Microwave Resonators April 2011 Mark W. Ingalls
Lumped-Element Resonators Series RLC Resonator A series RLC network, fig. 3.1, has the input impedance:
R
BW
.707
Eq. 3. 1
Z in = R series + jω L − j
ωC
Re{Z}
.
f0
Im{Z}
where ω = 2πf is the angular frequency.
|Z|
The real power dissipated in the resonator is proportional to the magnitude1 of the current squared:
R
L
C
Eq. 3. 2 Fig. 3. 1. Frequency response of a series RLC circuit.
2
Pdiss = I R
Similarly, there will be a quantity associated with the inductor and the capacitor with the dimensions of watts, called reactive power: Eq. 3. 3
1 2 Preact = I ω L − ω C This quantity is “imaginary” so it corresponds to the rate that energy is stored and then released per full cycle. In other words, each reactive element stores energy for one halfcycle and releases that energy for one half-cycle. To make this concept explicit, we can separate the reactive power into two half-cycles: Eq. 3. 4
Preact =
ω
1 ω 2 1 2 I L − 2 − I 2 − L 2 ω C 2 ω C 14 442444 3 14442444 3 first half −cycle
sec ond half − cycle
The energy stored in an inductor is: Eq. 3. 5
1 2 L I = WM 2 and the energy stored in a capacitor is:
1
RMS, not peak, voltage and current values are used throughout this discussion.
Microwave Resonators
page 3.1
Eq. 3. 6
1 1 1 2 I = CV 2 2ω C 2
2
= WE
So that on average the total energy stored in the resonator is the sum of stored magnetic and electric energy: Eq. 3. 7
Wave = WM + W E while the instantaneous rate of change in stored (or released) energy is the difference in stored energy plus the difference in released energy per cycle: Eq. 3. 8
Preact = 2 ω (WM − WE ) In other words, reactive power is the net rate of change of stored and released energy per cycle. An important figure of merit for resonators is the quality factor (Q), defined as the ratio of energy stored to power dissipated per cycle: Eq. 3. 9
Q =ω
W M + WE Pdiss
At resonance, the magnetic and electric energies are precisely equal so that: Eq. 3. 10
Q = ω0
2WM 2WE ω L 1 = ω0 = 0 = Pdiss Pdiss Rseries ω 0 RseriesC
where ω0 = 2πf0 is the angular resonant frequency. We can rearrange eq. 3.1 so that: Eq. 3. 11
1 Z in = R + jω L1 − 2 , ω LC which in the “neighbor hood” of ω0 is: Eq. 3. 12
ω 02 Z in = R + jω L1 − 2 ω At a frequency slightly higher than resonance (ω = ω0 + ∆ω), the term in brackets is approximately:
Microwave Resonators
page 3.2
Eq. 3. 13
ω 2 − ω 02 (ω − ω 0 )(ω + ω 0 ) = =K ω2 ω2 (ω − (ω − ∆ω ))(ω + (ω − ∆ω )) = K ω2 ∆ω (2ω − ∆ω ) 2∆ω ≅ ω ω2 …if ∆ω is small. So, the input impedance of a series RLC resonator just above resonance is approximately: Eq. 3. 14
Z in (ω 0 + ∆ω ) ≅ Rser + 2 j L∆ω This result will be useful when trying to model other types of microwave resonators with an equivalent lumped-element network.
Parallel GLC Resonator
G
The parallel combination of resistor, inductor and capacitor is the dual of the series RLC resonator. Its input impedance is:
C
L
R ×.707
Eq. 3. 15
Z in = G +
1 j ωC − ωL
−1
Re{Z}
BW
where
Im{Z} |Z|
Eq. 3. 16
G=
1 R parallel
is the conductance, in siemans, of a resistor in parallel with a capacitor and inductor, fig. 3.2.
Fig. 3. 2. Frequency response of a parallel GLC circuit, transformed to an impedance.
Invoking the duality principle, we may directly write: Eq. 3. 17
Q par =
ωC G
=
R par 1 = ωCR par = ωLG ωL
Eq. 3. 18
1 Z in (ω 0 + ∆ω ) = + 2 jC∆ω R par
−1
Q and Bandwith We will derive the approximate relationship between Q-factor and bandwidth of a series RLC resonator as an example. The relationship holds for all types of resonators, but we will not derive the equation for all other resonators.
Microwave Resonators
page 3.3
By KVL, we may write: Eq. 3. 19
VR = V
R ser R ser + j ( ωL − 1
ωC
for the series RLC resonator. At resonance, the reactive portion of Zin vanishes, so that there is no net voltage drop across the capacitor / inductor combination. Thus, maximum power is delivered to the resonator at ω0. Off resonance, the power delivered to the resonator will fall to half the maximum level when |VR|2 is V/2: Eq. 3. 20
R ser
=
1 2
R ser + j ( ω half L − 1
ω half C
We now square both sides, taking advantage of the magnitude operation in the denominator, and redistribute R: Eq. 3. 21
ω L 1 = half − 1 ω R ser C half R ser
2
The positive and negative roots will result in two half-power frequencies. We make the following substitution: Eq. 3. 22
ω ω 1 = half Q − 0 Q ω half ω0
2
and rearrange to express Q solely in terms of frequency: Eq. 3. 23
Q=
ω 0ω half ω 0ω half = 2 2 ω half − ω 0 (ω half − ω 0 )(ω half + ω 0 )
Now, let: Eq. 3. 24
ω half = ω 0 +
∆ω 2
∆ω , 2 ∆ω ω1 − ω 0 = , 2 ∆ω and ω 1 + ω 0 = 2ω 0 + 2
so that
ω 0ω half = ω 0 ω 0 +
The result is: Eq. 3. 25
Microwave Resonators
page 3.4
Q=
( ∆ω (ω
)≅ω + ∆ω ) ∆ω 4
ω 0 ω 0 + ∆ω 2 0
0
if ∆ω is small.
By a similar procedure, the parallel RLC Q-factor can also be expressed as the ratio of resonant frequency to bandwidth.
Transmission Line Resonators Quarter-wavelength short-circuit terminated line We know from the study of transmission lines that a quarter-wavelength transmission line terminated in a short circuit acts like a parallel GLC resonator at frequencies near resonance. We may write the input impedance as: Eq. 3. 26
Z in = Z 0 (α l + j tan(β l )) because the low loss assumption is valid for transmission line resonators. Often, we would like to express the input impedance in terms of an equivalent GLC network in order to understand how such a resonator can be used in a circuit. Thus, we want to obtain an equivalent Rparallel,, L, and C for the transmission line resonator. We write for βl: Eq. 3. 27
βl=
π ω 2 ω0
because the resonator is a quarter-wavelength long at ω0 and its wavelength is a linear function of frequency. Slightly above resonance, we expect the resonator to act like a lumped equivalent capacitance. Using a similar procedure to the one used on the parallel GLC circuit, we write: Eq. 3. 28
π ω π (ω 0 + ∆ω ) π π ∆ω = = + 2 ω0 2 ω0 2 2 ω0 By trigonometric identity and the small argument formula, we write: Eq. 3. 29
π ∆ω π ∆ω π 2 ω0 = − cot ≅ tan + 2 2 ω0 2 ω0 π ∆ω Thus, at frequencies just above the quarter-wave resonance, we may simplify eq. 3.26 to read: Eq. 3. 30
α l 1 π ∆ω Z in (ω 0 + ∆ω ) = +j Z0 2 ω 0 Z0
Microwave Resonators
−1
page 3.5
Comparing eq’s 3.18 and 3.30 we see that for the quarter-wavelength short-circuited resonator: Eq. 3. 31
R par =
Z0 αl
and Eq. 3. 32
1 π ∆ω Z0 2 ω 0
2C∆ω = Solving for C we find: Eq. 3. 33
C eq =
π 4ω 0 Z 0
Making use of the identity Eq. 3. 34
ω0 =
1 LeqC eq
we find: Eq. 3. 35
L eq =
4 Z0 π ω0
So, we see how in the neighborhood of its resonant frequency, the shorted λ/4 resonator behaves liked an equivalent lumped GLC network.
Microwave Resonators
page 3.6
Half-wavelength open-circuit terminated line We know from our study of transmission lines and the Smith chart that a half-wavelength open-circuited transmission line is just the inverse, or dual, of the short-circuited line. Thus, we may directly write: Eq. 3. 36
1 Z in = (α l + j tan(β l )) Z0
−1
Eq. 3. 37
Z in (ω 0 + ∆ω ) =
Z0
αl+ j
∆ωπ
ω0
Eq. 3. 38
R par =
Z0 αl
Eq. 3. 39
Ceq =
1
π
Z 0ω 0 2
Eq. 3. 40
Leq =
2 Z0
π ω0
Microwave Resonators
page 3.7
Waveguide Resonators Parallel Plate Waveguide Consider the parallel plate structure depicted in fig. 3.3. It has two conducting plates separated by a dielectric spacer. If we uniformly excite this structure along its left edge so that l1 is in the direction of propagation, it will act like an mλ/2 transmission line terminated in an open circuit. From our study of transmission lines, we know that the resonant frequencies will be:
σ ε V(l2))
V( l1))
l2
l1 Fig. 3. 3. Two parallel conducting plates separated by a dielectric spacer. The structure may be viewed as a resonant transmission line of length, l1, or as a resonant line of length, l2.
Eq. 3. 41
f m, 0 =
v ph m ε r 2 l1
where m is an integer number of half-wavelengths in the direction of l1, and the subscript, 0, indicates no variation in the direction of l2. Similarly, if we excite the structure from its front edge so that propagation is in the direction of l2, we know the resonant frequency will be: Eq. 3. 42
f0, n =
v ph
n ε r 2l 2
where n is an integer number of half wavelengths, independent of m, and the index,0, indicates no variation in the direction of l1. Recall from the study of transmission lines that the resonant frequencies, fm,0 and f0,n, come from the differential equation: Eq. 3. 43
∂ 2 v ( x ,t ) − ∂ x2
∂ 2 v ( x ,t ) LC = 0, ∂ t2 2
and that the solution was the product of a one-dimensional space function and a time function. Clearly, the problem we are currently facing is a two dimensional one:
Microwave Resonators
page 3.8
Eq. 3. 44 2 ∂ 2 v ( x , y ,t ) ∂ 2 v ( x , y ,t ) 2 ∂ v ( x , y ,t ) + − k =0 2 2 2 x y t ∂ ∂ ∂
(Note the substitution of k for LC .) The solution to eq. 3.44 is the product of two onedimensional functions and one time function. As before, we group the time dependence into a phasor, leaving the spatial dependence explicitly stated: Eq. 3. 45
V (x, y ) = X (x ) ⋅ Y (y ) , where V(x,y) is a phasor which has been decomposed into the product of one function, X(x), which depends only on x and one function, Y(y), which depends only on y. We are not interested so much in the total solution to eq. 3.45 as we are in the characteristic, or eigen- frequencies, which are defined by eq. 3.46. Eq. 3. 46
f m,n =
2
v ph 2 εr
m n + lx ly
2
According to eq. 3.46, a parallel plate resonator with a dielectric constant of 38 and l1 = l2 = 2.25 cm, driven as a one-port element will absorb energy at the frequencies shown in fig. 3.7. The current and voltage distribution of the f1, 1 resonance is plotted in fig 3.4. The dark area in the left half of the figure represents zero currents at the edge of the top and bottom conductors; the shading lightens with increasing current. In the right half, the dark and light areas represent voltage maxima and minima, while the gray shades represent a transition between maximum and minimum.
TEM21, TEM12 (degenerate)
TEM20, TEM02 (degenerate)
TEM22 TEM10, TEM01 (degenerate)
0
0.5
TEM11
1
1.5
2
2.5
3
3.5
Fig. 3. 4. Absorption spectrum of (lossless) parallel plate resonator with x = y = 2.25 cm and εr = 38.
Microwave Resonators
page 3.9
Note that when both m and n are non-zero, the resulting eigenfrequency does not correspond to either the length or width dimension of the resonator. When m = n, the eigen-frequency roughly corresponds to Fig. 3. 5. Voltage (left) and current (right) distribution of TEM1,1 mode of the diagonal of the resonator, as depicted parallel plate resonator. in figs. 3.5 and 3.6. When m≠n≠0, the eigen-wavelength doesn’t correspond to any recognizable dimension in the resonator, fig. 3.7
Fig. 3. 7. Voltage distribution of TEM21 mode of parallel plate resonator.
Fig. 3. 6. Voltage contour plot of TEM2,2 mode of parallel plate resonator.
Microwave Resonators
page 3.10
Waveguide Cavities By enclosing the ends and sides of the parallel plate resonator with metallic walls, the waveguide cavity resonator can be formed. In this case, the boundary conditions require that the voltage be zero at all six walls. Thus, TEM modes can not be supported in metallic waveguide.
σ
l3 l2
l1 Fig. 3. 8. Hollow conducting box filled with dielectric. The structure may be viewed as a resonant transmission line of length, l1, or as a resonant line of length, l2, or as a resonant transmission line of length l3. Consider the rectangular metallic cavity shown in fig. 3.8. The boundary conditions dictate zero voltage at the box walls, but the voltage in the center of the box is free to assume any value, as long as the function of voltage with position is continuous. This suggests that standing sine waves could exist inside the box. This indeed happens. The eigenequation for the allowable frequencies is: Eq. 3. 47
f m,n,p =
v ph 2 εr
2
2
m n p + + lx ly lz
2
Although time doesn’t permit detailed exploration of cavity resonators, it is important to mention that a resonant cavity can take any shape and have other boundary conditions, as long as a standing wave made up of sines or cosines can be supported. Common structures include circular cylindrical metallic “cans”, dielectric “pucks”, and “doughnuts,” and even facing confocal mirrors.
Resonator Coupling The resonant cavity in fig. 3.8 consists of six metallic walls enclosing a dielectric space– since the causality principle implies that nothing can happen in the cavity without an outside force, there is clearly a need to discuss coupling. In general, there are three ways to couple to a microwave resonator. The obvious way is to make a tight connection from the source to the resonator (and from the resonator to the load) with a transmission line (or waveguide). The two less obvious ways are to make a loose connection to the resonator by assimilating some the electric or magnetic field lines within the resonator.
Microwave Resonators
page 3.11
Loose coupling to a resonator in a circuit can be viewed either as a capacitor in series between the resonator and the rest of the circuit or as an inductive transformer connected between the resonator and the rest of the circuit. Due to the simplicity of the capacitor model, it is the model most often used for modeling. Alternatively, loose coupling can be viewed as a pair of antennas. If electric field coupling is used, the antennas are modeled as Hertzian dipoles. If magnetic field coupling is used, the antennas are viewed as Hertzian current loops.
Coupling Coefficient
2
Equation 3.10 gave the Q-factor of a series RLC circuit as:
Q=
ω0 L R series
When a series resonator is connected directly to a signal source and a load, the source and load resistances contribute directly to Rseries, loading down the resonator. When this problem occurs, the response bandwidth grows. The situation is analogous to over damping, under damping and critical damping in mechanical systems: If the desired effect is to create a timing mechanism, the oscillator must be as under damped as possible. This is done by de-coupling the R L C resonator from the driver and from the load. Consider the effect of adding extremely large capacitors in parallel with the source and load, as shown in fig. 3.9. Fig. 3. 9. A series RLC resonator with the addition of de-coupling The effective contri- capacitors at the source and load. bution of their capacitive reactance will be small, since reciprocal series capacitors add. At frequencies of interest, the de-coupling capacitors will short-circuit the source and load resistances, greatly reducing their effect on the circuit. The effect of coupling on the resonator is described by the following equation: Eq. 3. 48
Q0 ≡κ QL Where Q0 is the “unloaded” Q of the resonator, QL is the “loaded” Q of the resonator in combination with the rest of the circuit, and κ is defined to be the coupling coefficient.
2
D. Kajfez, Q Factor, Vector Fields, Oxford, MS, 1994, pp. 39–61.
Microwave Resonators
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