Download The varactor resonator shown is resonant at 135 MHz. In the circuit

1- The varactor resonator shown is resonant at 135 MHz. In the circuit, L = 150 nH, CT, = 12
pF, Cp = 5 pF, and Cj is the value required for resonance.
(a) Find the value of Cj that makes the circuit resonant at 135 MHz.
(b) Assume that the varactor diode has Cj = 20 pF and an unloaded Q of 100 at 50 MHz, and
that the inductor has an unloaded Q of 50 at 135 MHz. Find the loaded Q of the resonator at
135 MHz. Neglect anything that might be connected to the resonator terminals. Make
reasonable assumptions.
3- For the resonant series circuit shown below, determine the expression for quality factor, Q,
in terms of R, L, and C. Recall that Quality factor is defined as: Q = 2π(max energy
stored/total energy lost per cycle).
3- A series resonant circuit at 50 MHz has Rs = 0.2 ohms, and RL= 0.3 ohms. The source has
an available (maximum) power of 10 mW.
a) Specify the values of L and C that will result in a loaded Q of 20.
b) How much power is delivered to the 0.3 ohm resistor at resonance? Why is power not
equal to the Pavailabe ? What is the efficiency of the circuit?
c) Re-design the circuit to attenuate (relative to the fundamental frequency) the second
harmonic of 50 MHz by 40 dB. Give the values of L and C.
2- For the folowing circuit, determine an expression for Vo1 (t ) and Vo 2 (t ) as well as the
circuit bandwidth. You may begin with the assumptions made in class.
V01 (t )
C1
i (t ) = cos107 mA
10μH
C2
500 Ω
V02 (t )
C1 = C2 = 2000 pF
If we assume nQT ' QE > 100 and QE > 10 , we may replace the original circuit by the
following model
V01 (t )
i (t ) = cos107 mA
10μH
1000 pF
1: n
500 Ω V02 (t )
for which n =
C1
1
CC
= , C = 1 2 = 1000 pF , ω 0 =
C1 + C2 2
C1 + C2
Z11 ( jω 0 ) = Req =
1
= 107 rad/sec ,
LC
500
= 2 kΩ , Z12 ( jω 0 ) = nZ11 = 1 kΩ .
n2
In addition we note that QT ' =
Req
ω0 L
= 20 , QE = (500 Ω)ω 0 (C1 + C2 ) = 20 , and nQT ' QE = 200 .
These values justify the use of the model.
Since the current drive is at resonant frequency, where the input impedance appears purely
resistive ( Z11 ( jω 0 ) = Req ), Vo1 (t ) may be written directly in the form
Vo1 (t ) = Req cos107 t = 2 cos107 t V and hence Vo 2 (t ) = nVo1 (t ) = 1cos107 t V .
Finally, since the only loading in the circuit is across C2 , QT ' = QT and
BW =
ω0
QT
= 5 x107 rad/sec .
2- Aşağıdaki paralel RLC devresi için rezonans frekansını, yüklü ve yüksüz Q’larını
hesaplayın.
2- Consider the loaded parallel resonant circuit illustrated below. Compute the resonant
frequency in radians per second, unloaded Q, and the loaded Q of this circuit.
4- A circuit has a resonant frequency of 200 MHz and a bandwidth of 40 MHz. Determine
the Q of the resonant circuit and the two half-power frequencies.
f0 = 200 MHz, BW = 40 MHz,
f0 =
f1 f2 , f2 = f1 + BW ⇒ f0 =
f1 ( f1 + BW) ⇒ f02 = f12 + f1BW
−BW+ BW2 + 4 f02
⇒ f + f1BW − f = 0 ⇒ f1 =
⇒ f1 = 180.998 MHz
2
f
200
f2 = f1 + BW = 220.998 MHz, QL = 0 =
=5
BW
40
2
1
2
0
4- Rezonans frekansı 175 MHz ve bant genişliği 30 MHz olan bir rezonatör devresinin
yüklenmiş Q’sunu ve gücün yarıya düştüğü frekansları hesaplayın.
4- A circuit has a resonant frequency of 175 MHz and a bandwidth of 30 MHz. Determine the
loaded Q of the resonant circuit and the two half-power frequencies.
2- The crystal resonator in the figure below has a series resonant frequency of 5 MHz and an
anti-parallel resonant frequency of 5.4 MHz. If the package capacitance is 10 pF, determine
the values of L1 and C1.
2- Consider the circuit in the following figure:
a) For any value of resistance R, capacitance C, and frequency ω, find the value impedance Z1
which makes Zeq, purely real. What is the value of Zeq in that case?
−1 Z1 = j ωC since the three devices are in parallel. In that case Zeq = R
b) Find the minimum line length needed, to create the impedance Z1 with a shorted lossless
transmission line. Assume Z 0 = R = 50 Ω, ω = 2π ×7 ×106 Hz, C = 1 µF, c = 2×108 m s .
 l
For a shorted line Z L = jZ 0 tan 2π  .
 λ
We need −
 l
1
1
l
c
= j ωC → tan 2π  =
→ ≅ 0.25 → l =
= 7.14 m
 l 
 λ  Z 0 ωC
λ
4f
jZ 0 tan 2π 
 λ
4- Consider a series resonant circuit.
a) Find the expression for I (f ) I (f0 ) in terms of QL, f, and f0.
V0 =
Vin R
Vin
I (f )
1
→I =
→
=
f
f
f
I ( f0 )
f 
f 
f 
1 + jQL  − 0 
1 + jQL  − 0 
1 + jQL  − 0 
 f0
 f0
f 
f 
f 
 f0
b) The magnitude of the current in a series resonant circuit operating at a frequency of 95
MHz is 50 percent of the current at resonance. If the resonance frequency is greater than 95
MHz, what is the resonance frequency and what is the loaded Q (at resonance) of the
resonant circuit?
QL = 10, f = 95 MHz
2- Consider a lossy transmission line of characteristic impedance Zo and propagation
constant γ = α + jβ . A resonator is designed using a short-circuited transmission line section
of length l = 1λ. (Assume that the total loss of this resonator is much less than one neper).
(a) Will this section exhibit series resonance or parallel resonance? Draw the equivalent RLC
model for this resonator.
It will exhibit a series-type resonance when l=λ (similar to the short circuited λ/2 resonator).
(b) Find the input impedance near the resonance frequency.
We assume that the total loss of this resonator is much less than one neper. αl << 1. So we
can write tanhαl = αl.
(c) Find the RLC model parameters of the resonator.
(d) Find the Q of the resonator.