## Open Collection of Student Writing (OCSW)

• Area: Science, Engineering & Mathematics
• Program: Engineering Tech
• Type of Writing: Report
• Type of Writing: Scientific (writing to communicate scientific research results)
• Course Level: 2000
• Year: 2018
• Paper ID: CS.C.S.I.S.R.S.2.2.1367

# Circuit Analysis: an Op-Amp Based Graphic Equalizer.

The circuit that I endeavored to analyze was a graphic equalizer. Information for this circuit was mainly acquired in a supplemental to the application report, SBOA091B, by Texas Instruments. The circuit in question employs three op-amps and as such it will be appropriate to analyze in three stages.

The first stage is a simple inverting buffer. This serves to ensure that any unwanted and potentially destructive peaks saturate to ground. I selected two 100kΩ resistors for this stage. Furthermore, the non-inverting input is tied to reference which I’ll explain in detail later.

The next stage comes in the form of a band pass filter. At the heart of this stage lies an op-amp that functions as a simulated inductor. The Inductor circuit contains a feedback resistor I’ll refer to as Rfb. a reference resistor, Rref, and a capacitor which will henceforth be referred to as CL. The formula for determining the inductance is L=(Rref-Rfb)*Rfb*CL. Once the inductance is determined we can move on to calculating the center frequency ƒ0=1/(2π√LC)  (C being the paired capacitor CC in the RLC configuration) which allows us to calculate the reactance of the filter, XL= 2π*ƒ0*L.  Establishing XL allows us to determine the Q factor of the filter since Q= XL/Rfb. With these formulae in hand we begin to get an idea of what the circuit can do.

The potential benefits of using simulated induction are as follows: 1) Inductors are notoriously unpredictable where the capacitor op-amp pairing isn’t. 2) Generally, Inductors and capacitors are inversely paired, but using capacitors allows for a proportionality of 2 to 1 with the inductive capacitor being half that of the capacitive one. 3) In circuits where magnetic fields might cause disruptive interactions this isn’t a consideration since the capacitor uses the electric field to store charge. 4) a final interesting feature of this circuit is that the Q factor can be incrementally increased by adding further filter stages in series.

The circuit isn’t without it’s shortcomings however. Rfb will never allow for the highest performance. This is due to undesirable high-frequency roll-off for resistors under around 330Ω, and a low stop-band rejection that occurs above about 470Ω. These characteristics make the op-amp impractical for low-pass, high-pass, and notch filtration. Selecting 470Ω will work for band-pass operations since the response remains relatively stable at higher frequencies, but with a Q factor of around only 5 after 2 induction stages it’s less than optimal.

In my circuit 470Ω is the resistor chosen for Rfb given it’s an E-6 standard which is what I’ll be selecting for due to ready availability.  Rref is set to a value of 100kΩ. This means that for the upper end of the hearing range, 20khz, the capacitors would be about 1.5nF for CC and 680pF for CL. For the lower range, 20hz, a CC  of 2.2μF and CL of 1μF will be sufficiently close. Due to the limited values creating drastically varying ranges it’s advisable that all resistors should ideally be toleranced to 1%, though in practice I was able to get valid results from values off by more. One could even perhaps select for that to get closer to the 20hz.

In the third stage the inverting and non-inverting inputs are figured for a minimum gain of -17dB (1/7) and 17dB (7) respectively. To calculate the desired output the formulae R+=Rfb(17dB-1) and R=Rfb*-17dB. These both calculate to 2820, which is closest to 3.3kΩ so that’s the value used for those resistors. The inverting resistor functions as a feedback, and is tied in series to a 10kΩ pot in which the wiper runs through the filter to reference. Adjusting the pot increases or decreases the gain and Q of the output.

At the input and the output there are 10μF electrolytic capacitors with the positive terminals pointing inward. These serve to filter out any unwanted dc offset between sequential parts of the device. I put a 2.2MΩ resistor to ground at the terminus of the output to represent a load on the circuit.

The reference is Vcc/2. I set the positive rail supply of the op-amps to 15V and the negative rail to ground. In order to get a virtual ground reference of 7.5V I used two 1kΩ resistors in a voltage divider configuration on Vcc.

Simulating the circuit in LTspice gave a clear peak in AC analysis and a gain of roughly 10dB across most values of ƒ0. In practice the Q of the LM358 failed to meet the output potential of the LT1013 that I simulated with. This is likely due to less responsiveness from the rail. However, the oscilloscope still presented a clear gain within about a 1k range between both ƒco when the pot was maxed out. The simulation does, in fact, demonstrate the beginning of frequency roll-off to the far right.

I really enjoyed studying this circuit. It’s provided me with much more understanding of how audio is filtered and attenuated. I hope to implement this knowledge in making my own graphic equalizer in the future.

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