Capacitive Coupled RLC Circuits

Ralph and Rachel Rosenbaum and Avraham Semenkee

Computer-Electronic Laboratory

School of Physics and Astronomy

TelAvivUniversity

Room #301 ShenkarBuilding

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, RiceUniversity, Houston, Texas77005-1892, U.S.A. Dodds’s e mail address is: . 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.

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, Cc 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.

Figure 1

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: Vsec = isec/(jwC)or isec = jwCVsec.(3)

By eliminating ipri, I find the following:

Vpri = -wCcisec[wL – 1/wC – 1/wCc – jR][R – j(1/wC + 1/wCc – wL)] +

jisec/wCc;(4)

or: Vpri/isec = wCc[jA + B][A – jB] + j/wCc with

A = R and B = 1/wC + 1/wCc – wL.(5a)

Next:Vpri/Vsec = w2CCc(B2 – A2) – C/Cc + j2w2CCcAB,(5b)

or: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 and B = 1/wC + 1/wCc – wL(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.

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”.

Useful Comments on the Coupled Resonant Circuits

  1. 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.
  2. 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.
  3. 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 Circuits1