How much insertion loss does a critically coupled one-port resonator absorb at resonance?

A critically coupled one-port resonator absorbs essentially all incident power at the resonant frequency. Since the input is matched, the reflection coefficient is zero and the return loss is theoretically infinite (in practice 25 to 40 dB limited by tuning accuracy and conductor loss). All the power that is not reflected is dissipated in the resonator's loss, so a critically coupled cavity behaves as a near-perfect absorber at f0. This is the desired condition for reflection-type measurements and for matched terminations built from lossy resonators, but it is the opposite of what a low-loss bandpass filter needs, where each resonator is deliberately overcoupled to keep dissipation low.

Passive Components

Critically Coupled

Q: How do you tell from a Smith chart whether a resonator is critically coupled?

On a Smith chart referenced to the system impedance, plot the input reflection coefficient as frequency sweeps through resonance. A critically coupled resonator traces a loop that passes exactly through the center of the chart, meaning the input impedance at resonance equals the characteristic impedance and the reflection coefficient drops to zero. An undercoupled resonator traces a small loop entirely within the right (high-impedance) half and never reaches the center, while an overcoupled resonator traces a loop that encircles the center. The size and position of the loop directly indicate the coupling coefficient: a loop diameter equal to the full chart radius corresponds to k=1, that is, critical coupling.

Q: What is the relationship between loaded Q and unloaded Q at critical coupling?

At critical coupling the external Q set by the feed equals the unloaded Q of the resonator, so Qext = Qu. Because 1/QL = 1/Qu + 1/Qext, the loaded Q is exactly half the unloaded Q, QL = Qu/2. For example, a cavity resonator with an unloaded Q of 10,000 that is critically coupled to a 50 ohm line presents a loaded Q of 5,000. This halving is why a one-port reflection measurement of a critically coupled resonator shows a 3 dB bandwidth twice as wide as the intrinsic, undriven resonance, and it is the basis of the standard reflection method for extracting unloaded Q from a single critically coupled port.

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Describing the coupling condition where the external Q of a feed network is made equal to the unloaded Q of a resonator, so the coupling coefficient k equals 1. At this point the resonator input is impedance-matched to the source at resonance, the reflection coefficient drops to zero, and power transfer into the resonator is maximized. The loaded Q is then exactly half the unloaded Q (Q_L = Q_u/2), and the resonant response is single-peaked and dip-free, sitting precisely on the boundary between the undercoupled and overcoupled regimes. Critical coupling is the reference condition for reflection-method Q measurement and for matched lossy terminations.

Coupling coefficient: k = 1

Loaded Q: Q_u / 2

Return loss at f₀: > 25 dB (ideal ∞)

The Boundary Between Undercoupling and Overcoupling

Every resonator that is connected to a source dissipates energy two ways: internally, through conductor and dielectric loss described by its unloaded Q, and externally, through the coupling structure that delivers energy back into the feed line, described by the external Q. The coupling coefficient k is simply the ratio of these two loss rates, k = Q_u/Q_ext. When the external loading exactly balances the internal loss, the two Q values are equal, k = 1, and the resonator is critically coupled. This is the single operating point at which a one-port resonator presents a perfect match to the source at its resonant frequency and absorbs all of the incident power.

Below this point the resonator is undercoupled (k < 1): the coupling structure is too weak, most of the incident power is reflected, and the input impedance at resonance is higher than the line impedance. Above it the resonator is overcoupled (k > 1): the coupling is too strong, the input impedance at resonance falls below the line impedance, and again some power is reflected. Only at the critical point does the on-resonance input resistance equal the system impedance. For a series-fed resonator this is the condition R_series = Z₀; for a shunt-fed resonator it is the corresponding conductance match.

The practical consequence is that critical coupling is not universally desirable. It is exactly right for a reflection-type Q measurement, where a clean match lets the analyzer extract the unloaded Q from the loop diameter on a Smith chart, and for absorptive terminations. It is exactly wrong for a low-loss bandpass filter, where each resonator is deliberately overcoupled so that very little of the signal energy is dissipated as it passes through the network.

Governing Relationships

Coupling coefficient (one-port):
k = Q_u / Q_ext  →  critical coupling when k = 1

Loaded Q at critical coupling:
1/Q_L = 1/Q_u + 1/Q_ext  and  Q_ext = Q_u → Q_L = Q_u / 2

Reflection coefficient at resonance:
Γ₀ = (k − 1) / (k + 1)  →  Γ₀ = 0 when k = 1

Where Q_u = unloaded Q, Q_ext = external Q, Q_L = loaded Q, k = coupling coefficient. Example: a cavity with Q_u ≈ 10,000 critically coupled to a 50 Ω line presents Q_L ≈ 5,000 and Γ₀ ≈ 0.

Coupling Regime Comparison

Condition	Coupling k	Q_ext vs Q_u	Γ at f₀	Smith-chart loop	Best use
Undercoupled	k < 1	Q_ext > Q_u	Γ > 0 (positive)	Inside center, no encirclement	High-Q sensing, weak probes
Critically coupled	k = 1	Q_ext = Q_u	Γ = 0 (matched)	Passes through chart center	Reflection Q measurement, absorbers
Overcoupled	k > 1	Q_ext < Q_u	Γ < 0 (encircles)	Loop encircles chart center	Low-loss bandpass filters

Common Questions

Frequently Asked Questions

How do you tell from a Smith chart whether a resonator is critically coupled?

Sweep the input reflection coefficient through resonance and watch the loop it traces. A critically coupled resonator passes exactly through the center of the chart, meaning the input impedance equals Z₀ and Γ = 0. An undercoupled resonator traces a small loop entirely within the high-impedance half and never reaches center; an overcoupled resonator traces a loop that encircles the center. A loop diameter equal to the full chart radius corresponds to k = 1.

What is the relationship between loaded Q and unloaded Q at critical coupling?

At critical coupling Q_ext = Q_u. Because 1/Q_L = 1/Q_u + 1/Q_ext, the loaded Q is exactly half the unloaded Q: Q_L = Q_u/2. A cavity with Q_u = 10,000 critically coupled to a 50 Ω line shows Q_L = 5,000, so its measured 3 dB bandwidth is twice the intrinsic width. This is the basis of the standard one-port reflection method for extracting unloaded Q.

How much power does a critically coupled one-port resonator absorb at resonance?

Essentially all of it. With the input matched, Γ = 0 and return loss is theoretically infinite (25 to 40 dB in practice, limited by tuning and conductor loss). All non-reflected power is dissipated in the resonator's internal loss, so a critically coupled cavity acts as a near-perfect absorber at f₀. That is ideal for reflection measurements and matched terminations, but the opposite of what a low-loss filter needs, where resonators are deliberately overcoupled.

Related Terms

← Critical Temperature CRLH →