition, pp. 1483-1484, Taylor & Francis, 2013 ISBN 1136034102.
- Phillip R. Geffe, "LC filter design", in, John Taylor, Qiuting Huang (eds), CRC Handbook of Electrical Filters, pp. 76-77, CRC Press, 1997 ISBN 0849389518.
This page was last edited on 28 April 2023, at 01:11