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: a drain that pulls that harmonic current out of the network. But the filter's inductance also adds to the source inductance, and together they create a new parallel resonance — always below the order you tuned for.
The idea
Put an inductor and a capacitor in series and their reactances fight each other: the inductor's grows with frequency, the capacitor's shrinks. At one frequency they are equal and opposite and cancel exactly, leaving only the small resistance of the winding. That frequency is the branch's series resonance, and at it 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, and that path is now the filter, not the network. The current still exists (the drive still draws it), but it circulates locally between load and filter instead of flowing out into the system and distorting 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 supplies useful reactive power at the fundamental as a bonus. The tuning order then fixes the reactor: exactly the size that cancels the capacitor at that order. Size and tuning are independent choices.
Now for the effect that is easy to miss. That capacitor is still a capacitor at every other frequency, and it still sits across an inductive network, the same tuned-circuit pair as in resonance. But it now faces the source inductance plus the filter's own reactor, in series, and a larger inductance always means a lower resonant frequency. The parallel resonance the filter creates therefore always lands below the order the filter drains. Tune to the 5th and the peak appears near the 4th. Tune to the 7th and it appears near the 6th — uncomfortably close to the 5th.
So a filter never simply removes a problem. It creates a low-impedance drain at one order and a high-impedance amplifier at a lower one, and the worth of the design depends entirely on what the loads inject at that lower order. Filter design is a frequency-scan exercise: you check the whole curve, not just the notch you asked for.
Try it
Move the tuning order and watch two things at once: the notch follows the slider exactly, and the peak stays below it. Push 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 slides down, possibly onto 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. A real 5th filter is tuned to 4.7, not 5.0. Capacitors lose capacitance as they age and the notch drifts upward. A low tuning absorbs that drift, and it pushes the parallel peak further from the order below.
- A sharp filter is not automatically a better filter. A higher quality factor deepens the notch and makes the parallel peak taller and narrower. A low-Q filter drains less current but leaves a lower peak. On a bus with uncertain load damping, that is often the right trade.
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 for the order and you have the tuning rule. Size the reactor to put the notch where you want it:
the tuning order
n = √( XCXL ) · XL = XCn²
The branch resistance sets the depth of the notch, and it comes from the reactor's quality factor 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, 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, the tuning order by XL alone, and the larger denominator gives the smaller square root. So npar sits below n for any source reactance above zero. No filter design escapes this.
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 collapses to R and the numerator goes very small. At the parallel order the two reactive parts cancel in the denominator instead, and the impedance rises to a peak.
Two teaching approximations sit behind this page. The source is a pure inductance with no resistance and no load damping, so the peak is as tall as the arithmetic permits; a real bus gives a flatter one. And every reactance is given at the fundamental and scaled by h or 1/h, which ignores how the impedance of cables and transformers really changes with frequency.
See it in Phasor
Phasor models a filter as an ordinary shunt branch with its R, L and C values, so the frequency scan shows both the notch and the parallel peak with no special handling. Scan the bus before you add the filter, add it, and scan again — the second curve tells you whether the design is finished.