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Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Transmission Line Resonators
S. R. [email protected]
School of Electronics EngineeringVellore Institute of Technology
April 29, 2013
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Outline
1 Implementation of Capacitors and Inductors
2 Series and Parallel Resonators
3 Implementation of Resonators using TX Lines
4 Summary
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Outline
1 Implementation of Capacitors and Inductors
2 Series and Parallel Resonators
3 Implementation of Resonators using TX Lines
4 Summary
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Using Short Circuited TX Line
tangent
Inductor
CapacitorCapacitor
Inductor
+
-
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Using Open Circuited TX Line
Inductor Inductor
CapacitorCapacitor
+
-
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Outline
1 Implementation of Capacitors and Inductors
2 Series and Parallel Resonators
3 Implementation of Resonators using TX Lines
4 Summary
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Series and Parallel Resonators
(Series Resonator) (Parallel Resonator)
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Series ResonatorFor the series resonator shown in Fig. X,
Zin = R + jωL +1
jωC.
If it is a loss-less series resonator, R = 0. In either case, resonance occurs when imaginary componentof the impedance becomes zero. So, resonant frequencya
ω0 is given by
jω0L +1
jω0C= 0
⇒ ω0=1√LC
. (1)
One more important definition for resonators is quality factor, which is defined as
Q0 =ω0L
R=
1ω0CR
. (2)
For ideal loss-less resonators quality factor tends to infinity.
aAnother definition of resonant frequency could be the frequency at which stored electric andmagnetic energies are equal.
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Series Resonators - Approximation of Inputimpedance
We have seen that the input impedance of a series resonator is given as
Zin = R + jωL +1
jωC.
This equation can be approximated within the neighborhood of resonant frequency ω0 as shownbelow:
Zin = R + jωL +1
jωC= R + j (ω0 +4ω) L +
1j (ω0 +4ω)C
= R + jω0L + j4ωL +
(1 + 4ω
ω0
)−1
jω0C
≈ R + j4ωL + jω0L +1
jω0C︸ ︷︷ ︸this term is zero
−4ωω0
jω0C= R + j4ωL−
4ωω0
jω0C= R + j
4ω
ω0ω0L−
4ωω0
jω0C
= R[
1 + j4ω
ω0
ω0LR
+ j4ω
ω0
1ω0CR
]= R
[1 + j
4ω
ω0Q0 + j
4ω
ω0Q0
]= R
[1 + j2
4ω
ω0Q0
](3)
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Parallel ResonatorFor the parallel resonator shown in Fig. X,
Yin =1R
+1
jωL+ jωC.
If it is a loss-less parallel resonator, R → ∞. In either case, resonance occurs when imaginary com-ponent of the admittance becomes zero. So, resonant frequencya
ω0 is given by
1jω0L
+ jω0C = 0
⇒ ω0=1√LC
. (4)
One more important definition for resonators is quality factor, which is defined as
Q0 =R
ω0L= ω0CR. (5)
For ideal loss-less resonators quality factor tends to infinity.
aAnother definition of resonant frequency could be the frequency at which stored electric andmagnetic energies are equal.
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Parallel Resonators - Approximation of Inputimpedance
We have seen that the input admittance of a parallel resonator is given as
Yin =1R
+1
jωL+ jωC.
So, the equation for input impedance can be approximated within the neighborhood of resonantfrequency ω0 as shown below:
Zin =
[1R
+1
jωL+ jωC
]−1
=
[1R
+1
j (ω0 +4ω) L+ j (ω0 +4ω)C
]−1
=
1R
+
(1 + 4ω
ω0
)−1
jω0L+ jω0C + j4ωC
−1
≈
1R
+
(1− 4ω
ω0
)jω0L
+ jω0C + j4ωC
−1
=
1R−
4ωω0
jω0L+
1jω0L
+ jω0C + j4ωC
−1
=
1R
1−R4ω
ω0jω0L
+ j4ω
ω0ω0CR
−1
= R[
1 + jR
ω0L4ω
ω0+ jω0CR
4ω
ω0
]−1
= R[
1 + jQ04ω
ω0+ jQ0
4ω
ω0
]−1
=R
1 + j2Q04ωω0
(6)
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Outline
1 Implementation of Capacitors and Inductors
2 Series and Parallel Resonators
3 Implementation of Resonators using TX Lines
4 Summary
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Simple Explanation (Assuming Loss-less TX Lines)
CASE 1
CASE 3
CASE 2
CASE 4
• For CASE 1 and 4, input impedance Zin = 0 ... So, they both are series resonators
• For CASE 2 and 3, input impedance Zin → ∞ ... So, they both are parallel resonators
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
λ02 Short Circuited TX Line (CASE 4)
This time, it is assumed that the TX line is made up of lossy elements so that α 6= 0. In such case,input impedance is given as
Zin = Z0
[ZL + Z0 tanh γlZ0 + ZL tanh γl
]= Z0 tanh γl, ∵ ZL = 0
= Z0 tanh (αl + jβl)
= Z0tanh αl + tanh (jβl)
1 + tanh αl tanh (jβl)
≈ Z0αl + j tan βl
1 + jαl tan βl, ∵ tanh αl ≈ αl (7)
Within the neighborhood of center frequency ω0, βl can be written as
βl =2π
λl =
2π
λ
λ0
2= π
ω
ω0= π
(ω0 + ∆ω)
ω0= π
(1 +
∆ω
ω0
).
So,
tan βl = tan(
π∆ω
ω0
)≈ π
∆ω
ω0. (8)
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
λ02 Short Circuited TX Line (CASE 4) ... Contd
Substituting (8) in (7) gives
Zin ≈ Z0
αl + jπ ∆ωω0
1 + jαlπ ∆ωω0
= Z0
(αl + jπ
∆ω
ω0
), ∵ αl
∆ω
ω0� 1 (9)
Comparing the above equation with the input impedance of series resonator Zin ≈ R[1 + j24ω
ω0Q0
]gives
{R = Z0αlQ0 = π
2αl =β02α , since β0l = π
. (10)
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
λ04 Short Circuited TX Line (CASE 3)
Input impedance of a lossy TX line is given as
Zin = Z0
[ZL + Z0 tanh γlZ0 + ZL tanh γl
]= Z0 tanh γl, ∵ ZL = 0
≈ Z0αl + j tan βl
1 + jαl tan βl(11)
= Z0−j cot βlαl + 1−j cot βl + αl
. (after multiplying with −j cot βl) (12)
Within the neighborhood of center frequency ω0, βl can be written as
βl =2π
λl =
2π
λ
λ0
4=
π
2ω
ω0=
π
2(ω0 + ∆ω)
ω0=
π
2
(1 +
∆ω
ω0
).
So,
cot βl = − tan(
π
2∆ω
ω0
)≈ −π
2∆ω
ω0. (13)
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
λ04 Short Circuited TX Line (CASE 3) ... Contd
Substituting (13) in (11) gives
Zin ≈ Z0−j cot βlαl + 1−j cot βl + αl
= Z0
j(
π2
∆ωω0
)αl + 1
j(
π2
∆ωω0
)+ αl
=Z0[
αl + j(
π2
∆ωω0
)] , ∵(
π
2∆ω
ω0
)αl� 1. (14)
Comparing the above equation with the input impedance of parallel resonator Zin ≈ R[1+j24ω
ω0Q0]
gives
{R =
Z0αl
Q0 = π4αl =
β02α , since β0l = π
2
. (15)
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Outline
1 Implementation of Capacitors and Inductors
2 Series and Parallel Resonators
3 Implementation of Resonators using TX Lines
4 Summary
Vector Calculus EE208, School of Electronics Engineering, VIT
Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary
Summary
Series Resonators:
• ω0 = 1√LC
• Q0 =ω0L
R = 1ω0CR
• Zin ≈ R[1 + j24ω
ω0Q0
]Parallel Resonators:
• ω0 = 1√LC
• Q0 =ω0L
R = 1ω0CR
• Zin ≈ R[1 + j24ω
ω0Q0
]
λ02 Short Circuited TX Line:
• R = Z0αl
• Q0 = π2αl =
β02α
• Zin ≈ Z0
(αl + jπ ∆ω
ω0
)λ04 Short Circuited TX Line:
• R =Z0αl
• Q0 = π4αl =
β02α
• Zin ≈Z0[
αl+j(
π2
∆ωω0
)]
Vector Calculus EE208, School of Electronics Engineering, VIT