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| . In the figure above, ''L,,p,,'' represents the inductance of primary coil, and ''L,,s,,'' represents that of secondary coil. Generally the signal source has an output impedance, which is 50 Ohm in this case. One of the most important role of the transformer is changing the impedance. For example the load impedance looks reduced by 1/''n''^2^ if we look at the impedance from primary side. Thus the source voltage will be shared by 50 Ohm and reduced load impedance ''Z'',,L,,/''n''^2^, then the output voltage ''V'',,out,, will be Vout=nVin*Z/(). Note that the coefficient ''n'' directly represents step-up gain. Then the maximum ''n'' will be find if you change the value in this equation. | . In the figure above, ''L,,p,,'' represents the inductance of primary coil, and ''L,,s,,'' represents that of secondary coil. Generally the signal source has an output impedance, which is 50 Ohm in this case. One of the most important role of the transformer is changing the impedance. For example the load impedance looks reduced by 1/''n''^2^ if we look at the impedance from the primary coil side. Thus the source voltage ''V'',,in,, will be shared by 50 Ohm and reduced load impedance ''Z'',,L,,/''n''^2^ in series, then the output voltage ''V'',,out,, will be expressed as |
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| . Note that the coefficient ''n'' directly indicates the step-up gain. The total gain ''G'' will get its maximum when the impedance matching is satisfied. Namely, attachment:equation2.png . is the condition of the impedance matching. In this case the maximum gain is described as; attachment:equation3.png |
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| == 2.2 resonance with LC == . Let's begin with one of the simplest resonant circuit. The circuit has a ''L'' and ''C'' in parallel. Inductance ''L'' has a angle of 90 deg from real axis, and capacitor ''C'' has a angle of -90 deg from real axis in terms of impedance. In addition their amplitude in real-imaginary plane grow up along with frequency. The total impedance is expressed as; |
== 2.2 LC resonant circuit == . In this section the LC resonant circuit is introduced because multi-resonant circuit can be essentially considered as the set of the LC resonant circuit. Let's begin with one of the simplest resonant circuit. The circuit has a ''L'' and ''C'' in parallel. Inductance ''L'' has a angle of 90 deg from real axis, and capacitor ''C'' has a angle of -90 deg from real axis in terms of impedance. In addition their amplitude in real-imaginary plane grow up along with frequency. The total impedance is expressed as; |
Overview
- A set of phase modulation sidebands are imposed to the main laser beam for control of the mode cleaner cavity and the main interferometer. The frequency of the sidebands are determined from the signal extraction and control scheme: 11MHz, 23.5MHz, and 55MHz.
- We are investigating a multiply resonant circuit to be attached to EOM in order to create those three pairs of sidebands with a single crystal.
- (What is motivation? Why single crystal? Why three eoms are not enough? Why triple electrode EOM is not enough?)
The feasibility and the basic ideas of the triple resonant circuit was investigated by Stephanie Erickson, a 40m SURF student in 2009 Summer. Link to Stephanie's descriptions (What did she demonstrated?)
Steps for Development
studying the principle of LC circuit + transformer
investigation of loss effect in LC circuit
- make a test circuit (single resonant) with highest impedance ( - Dec.20)
- make a new transformer for matching the impedance of the test circuit ( - Dec. 25)
- performance test by using optical spectrum analyzer ( - Dec. 31)
- study of triple resonant circuit (- Jan.5)
- investigation of loss effect in the multi-resonant circuit ( - Jan.12)
- make a prototype circuit with highest impedance ( - Jan.13)
- make a new transformer for prototype ( - Jan. 20)
- performance test of prototype by optical analyzer ( - Jan. 25)
- final design and ordering (- Jan.31)
- performance test with ordered circuit
- installation
1. Conceptual idea
- The goal of this development is to make a triple-resonant circuit that operates an EOM. And we should make a high gain circuit to achieve sufficient amount of modulation depth. The conceptual design has been already investigated by Stephanie, a SURF student in 2009.
2. Fundamentals
2.1 Transformer
- The reason why we use a transformer are mainly follower;
handling high voltage, - step up the voltage and
- impedance matching.
2.1.1 handling high voltage
- Our EOM has the modulation efficiency of ~13mrad/V and the desired modulation depth is typically 0.1-1rad. Therefore the voltage inputted onto the EOM will reach 10-100V. In such high voltage regime the ordinary op-amplifiers are no longer works because those output voltage are limited to around +/- 15V which corresponds to the source voltage of the amplifiers. In contrast transformers are even available in such high voltage regime because the they works passively.
2.1.2 step-up gain and impedance matching
For the transformer alone, the step-up gain is equal to its turn ratio n. Therefore in order to make a high gain circuit with a transformer, big number of n is simply required. However there is a limitation for the maximum turn ratio nmax because of the impedance matching issue.
The principle of impedance matching with transformer is explained as follower. At first consider the simple case, a transformer with load impedance ZL. The figure below shows the schematic of the circuit with the transformer.
In the figure above, Lp represents the inductance of primary coil, and Ls represents that of secondary coil. Generally the signal source has an output impedance, which is 50 Ohm in this case. One of the most important role of the transformer is changing the impedance. For example the load impedance looks reduced by 1/n2 if we look at the impedance from the primary coil side. Thus the source voltage Vin will be shared by 50 Ohm and reduced load impedance ZL/n2 in series, then the output voltage Vout will be expressed as
- attachment:equation1.png
Note that the coefficient n directly indicates the step-up gain. The total gain G will get its maximum when the impedance matching is satisfied. Namely,
- attachment:equation2.png
- is the condition of the impedance matching. In this case the maximum gain is described as;
- attachment:equation3.png
For those reasons, the higher ZL provide the higher gain. That's the reason why impedance matching is so important. Moreover there are another merits in impedance matching. An impedance matched system can avoid the reflection of signal, which might be harmful for some circuits, and avoid the standing wave, which makes behavior of the circuit more complicated.
2.2 LC resonant circuit
In this section the LC resonant circuit is introduced because multi-resonant circuit can be essentially considered as the set of the LC resonant circuit. Let's begin with one of the simplest resonant circuit. The circuit has a L and C in parallel. Inductance L has a angle of 90 deg from real axis, and capacitor C has a angle of -90 deg from real axis in terms of impedance. In addition their amplitude in real-imaginary plane grow up along with frequency. The total impedance is expressed as; Z=1jL*omega +
The circuit resonates when the angular frequency is [ , where the denominator gets to completely zero. An interesting thing will be found if you look at the impedance in frequency-impedance domain (like a bode plot). There are two solid line, one with f1 dependence represents impedance of L, the other solid line with 'f'-1 represents that of C. Then the resonance occurred where those two lines are crossing. This phenomenon physically indicates a [. According to equation above, the resonant impedance goes to infinity thanks to have a pure imaginary number. However if impedance of L and C have an extra real part, the cancellation does not complete and eventually the resonant impedance is no longer infinite.
2.3 Loss
In the ideal case, where no losses, we can get an infinite impedance. But in the real circuit the impedance must be limited. The one of the component which makes impedance lower is losses. Thus we have to think about the effect of loss toward the impedance. Begin with simple model shown in fig.2 The circuit is parallel LC resonant circuit and it has an DC resistance each in serial to L and C.
3. Study of Single resonant circuit
- In order to understand how the circuit works and how losses affects, we started with a single resonant circuit. The circuit is composed by L, C and transformer in parallel.
3.1 resonant impedance
- According to our measurement....
3.2 loss effect
- The loss effect has been directly observed via the impedance measurement.
3.3 design single-resonant circuit
- Based on the result described above, the single-resonant circuit is designed.
3.4 optical test
- So far we only think about electrical aspect, here the optical aspect is considered. The motivation of optical test is
3.5 conclusion
- Through whole measurement there are some suggestion for designing better circuit.
4. multiply resonant circuit
4.1 what's difference from single-resonance ?
- Here we consider the difference between multi-resonant and single-resonant. This
4.2 design
- The requirements for the multiply-resonant circuit are follower;
4.3 performance test
- Here we will report the resultant performance of the circuit both in electrical and optical test.
- As we expected....
App. impedance measurement
- In this chapter the technique for measuring the impedance by using Network Analyzer (Agilent AG4395A) is explained. The point is to reduce the reflective waves.