Transformers in Single-Tuned and Double-Tuned Filters
14.15
between 995 kHz and 1005 kHz. The band-pass filters in the RF stage in the radio receiver have to be
tuned to 1 MHz centre frequency with a bandwidth of 10 kHz to receive this station. Tuned amplifiers
make use of inductances in the signal coupling transformer and variable capacitors to realise the
required band-pass filter.
14.5.1
Single-tuned amplifier
Single-tuned amplifier uses a tightly coupled transformer and a capacitor. The capacitor is connected in
parallel to the primary of the transformer. Next stage amplifier is connected to the secondary and it may be
modeled by the input resistance of the next stage. The transformer primary in parallel with the capacitor
forms a parallel resonant circuit and the transistor drives the signal
current into this parallel combination.
example: 14.5-1
An RF transformer with
k
≈
1 and
n
=
0.5 is passively terminated
by a resistor of 800
W
. It is driven by a current source
i
S
(
t)
=
I
m
cos
w
t A in parallel with a capacitor of 220 pF at the primary
side. The primary self-inductance of the transformer is 115
m
H.
Derive an expression for the ratio of output voltage phasor to
input current phasor.
Solution
The phasor equivalent circuit
of the circuit is shown in
Fig. 14.5-2. The transformer is replaced by its equivalent
circuit in this diagram by reflecting the 800
W
in the
secondary as 800/(0.5)
2
=
3.2 k
W
in the primary.
The ratio of output voltage phasor to input current
phasor is just the impedance
of parallel combination
of inductor, capacitor and resistor.
Using the parameter
symbols
R,
L and
C,
1
1
1
1
2
2
Z
R
j L
j C
j L R
LCR
j RL
LC
j L R
j L
V
I
Z
j L
o
s
= +
+
=
+ −
=
−
+
∴
= =
w
w
w
w
w
w
w
w
w
(
)
((
)
1
2
−
+
w
w
LC
j L R
Substituting the numerical values for circuit parameters,
V
I
j
f
f
j
f
o
s
=
−
+
722 6
1
0 226
2
.
(
)
.
, where
f is the cyclic frequency of input in MHz. The
magnitude and
phase of this ratio function is given by
Magnitude of
in MHz
Phase of
Z
f
f
f
f
Z
=
−
+
=
722 6
1
0 226
2 2
2
.
(
)
( .
)
;
pp
2
0 226
1
1
2
−
−
−
tan
.
;
f
f
f in MHz
Fig. 14.5-1
Circuit for
Example: 14.5-1
220 pF
115
µ
H
800
Ω
+
–
i
S
(
t
)
v
O
(
t
)
n = 0.5
k
= 1
Fig. 14.5-2
Phasor equivalent
circuit for the circuit
in Fig. 14.5-1
3.2 k
Ω
Ω
+
–
I
S
V
O
–
j
4.545
10
-9
j
115
10
-6
Ω
ω
ω
14.16
Magnetically Coupled Circuits
w
=
1
LC
is the angular frequency at which the susceptance of inductor and capacitor will be equal
in magnitude but opposite in sign. Therefore, these two susceptance values will cancel each other
at that frequency and leave
R as the impedance value. At all other frequencies, some susceptance –
inductive or capacitive – will shunt
R thereby bringing down the magnitude of impedance below
R.
Thus, impedance is a maximum of
R at
w
=
1
LC
. Normalising the impedance function by dividing
it by value of
R,
Z
R
f
f
f
Z
R
f
f
=
−
+
∠
= =
−
−
−
722 6
1
0 226
2
0 226
1
2 2
2
1
2
.
(
)
( .
)
tan
.
and of
q
p
;
f in MHz
Fig. 14.5-3 shows the plots for these functions.
f
in MHz
in rad
θ
0.5
0.5
–0.5
1
1
–1
2
1.5
1.5
–1.5
1
0.8
0.6
0.4
0.5 1
2
f
in MHz
|
Z
|
R
1.5
0.2
Fig. 14.5-3
Magnitude and phase plots of normalised impedance function in Example: 14.5-1
The plots reveal that it is a highly
frequency-selective circuit. The circuit produces large magnitude
output if the frequency of sinusoidal current source is at 1 MHz or nearby. The response is small if the
frequency is far away from 1 MHz on either side. Such a response characteristic is called a
band-pass
characteristic or
tuned characteristic. A circuit with band-pass nature
can extract some frequency
components selectively from a mixture of sinusoidal waveforms at different frequencies presented to
it at the input. Such circuits find wide application in communication and signal processing circuits.
The circuit we discussed in this example illustrates the principle of
tuned amplifiers that are used in
all sorts of electronic communication circuits starting from the common transistor radio receiver. The
current source is realised by a transistor circuit in a tuned amplifier.
Single-tuned amplifier is an example where the inductance of a transformer is intentionally
designed to resonate with a chosen capacitor at a desired frequency.
The cut-off frequencies of a band-pass filter are the frequencies at which the magnitude of transfer
gain of the filter is
1
2
0 707
=
.
times the value at centre frequency. They are 0.9 MHz and 1.1 MHz
in this example. The difference between the two cut-off frequencies is defined as the bandwidth of
the band-pass filter. Hence, the band-pass filter in this example has centre frequency of 1MHz and
bandwidth of 0.2 MHz.
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