Capacitive Coupled RLC Circuits
Ralph and Rachel Rosenbaum and Avraham
Semenkee
Computer-Electronic Laboratory
School of Physics and Astronomy
Tel Aviv University
Room #301 Shenkar Building
Ramat Aviv, 69978
Introduction
Two RLC Circuits are coupled to one another through a common coupling
capacitor Cc, whose magnitude we can change. The circuit diagram is shown in Figure
1 below. This clever idea comes from the laboratory write up by Prof. S. A. Dodds,
Physics Department, Rice University, Houston, Texas 77005-1892, U.S.A. Dodds’s
e mail address is: dodds@rice.edu. I have derived the equation for Vsec/Vpri. The
expression is really complicated and is most easily studied by programming it, using
EXCEL or some other similar scratch pad program.
The classical example of two oscillators coupled to one another is two
pendulums joined by a spring. If one pendulum is started swinging with small
amplitude, the other slowly builds up an amplitude as the spring feeds energy from
the first one into the second. Then the energy flows back into the first and the cycle
repeats. The general behavior can be complicated, depending on the initial excitation
as well as the system parameters.
Capacitive Coupled RLC Circuits
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A particular simple situation can be set up for two identical pendulums. If you
start the two swinging together, they will continue to swing in unison at their natural
frequency. Alternatively, if they are started exactly l80 degrees out of phase (swinging
in opposite directions), they will maintain this motion but at a higher frequency than
they would if uncoupled. These two possibilities are called the normal modes of the
system. When the pendulums are not identical, there are still two normal modes, but
the motions are more complicated and neither mode is at the uncoupled frequency.
We will study the electronic analog of a pendulum system, a pair of coupled
RLC circuits as illustrated in Figure 1. The components R, L, and C form one of two
resonant circuits. The capacitor Cc couples the two circuits. In the limit as Cc  ∞,
the two circuits are uncoupled since the capacitor acts like a “short” and could be
replaced by a wire, as far as AC signals are concerned. The other limit, C c  0,
corresponds to strong coupling, and the circuits behave like a single RLC circuit with
Leff = L + L = 2L and Ceff = 1/(1/C + 1/C) = C/2. The intermediate range of relatively
weak coupling is most interesting.
ipri
C
R
Vpri
isec
ipri-isec L
L
R
C
Cc
Loop A
Loop B
Figure 1
Capacitive Coupled RLC Circuits
Vsec
2
I find the following equations using Kirchhoff’s voltage law (see Figure 1):
Loop A:
Vpri = ipri[R + 1/(jwC) + jwL] + (ipri – isec)/(jwCc)
;
(1a)
Rearranging: Vpri = ipri[R + 1/(jwC) + 1/(jwCc) + jwL] - isec/(jwCc) ;
(1b)
Rearranging: Vpri = ipri[R - j(1/(wC) + 1/(wCc) - wL] + jisec/(wCc) .
(1c)
Loop B:
0 = isec[-R – jwL – 1/(jwC)] + (ipri – isec)/(jwCc)
;
(2a)
Rearranging: 0 = isec[-R – jwL – 1/(jwC) – 1/(jwCc)] + ipri/(jwCc) ;
(2b)
Rearranging: 0 = isec[-R – j(wL – 1/(wC) – 1/(wCc))] - jipri/(wCc) .
(2c)
Also:
(3)
Vsec = isec/(jwC)
or
isec = jwCVsec .
By eliminating ipri, I find the following:
Vpri = -wCcisec[wL – 1/wC – 1/wCc – jR][R – j(1/wC + 1/wCc – wL)] +
jisec/wCc
or:
;
Vpri/isec = wCc[jA + B][A – jB] + j/wCc
A = R and
(4)
with
B = 1/wC + 1/wCc – wL
Next: Vpri/Vsec = w2CCc(B2 – A2) – C/Cc + j2w2CCcAB
or:
.
(5a)
,
(5b)
Vsec/Vpri = 1/[ w2CCc(B2 – A2) – C/Cc + j2w2CCcAB ].
(5c)
Finally the magnitude Vsec/Vpri becomes { remember c = [real2 + imag2]1/2 }:
Vsec/Vpri = 1/[ {w2CCc(B2 – A2) – C/Cc}2 + {2w2CCcAB}2 ]1/2 .
with: A = R
B = 1/wC + 1/wCc – wL
and
(5d)
Wee, I hope Equation (5d) is correct! In Figure 2, we plot the frequency response of
the coupled RLC circuits using different values of Cc and C = 1 F, L = 0.72 mH and
R = 5 . The responses are certainly surprising and amazing.
Capacitive Coupled RLC Circuits
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3.0
Vsec/Vpri Using Cc = 10 uF
Vsec/Vpri Using Cc = 3 uF
Vsec/Vpri Using Cc = 1 uF
Vsec/Vpri Using Cc = 0.30 uF
Vsec/Vpri Using Cc = 0.15 uF
2.5
Vsec /V pri
2.0
1.5
1.0
0.5
0.0
0
20000
40000
80000
60000
100000
120000

Figure 2
The Experiment
Build the circuit according to Figure 1, using either a “plug-in” socket board,
or using a soldering iron and a printed circuit board. Ask Avraham for the circuit
components (two C’s = 1 uF, two L’s = 34 mH, and one Cc = 0.01, 0.1, 1 and 34 uF).
One circuit board is already prepared with “plug-in” Cc capacitors. Check the
response of the circuit manually using a HP function generator and a HP DMM.
Remember the highest AC frequency that the HP DMM can measure is 300 kHz.
If the circuit response properly, use either LabVIEW or VEE to computer
control the function generator and DMM. Display the collected data on the computer
monitor screen (on the “front panel” if you are using LabVIEW). Also use some of
the icons to determine the frequencies were the resonant peaks occur. Calculate
Vsec/Vpri from Equation (5d). Remember to convert “w’s” back to frequency “f’s”.
Capacitive Coupled RLC Circuits
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140000
Useful Comments on the Coupled Resonant
Circuits
A.
It is very important to normalize your measurment output
voltage vsec by the input voltage vpri. The reason is a design
mistake in the Hewlett Packard function generator – namely its
output voltage is a function of the load impedance of the circuit
connected to it. Since at the two resonant frequencies, the input
impedance of our RLC circuit drops by almost a factor of ten,
then so will the magnitude of the output voltage from the
function generator! If you don’t believe me, try to monitor the
output voltage of the function generator with another DMM as
you sweep through one of the resonant frequencies of the
circuit!! Thus, it is important to measure not only the output
voltage vsec but also the input voltage vpri with a second DMM
and then to take the ratio vsec/vpri.
B.
In fitting one of the theories to your data, remember that you
have two fitting parameters, one is the coupling capacitor Cc or
the Mutual Inductance M, and the second one is the coil
resistance R, that controls the magnitude of the resonance
peaks. Remember to vary R for the best agreement between
theory and your data.
C.
When you assign a label to variable using LABView, you must
use a small alpha numeric letter (plus a number, if you wish).
For example let our variable in “pi” =  = 3.14; then:
c = 3.14, this is FINE and OK.
C = 3.14, this is BAD because the alpha numeric character is a “capital” letter.
c1 = 3.14, this is FINE and OK.
Capacitive Coupled RLC Circuits
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