RLC Parallel Circuit: Anti-Resonance and Impedance Peak

A parallel RLC tank is the electrical dual of the series RLC circuit, and it shows up everywhere impedance matching or frequency selection is needed without series insertion loss: antenna matching networks, oscillator tank circuits, crystal equivalent circuits, and power-line filters. Unlike the series RLC, which presents minimum impedance at resonance, the parallel RLC presents maximum impedance — a fact that trips up engineers who reason about resonance only from the series case. Understanding why requires tracking currents rather than voltages, and recognizing that the circulating tank current can be far larger than what the source ever sees.

Circuit and Admittance Formulation

Consider an ideal resistor, inductor, and capacitor all connected across the same two nodes, driven by a current source I (current drive is the natural choice here, just as voltage drive is natural for series RLC).

        +----+----+----+
        |    |    |    |
       (I)   R    L    C
        |    |    |    |
        +----+----+----+
              (ref)

Because all three elements share the same voltage, it's easiest to work in admittance, Y = 1/Z:

Y(jω) = 1/R + 1/(jωL) + jωC
      = 1/R + j(ωC − 1/(ωL))

The total impedance is Z(jω) = 1/Y(jω). Note the sign pattern is the mirror image of the series RLC impedance Z = R + j(ωL − 1/(ωC)) — that's the series/parallel duality: R ↔ 1/R (or G), L ↔ C, voltage ↔ current.

Anti-Resonance Condition

Resonance in the parallel tank is defined as the frequency where the imaginary part of Y vanishes — the susceptance terms cancel:

ωC − 1/(ωL) = 0
ω₀ = 1/√(LC)

This is the same ω₀ formula as the series case (expected, since it comes purely from L and C), but the consequence is opposite. At ω₀, Y(jω₀) = 1/R (purely real, minimum magnitude), so:

Z(jω₀) = R

Impedance is purely resistive and at its maximum magnitude — this is why the phenomenon is called anti-resonance: the circuit behaves oppositely to the series case, where impedance is minimum and purely resistive at resonance. Away from ω₀, the reactive term adds magnitude to Y, so |Z| only decreases.

At frequencies well below ω₀, the inductor's low reactance dominates admittance (Y is large, dominated by 1/(jωL)), so Z is small and inductive. Well above ω₀, the capacitor dominates admittance, and Z is small and capacitive. The impedance peak at ω₀ is a narrow window between these two low-impedance regions.

Quality Factor and Bandwidth

For the parallel RLC, Q is defined from energy stored versus energy dissipated per cycle, same definition as series RLC, but the formula inverts because R plays the opposite role (a large parallel R means low loss, unlike series R where large R means high loss):

Q = R / (ω₀L) = ω₀RC = R·√(C/L)

Bandwidth (the −3 dB width of the impedance peak, in the same sense as the series circuit's current peak) is:

BW = ω₀ / Q  (rad/s)     or     f_BW = f₀ / Q  (Hz)

High Q means a tall, narrow impedance peak — R is large relative to the reactances, so little energy leaks out per cycle. Low Q means a broad, shallow peak — R loads the tank heavily.

Worked Example

Take R = 10 kΩ, L = 100 µH, C = 100 pF, driven by an ideal current source.

Step 1 — resonant frequency:

ω₀ = 1/√(LC) = 1/√(100×10⁻⁶ × 100×10⁻¹²)
   = 1/√(1×10⁻¹⁴) = 1/(1×10⁻⁷) = 1×10⁷ rad/s
f₀ = ω₀/(2π) ≈ 1.592 MHz

Step 2 — Q:

Q = R·√(C/L) = 10,000 · √(100×10⁻¹² / 100×10⁻⁶)
  = 10,000 · √(1×10⁻⁶) = 10,000 · 1×10⁻³ = 10

Step 3 — bandwidth:

f_BW = f₀/Q = 1.592 MHz / 10 ≈ 159.2 kHz

Step 4 — peak impedance and voltage: at ω₀, Z = R = 10 kΩ (purely resistive). For a 1 mA (rms) drive current, the voltage across the tank at resonance is:

V = I·Z = 1 mA × 10 kΩ = 10 V (rms)

Check — branch currents at resonance: with V = 10 V across each element,

I_R = V/R = 10 V / 10 kΩ = 1 mA
I_L = V/(ω₀L) = 10 V / (1×10⁷ × 100×10⁻⁶) = 10 V / 1000 Ω = 10 mA
I_C = V·ω₀C = 10 V × 1×10⁷ × 100×10⁻¹² = 10 V × 1×10⁻³ = 10 mA

I_L and I_C are equal in magnitude (10 mA) and 180° out of phase, so they cancel exactly in KCL at the top node, leaving only I_R = 1 mA to match the source. This confirms the circuit: the source only needs to supply 1 mA even though 10 mA is circulating between L and C — a tenfold current magnification inside the tank, consistent with Q = 10. This circulating current is exactly what makes parallel resonance useful for tank circuits in oscillators: energy sloshes between L and C internally while only replenishing resistive losses from the source.

Practical Implications

• Impedance peaking, not current peaking: unlike series RLC where current spikes at resonance, the parallel tank's impedance spikes — current from the source is minimized at ω₀, while internal circulating current is maximized. Confusing the two leads to sizing errors in tank inductors/capacitors that must handle the true circulating current, not the terminal current.
• Real inductors bring their own series resistance, which converts the "ideal" pure parallel model into a more complex network; the effective parallel R at resonance is often expressed via the inductor's own Q (R_p ≈ ω₀L·Q_L), a common shortcut in RF matching network design.
• Oscillator tank circuits rely on this impedance peak to set the oscillation frequency and provide the necessary phase shift and gain-limiting resistance at ω₀.
• Notch/trap filters use parallel RLC branches to reject a specific frequency by presenting a high impedance in series with the signal path (blocking that frequency) while passing others.
• Crystal and ceramic resonator models are parallel RLC (plus a shunt capacitance) — anti-resonance defines the crystal's parallel resonant frequency, distinct from its series resonant frequency, a distinction critical to correct oscillator circuit design.
• Q and R are directly linked — in a parallel tank, adding a loading resistor in parallel lowers R and therefore lowers Q (opposite of the series case, where adding series R lowers Q by increasing loss). This is why lightly loaded tank circuits (high R_load) preserve high Q, while heavily loaded ones damp the resonance.

Key Takeaways

• Parallel RLC resonance (anti-resonance) occurs at the same ω₀ = 1/√(LC) as series resonance, but produces a maximum impedance (= R) instead of a minimum.
• At ω₀, the inductor and capacitor admittances (susceptances) cancel, leaving a purely resistive, maximum-magnitude impedance.
• Q = R·√(C/L) = R/(ω₀L) = ω₀RC — larger parallel R means higher Q, the inverse relationship from series RLC.
• Bandwidth follows the same ω₀/Q relationship as the series case, defining the −3 dB width of the impedance peak.
• Branch currents in L and C can be Q times larger than the source current at resonance — a circulating current that must be accounted for in component ratings, even though the source sees minimum current.
• This behavior underlies tank circuits in oscillators, notch filters, and crystal resonator models, where the distinction between series and parallel resonant frequencies is functionally important.