1
Radioengineering Department
Laboratory of the circuits and signals
Laboratory work No.3
COUPLED RESONANT CIRCUITS
Work objectives: to get acquainted with methods to calculate and investigate coupled resonant
circuits using MathCAD and EWB packages, to investigate properties of the coupled resonant
circuit.
Theory
Examples of resonant circuits with inductive coupling are shown in the figure 1. Such circuits
are called as series or voltage resonant circuit. Their voltage transfer function can be expressed as
follows:
H ( jω ) = V C 2 / V IN
It is possible to simplify this transfer function using secondary parameters: coupling power A,
general mistuning ζ and quality factor Q:
H ( jω ) =
AQ
.
1 − ζ + A 2 + j 2ζ
2
Here A- coupling power and can be expressed as follows:
A=
XC
R1 R2
.
Here XC – coupling reactance, R1 and R2 – input and output contour resistances.
Value A is called coupling power. If X 12 = ωM , then
A=
ωM
.
R1 R2
When both contours have the same resonant frequency ( ω 01 = ω 02 = ω 0 ) and general frequency ω ≈ ω 0
A2 =
Here k =
ρ ρ
ω 2 M 2 ω 0 L1ω 0 L2 M 2
≅
= 1 2 k 2.
R1 R2
R1 R2 L1 L2 R1 R2
M
– coupling coefficient.
L 1L 2
2
So, from equation above follows, that coupling power depends on coupling coefficient and coupled circuits
quality factors: A = k Q1Q 2 . If both contours have the same parameters, then A = kQ .
Thus, as we can see from equations above, coupling power can be expressed in terms of coupling
coefficient k. Such expression is more convenient than expression in terms of mutual or coupling
impedances. In addition coupling coefficient k can be expressed as geometrical average of coefficients of
voltage transfer from first circuit to second k1 and from second to first k2:
k = k 1k 2 .
In the coupled networks where coupling is realized using inductive connections (Fig. 1a)
k1 =
M
,
L1
k2 =
M
,
L2
k=
M
;
L1L2
When coupling is build using autotransformative-conductive connections (Fig.1b)
k1 =
L12
,
L1 + L12
k2 =
L 12
,
L 2 + L12
k =
L12
;
( L 1 + L 12 )( L 2 + L12 )
When both contours are identical, Q >> 1 and XC=ωM, it is possible to rewrite previous equation:
ω M
XC
≅ 0 = k Q1Q2 .
A=
R
R1 R2
During the work:
1. Receive initial data from tutor and fill table 1.
R, ohms
C1
Vin
R1
L1
ω0, krads/sec
M
Q
C1
R2
L2
C2
V
C2
Vin
Table 1
C, µF
L, mH
R1
L1
L2
L12
R2
C2
a
b
Fig.1. Resonant network with inductive coupling (a) and its equivalent with autotransformative
coupling (b)
V
C2
3
2. Calculate frequency responses dependence on coupling power in the coupled resonant
circuits using MathCAD. Results of calculation put in to table 2.
A
0.5
1
1.5
2.5
3.5
2.5
Frequency
responses 0.75 0.8
H(ω)
φ(ω)
H(ω)
φ(ω)
H(ω)
φ(ω)
H(ω)
φ(ω)
H(ω)
φ(ω)
HM(ω)
φM(ω)
0.85 0.9
ω /ω0
0.95 1
1.05 1.1
1.15 1.2
1.25
Draw the graphs of these frequency responses.
3. Turn on EWB package and build the circuit shown in the figure 2. Calculate parameters of
inductances corresponding to coupling power equal to 2.5. Calculate frequencies,
corresponding to ratios of ω / ω 0 in the table 2, measure magnitude and phase of transfer
function and put these values in to last rows of the table.
Figure 2.
Work report
1. Work objectives.
2. Formulas, used to calculate frequency responses of the coupled resonant circuit.
3. Results of coupled circuit investigation: table with calculated and measured values and
graphs with magnitude and phase frequency responses.
4. Conclusions.
4
Control questions
1.
2.
3.
4.
5.
6.
7.
8.
9.
Name types of coupled resonant networks and sketch their circuit diagrams.
Explain, what is necessary to add into the circuit diagram shown in the figure 1a.
Explain, what is coupling power and coupling coefficient.
Sketch equivalent circuit diagram of the coupled resonant network and explain mean of its
components.
Show relations between circuit diagrams shown in the figure 1.
Explain influence of coupling power into coupled network properties.
Explain, what is resonant frequency, pass-band, bandwidth, central frequency.
Explain, what is frequency responses of the circuit.
Explain advantages of coupling in the resonant circuit.