Passive harmonic filters
The key idea
A single-tuned filter is an inductor and a capacitor in series on the bus. At one frequency their reactances cancel and the branch becomes a very low impedance. The branch is a drain that pulls that harmonic current out of the network. But the filter's inductance also adds to the source inductance. Together they create a new parallel resonance. That resonance is always below the order that you tuned for.
The idea
Put an inductor and a capacitor in series and their reactances oppose each other. The reactance of the inductor increases with frequency. The reactance of the capacitor decreases with frequency. At one frequency the two are equal and opposite, and they cancel exactly. Only the small resistance of the winding remains. That frequency is the branch's series resonance, and at that frequency the branch is close to a short circuit.
Connect such a branch to a bus and you have a single-tuned filter. Harmonic current at that order takes the path of least impedance. That path is now the filter, not the network. The current still exists, because the drive still draws it. But the current circulates locally between the load and the filter. It does not flow out into the system and distort the voltage at every bus.
Two numbers set the design. The filter size is the Mvar rating, and it fixes the capacitor. A larger filter has a smaller capacitive reactance, and the bank also supplies useful reactive power at the fundamental. The tuning order then fixes the reactor. The reactor must be the exact size that cancels the capacitor at that order. Size and tuning are independent choices.
The next effect is easy to miss. That capacitor is still a capacitor at every other frequency, and it still sits across an inductive network. This is the same tuned-circuit pair as in resonance. But the capacitor now faces the source inductance plus the filter's own reactor, and the two are in series. A larger inductance always gives a lower resonant frequency. The parallel resonance that the filter creates is therefore always below the order that the filter drains.
Tune the filter to the 5th and the peak appears near the 4th. Tune it to the 7th and the peak appears near the 6th, which is close to the 5th.
A filter therefore never simply removes a problem. It creates a low-impedance drain at one order. At a lower order it creates a high impedance that amplifies the current there. The value of the design depends entirely on what the loads inject at that lower order. Filter design is therefore a frequency-scan exercise. You check the whole curve, not only the notch that you asked for.
Try it
Move the tuning order and watch two things at the same time. The notch follows the slider exactly, and the peak stays below it. Move the tuning above 5.5 and the peak arrives at the 5th.
Series notch
h = 4.7
|Z| = 0.070 pu — a drain
Parallel resonance
h = 4.25
peak 2.59 pu — an amplifier
Bus impedance at the 5th
0.178 pu
0.50 pu with no filter
The model works in per unit, with the grid held at Xs = 0.10 pu. It puts a pure-inductance source in parallel with one series R-L-C branch and scans it order by order. The notch always sits at the tuning order, and the peak always sits below it. For this reason, engineers tune a real filter a little below the harmonic that it targets. Component tolerance and capacitor aging move the notch upward with time. A low tuning also keeps the parallel peak further from the order below.
Why it matters
- A filter is a system change, not a local fix. The notch is a property of the branch. The peak is a property of the branch and the grid behind it. Move the same filter to a weaker bus and the peak moves down. It can then arrive at an order that was safe before.
- Multiple filters interact. A 5th filter and a 7th filter on the same bus do not keep their separate scans. The capacitors add, the peaks move, and new peaks appear between the branches. Each new filter needs a new study.
- Designers detune filters on purpose. Engineers set a real single-tuned filter a few percent below its target order. A 5th filter is tuned to 4.7, not to 5.0. Capacitors lose capacitance as they age, and the notch moves upward. A low tuning absorbs that drift, and it also moves the parallel peak further from the order below.
- A sharp filter is not automatically a better filter. A higher quality factor makes the notch deeper. It also makes the parallel peak taller and narrower. A low-Q filter drains less current, but it also gives a lower peak. On a bus with uncertain load damping, that trade is often the correct one.
The math, if you want itOptional — the page reads completely without it
The branch resonates in series where the two reactances are equal in magnitude:
the series-resonance condition
h · XL = XCh
Solve it for the order and you have the tuning rule. You size the reactor to put the notch at the order that you want:
the tuning order
n = √( XCXL ) · XL = XCn²
The branch resistance sets the depth of the notch. That resistance comes from the quality factor of the reactor at the tuning frequency:
branch resistance from the quality factor
R = n · XLQ
The parallel resonance is the capacitor against all the inductance in the loop. That inductance is the source reactance and the filter's own reactor in series:
the new parallel-resonance order
npar = √( XCXs + XL )
Compare the two denominators. The parallel order divides by Xs + XL, but the tuning order divides by XL alone. The larger denominator gives the smaller square root. Therefore npar is below n for any source reactance above zero. No filter design avoids this result.
At each order the harmonic current source sees the two paths in parallel. The source path is Zs = j·h·Xs, and the filter path is Zf = R + j(h·XL − XC/h):
bus impedance at order h
|Zbus| = |Zs| · |Zf||Zs + Zf|
At the tuning order Zf falls to R, and the numerator becomes very small. At the parallel order the two reactive parts cancel in the denominator instead. The impedance then rises to a peak.
This page makes two teaching approximations. First, it treats the source as a pure inductance with no resistance and no load damping. The peak is therefore as tall as the arithmetic permits, and a real bus gives a flatter peak. Second, it gives every reactance at the fundamental and scales it by h or 1/h. This scaling ignores how the impedance of cables and transformers changes with frequency.
See it in Phasor
Phasor models a filter as an ordinary shunt branch with its R, L and C values. The frequency scan therefore shows both the notch and the parallel peak with no special handling. Scan the bus before you add the filter. Add the filter and scan again. The second curve tells you whether the design is finished.