Outline

Prototype series and parallel resonant circuits- Near resonance, a microwave resonator can usually be modeled by either a series or parallel RLC lumped-element equivalent circuit.

Transmission line resonators- Resonators composed of half- or quarter-wavelength transmission line.

Excitation of resonators

Series and parallel resonant circuitsSeries Resonant Circuit

The input impedance of a series resonant circuit is

The complex power delivered to the resonator is

Series and parallel resonant circuits

The power dissipated by the resistor is

The average magnetic energy stored in the inductor is

The average electric energy stored in the capacitor is

(6.3)

where Vc is the voltage across the capacitor.

The input impedance can be rewritten as

Series and parallel resonant circuits

Resonance occurs when the average stored magnetic and electric energies are equal, or Wm = We. (energy thus conserved)Then the input impedance at resonance is

which is a purely real impedance. With Wm = We , the resonant frequency must be defined as

Series and parallel resonant circuitsThe quality factor, or Q is defined as

which is a measure of the loss of a resonant circuit.

Lower loss implies a higher Q.

From (6.7) using (6.3), and the fact that Wm = We at resonance, we have

(6.8)

which shows that Q increases as R decreases.

average stored energy2energy loss in one period

π=

Series and parallel resonant circuitsInput impedance near resonanceInput impedance near resonance

The input impedance can first be rewritten from (6.1) as

sinceLet , where is small. Thus we have

The input impedance is therefore given by

6.9This form will be useful for identifying equivalent circuits with distributed element resonators.

ωωω ∆+= 0 ω∆

LC120 =ω

( )( ) ( ) ωωωωωωωωωωω ∆≅∆−∆=+−=− 220020

2

Series and parallel resonant circuitsHalfHalf--power fractional bandwidth of the resonatorpower fractional bandwidth of the resonator

When the frequency is such that , then the average (real) power delivered to the circuit is one-half that delivered at resonance. If BW is the fractional bandwidth, at the upper band edge. Then using (6.9) gives

6.11

22in 2RZ =

0 / 2BWω ω∆ =

21Re | |2in

in

VP RZ

=

Series and parallel resonant circuitsParallel Resonant Circuit (Anti-resonant circuit)

The input impedance of a parallel resonant circuit is

The complex power delivered to the resonator is

Series and parallel resonant circuits

The power dissipated by the resistor is

The average magnetic energy stored in the capacitor is

The average electric energy stored in the inductor is

(6.3)

where Vc is the voltage across the capacitor.

The input impedance can be rewritten as

Series and parallel resonant circuits

Resonance occurs when the average stored magnetic and electric energies are equal, or Wm = We. (energy thus conserved)The input impedance at resonance is

The resonant frequency is also defined as

The Q of the parallel resonant circuit can be expressed as

since Wm = We at resonance. The Q increases as R increases.

Series and parallel resonant circuitsInput impedance near resonanceInput impedance near resonance

Let , where is small. The input impedance can then be rewritten from (6.12) as

since

When R=∞, it reduces to

ωωω ∆+= 0 ω∆

LC120 =ω

1

0

0

1 1 (1 )

(1 )

oo

o

ωωω ω ω ω

ωωω

−∆= = +

+ ∆

∆≅ −

LC120 =ω

HalfHalf--power fractional bandwidth of the resonatorpower fractional bandwidth of the resonator

When the frequency is such that the average (real)power delivered to the circuit is one-half that delivered at resonance. This also implies that

6.21

Series and parallel resonant circuits

2*

2 2

1 1 | |Re ( )2 2

1 1| | | |2

in

in

V VP VR R

I ZR

= =

=

Series and parallel resonant circuitsLoaded and unloaded Q Loaded and unloaded Q

The Q previously defined is a characteristic of the resonant circuit itself.This Q is in the absence of any loading effects caused by external circuitry, and is called the unloaded Q.

In practice, a resonant circuit is invariably coupled to other circuitry, which will always have the effect of lowering the overall, or loaded Q, QL, of the circuit.

Series and parallel resonant circuitsIf the resonator is a series RLC circuit, the load resistor RL adds in series with R so that the effective resistance in (6.8) is R+ RL. If the resonator is a parallel RLC circuit, the load resistor RLcombines in parallel with R so that the effective resistance in is

.

If we define an external Q, Qe, as

6.22The loaded Q can be expressed as

6.23

( )LL RRRR +

Transmission Line Resonator

Ideal lumped elements are usually unattainable at microwave frequencies, so distributed elements are more commonly used. Here we consider transmission lines sections as resonator.Since the Q of these resonators is kind of interest, we must consider lossy transmission lines.

Short-circuited λ/2 lineOpen-circuited λ/4 line

Open-circuited λ/2 lineShort-circuited λ/4 line

series resonator

parallel resonator

Transmission Line Resonator

ShortShort--Circuited Circuited λλ/2 Line/2 LineSeries type of resonance can be achieved using a short-circuited transmission line of length λ/2.

At the frequency ω=ω0, the length of the line is l =λ/2, where λ=2π/β. The input impedance is thus from (2.91),

or

Observe that if α=0 (no loss).ljZZin βtan0=

Transmission Line Resonator1) In practice, most transmission lines have small loss, so we can

assume that , and so .

2) Now let , where is small, and assume a TEM line, we have

where vp is the phase velocity of the transmisssion line. Since for , we have

and then

1<<lα ll αα ≈tanh

ωωω ∆+= 0 ω∆

02 pl vλ π ω= = 0ω ω=

Thus

since is a higher order term.Equation (6.25) is of the form

which is the input impedance of a series RLC resonant circuit, as given by (6.9).

0 1lωα ω∆

Transmission Line Resonator

Transmission Line Resonator

The resistance of the equivalent circuit is

6.26aThe inductance of the equivalent circuit is

6.26bThe capacitance of the equivalent circuit is

6.26cThis resonator thus resonates for ∆ω=0, and its input impedance at this frequency is .

=

lZRZin α0==

Transmission Line ResonatorThe voltage distributions for the n = 1 and n = 2 resonant modes

The Q of this resonator can be found from (6.8) and (6.26) as

6.27since βl =π at the first resonance. This result shows that the Q decreases as the attenuation of the line increases.

Transmission Line ResonatorShort-Circuited λ/4 Line

Parallel type of resonance can be achieved using a short-circuited transmission line of length λ/4.

The input impedance of the shorted line of length λ/4 is

6.28

Transmission Line Resonator1) For small loss, 2) Assume that l = λ/4 at ω=ω0. and let .

For a TEM line,

so

Thus, the input impedance is

where is a higher order term.

ωωω ∆+= 0

ll αα ≅tanh

12/ 0 <<∆ ωωπαl

inZ

Transmission Line ResonatorThis result is of the same form as the impedance of a parallel RLC circuit.

Then we can identify

This resonator thus has a parallel type resonance for l =λ/4, with an input impedance at resonance of Zin=R=Z0/αl.The Q of this resonator is

6.31since l = π/2β at resonance.

=

Transmission Line ResonatorOpen-Circuited λ/2 Line

Parallel type of resonance can be also achieved using a open-circuited transmission line of length λ/2.

The input impedance of the opened line of length λ/2 is

Transmission Line Resonator1) For small loss,

2) Assume that l = λ/2 at ω=ω0. and let . For a TEM line,

so

Thus, the input impedance is

which is also of the form of the input impedance of a parallel resonant The Q is given by

ωωω ∆+= 0

ll αα ≅tanh

Transmission Line ResonatorVarious types of microstrip resonator

Transmission Line ResonatorEXAMPLEEXAMPLE

A microstrip resonator constructed from a λ/2 length of 50Ω open-circuited microstrip line. The substrate thickness is 0.159cm, with εr=2.08 and tanδ = 0.0004. The conductors are copper. Compute the length of the line for resonance at 5 GHz. Ignore fringing fields at the end of the line.

SolSolThe width of a 50Ω microstrip line on this substrate is W=0.508cm,The effective permittivity is εe=1.80Then the resonant length can be calculated as

This length should be shorten by 2∆l if the fringing fields must take into account.

2.24cm

Excitation of resonators

How the resonators can be coupled to external circuitry?How the resonators can be coupled to external circuitry?Electric couplingMagnetic couplingMixed coupling

Excitation of resonatorsCritical CouplingCritical Coupling

To obtain maximum power transfer between a resonator and a feedline, the resonator must be matched to the feed at the resonant frequency.

For example, the input impedance near resonance of the series resonant circuit is

6.71

and the unloaded Q is,

6.72

At resonance, ∆ω=0, the input impedance is Zin = R. Thus,

and the unloaded Q becomes

Excitation of resonatorsBut from (6.22), the external Q is

6.75which shows that the external and unloaded Q are equal under the condition of critical coupling.It is useful to define a coefficient of coupling, g ,as

which can he applied to both series ( g = Z0/R) and parallel ( g = R/Z0) resonant circuits.

1. g < 1 The resonator is undercoupled to the feedline (R>Z0).2. g = 1 The resonator is critically coupled to the feedline (R=Z0).3. g > 1 The resonator is said to be overcoupled to the feeclline (R<Z0).

Excitation of resonators

Smith Chart Representation

Excitation of resonatorsEquivalent circuit of a Gap-Coupled Microstrip Resonator

Now consider open-circuited microstrip resonator coupled to a microstrip feedline.

The gap in the microstrip line can be approximated as a series capacitor.

The normalized input impedance seen by the feedline is

where bc = ωZ0C is the normalized susceptance of the coupling capacitor, C.

Excitation of resonatorsAt resonance, Im z must equals to zero (We=Wm). Thus

6.85The transcendental equation are sketched in the figure. In practice, bc<< 1, so that the first resonant frequency, ω1, will be close to the frequency for which βl = π (the first resonant frequency of the unloaded resonator).

In this case the coupling of the feedline to the resonator has the effect of lowering its resonant frequency.

ω1

Excitation of resonatorsExpanding z(ω) in a Taylor series about the (unloaded) resonant frequency, ω1, and assuming that bc is small.

6.86Since z(ω1) =0, we have

where bc << 1 and . vp: the velocity of the transmission line (assumed TEM).The normalized impedance is then (for lossless resonator)

6.87

1/ωπ pv≅

z(ω)=

Excitation of resonatorsNow, include the losses for a high Q resonator by replacing the resonance frequency (pp. 268)

we have

An uncoupled λ/2 open-circuited transmission line resonator looks like a parallel RLC circuit near resonance, but the present case of a capacitive coupled λ/2 resonator looks like a series RLC circuit near resonance.

The series coupling capacitor is thus the so-called inverter.

1 1(1 )2

jQ

ω ω↔ +

12

1

( )( )2 c c

z jQb b

π ω ωπωω−

= +

2

inL

KZZ

=

Excitation of resonatorsAt resonant, the input resistance is .For critical coupling we must have R=Z0, or

6.82The coupling coefficient of (6.83) is

6.83< 1 and the resonator is undercoupled.

> 1 and the resonator is undercoupled.

2 20 / 2 1/c cR Z Qb bπ= ∝

Qbc 2π>

Qbc 2π<

(the gap width)cQ b C W→ → →∆