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@@samuelk3076 You're welcome! and thank you. Glad that you liked this rectifier video. Here are links to related op amp circuit videos: AC/DC converter Https://th-cam.com/video/4aG5NYX8tGo/w-d-xo.html Precision Rectifier th-cam.com/video/2eAsJbTBkQA/w-d-xo.html Triangle Oscillator circuit th-cam.com/video/JF5Up_cuL9k/w-d-xo.html I hope these videos are interesting as well.

@@horacepoon7044 You're welcome! Glad that this power amplifier video is helpful. Related videos are: Push-Pull Power Amplifier Design with Darlington and Sziklai transistor pairs th-cam.com/video/866MYibo8yE/w-d-xo.html Voltage Regulator Design with push pull output th-cam.com/video/CJl-urzeiTo/w-d-xo.html More Power Amplifiers & Voltage Regulators Videos are in th-cam.com/play/PLrwXF7N522y4vWr-8XRBqxpi5idFDE9BV.html video playlist.

@@ibrahimcetin153 You are welcome. Glad that this video is helpful. The Generalization of Karatsuba Algo, Toom-Cook (Toom3) is also explained with Examples in this follow-up video th-cam.com/video/1XiSyNzMX6Q/w-d-xo.html that I hope is interesting as well.

You're welcome. Glad that you like the explanation in this video. For more Electronic Amplifier examples see: Instrumentation Amplifier with Electronic Gain Control th-cam.com/video/C4tghZ-q6Zs/w-d-xo.html VCA Electronic Gain Control explained th-cam.com/video/cFzYZsPEtP0/w-d-xo.html Analog Multiplier Circuit th-cam.com/video/VP53A2zpVMQ/w-d-xo.html Push-Pull Power Amplifier Design with Op Amp, Sziklai Darlington Transistors th-cam.com/video/8BFzsi7-Vbs/w-d-xo.html I hope these Circuit videos are interesting as well.

Glad that this Digital Stethoscope Amplifier is helpful. I highly recommend using a zero offset low noise CMOS Op Amp for this design. Here are a few related signal conditioning videos: EKG ECG Amplifier with Right Leg Drive Explained th-cam.com/video/1c7KGXPs4do/w-d-xo.html Electric Guitar Amplifier to XLR Audio Signal th-cam.com/video/X4y8cwZdGEk/w-d-xo.html Bridge Audio Amplifier Explained th-cam.com/video/EDpu6urAtHA/w-d-xo.html Thermocouple Amplifier with Cold Junction Compensation Explained th-cam.com/video/-BsDLBI166U/w-d-xo.html

@m.sheriff1 You are welcome. Here are a few related circuit videos: Op Amps and Resistance Ladder Network th-cam.com/video/dx95Y0CBtnU/w-d-xo.html Resistance of infinite ladder resistor network th-cam.com/video/w96ooJty1Ic/w-d-xo.html Digital to Analog Converter Design with Resistor Ladder Explained th-cam.com/video/Y1H8HrrD4mQ/w-d-xo.html I hope these videos are interesting as well.

My pleasure. Glad that you like this Voltage to Current Converter Video. For another MOSFET circuit Video see "How to design a charger voltage regulator" th-cam.com/video/NdfyHoxjKrY/w-d-xo.html . Regarding your question, NMOS and PMOS transistors are active electronic devices (that require to be properly biased to work) in contrast with passive components like resistors and capacitors. Please watch the video at minute 11:28 when source terminal of NMOS and source terminal of PMOS are marked on the circuit diagram. For NMOS, the gate voltage needs to be higher than the source voltage by at least Threshold voltage (Vt) to turn on the NMOS. But for PMOS, the gate voltage needs to be less than the source voltage by at least the the magnitude of the threshold voltage in order to turn on the PMOS transistor.

Depends on your target application. Say, for Audio Signal Processing, the signal frequency range is ~50 Hz to 20 kHz. So, a clock frequency of say 500 kHz or higher should be proper. Capacitors in the range of ~5nF to 10nF should be fine assuming switch-on resistance of NMOS transistor less than 100 Ohms (check your transistor datasheet). To be on the safe side, your NMOS switching transistors should have a max switching frequency of at least a few MHz. Lastly, given the clock frequency is not that high, conservatively select a zero offset low noise CMOS Op Amp with a slew rate 20 times faster than your clock frequency. For example, if your clock rate is 500 kHz, then select Op Amp with slew rate of peak-peak (say 5 volts) per 100 ns or faster. See more amplifier videos in th-cam.com/play/PLrwXF7N522y5679YAO-lFrNVYqV9XMNTr.html op amp amplifiers videos playlist.

You are welcome. Glad that this Oscillator video is useful. Please elaborate on your question. What do you mean by applying impedances to the RC circuit below?

Glad that you are interested in this Toom3 Algorithm Tutorial video. You can find the Toom-Cook Algo Pseudocode online. In case helpful, here is a video of Toom2 aka Karatsuba Algorithm Explained with Examples th-cam.com/video/FEzBs2rrLqs/w-d-xo.html

While F=0.5 Hz is a considerably low cut-off frequency for the highpass filter, using equations explained at 18:40 , with Damping Factor set to 1/sqrt(2) we get R1=2*R2 and 2*pi*F=1/(C*sqrt(R1*R2)) . Hence we get R2 = 1/(2*pi*1.41*F*C) , if we set F=0.5 Hz and say C=1.2uF then we get R2=200 kOhm , R1=400 kOhm. Now, similarly, for lowpass filtering portion of this Sallen Key Filter, using equations explained at 26:35 , with R=100kOhm we can derive C1=11nF, C2=22nF. I hope this is helpful.

@@robr8554 You are welcome! Glad that you enjoy my videos. In case of single supply, the grounded input terminal of each op amp needs to be biased at 0.5*supply voltage. Then your output Sawtooth waveform oscillates between 0 and supply. For more Oscillator circuit videos see th-cam.com/play/PLrwXF7N522y4ee_0GL2EdguM-kLtJPxpU.html Oscillators and Signal Generators playlist. I hope this is helpful.

@@Chupacabras222 Thank you. Glad that this video is interesting and useful. Here are a few related circuit videos: Feedback Amplifier with JFET & BJT Transistors th-cam.com/video/NB1-mYglXsY/w-d-xo.html Feedback Amplifier with BJT: How to find DC Bias and AC Gain th-cam.com/video/6dLMxpLNKv4/w-d-xo.html I hope these videos are interesting as well.

Depends on your target application. In case of Audio Signal Processing, the signal frequency range is ~50 Hz to 20 kHz. So, I suggest you select your clock frequency conservatively say 400 kHz. Capacitors in the range of ~5nF to 10nF should be fine assuming switch-on resistance of NMOS transistor less than 100 Ohms (check your transistor datasheet). To be on the safe side, your NMOS switching transistors should have a max switching frequency of at least a few MHz. Lastly, given the clock frequency is not high, conservatively select a zero offset low noise CMOS Op Amp with a slew rate 20 times faster than your clock frequency. For example, if your clock rate is 400 kHz, then select Op Amp with slew rate of peak-peak (say 5 volts) per 100 ns or faster. More op amp amplifier videos are published in the th-cam.com/play/PLrwXF7N522y5679YAO-lFrNVYqV9XMNTr.html playlist. I hope this is helpful.

Thanks for watching, sharing your thoughts and asking questions. In this Band-gap voltage reference the series connection of VBE3 and R2 sets and defines the voltage drop Vdiode across the top and bottom rails of the circuits. The collector-emitter voltages (Vce) of BJT transistors T3, T4 and T5 will adjust accordingly to accommodate the forced voltage by the sum of R2.I2 and VBE3. Two related circuit videos are, Bandgap Voltage with Op Amp & Widlar Current Mirror Circuit th-cam.com/video/esMlNx5w9Jw/w-d-xo.html Voltage Regulator Design with BJT Darlington Transistors and Zener Diodes th-cam.com/video/ArisQp7V0Ac/w-d-xo.html I hope this explanation is helpful.

Thanks for watching and your follow-up question. This schematic is in a compact form (Hexagonal circuit structure) and as you noted is not the nominal straightforward (standard) way of drawing a circuit that would require a systematic flow of circuit components from left to right. The following videos are examples of third order and second order circuits with standard circuit schematic: th-cam.com/video/ENq39EesfPw/w-d-xo.html Analog Computer circuit that solves Differential Equation. th-cam.com/video/HeZRtnRXpEI/w-d-xo.html Quad Op Amps Analog Computer that solves a second order differential equation. I hope this clarification is helpful.

@@ЁбрагимИпатенкоибнАдхарма Thank you! Glad that you liked this Wilson Current Source Circuit video. Alternative design techniques are discussed in th-cam.com/play/PLrwXF7N522y48AAPxFaQlowim4-8gYoWz.html Hope this circuit playlist is interesting as well.

Glad that this op amp circuit of the Third-order Butterworth Filter Transfer Function is useful. I suggest you review your calculation of the first transfer function to make sure your derivation is correct. For more Op Amp Circuit and Analog Filter examples please see: Butterworth Analog Filter Design with Op Amp th-cam.com/video/SIMg5TOIgrA/w-d-xo.html Band-Pass Sallen-Key Filter Design th-cam.com/video/lsM09pcrRVE/w-d-xo.html Sallen-Key Filter Design Tutorial th-cam.com/video/KwUnQXbk7gM/w-d-xo.html Op Amp Analog Computer Differential Equation Solver th-cam.com/video/ENq39EesfPw/w-d-xo.html Universal Analog Filter Design th-cam.com/video/2J-0msXZE2o/w-d-xo.html Laplace Transform Example and S-domain circuit analysis: th-cam.com/video/ps8N5TPM_qU/w-d-xo.html

As explained in the video (see 25:00) by varying n (and m) we can control Damping Factor (zeta) and natural frequency (Omega) of the filter. That will allow us to control and adjust filter's response. Here are a few related examples: Laplace Transform Example and S-domain circuit analysis: th-cam.com/video/ps8N5TPM_qU/w-d-xo.html Op Amp circuit Bode Frequency plot th-cam.com/video/BLVzuuqAlZs/w-d-xo.html Bode Plot, Frequency Response, Transfer Function th-cam.com/video/vZFkPeDa1H8/w-d-xo.html Hope this explanation is helpful.

@umutaydn6184 You are very welcome. Glad that this Sallen-Key Filter Design Tutorial video is helpful. More Filter circuit videos are published in th-cam.com/play/PLrwXF7N522y7SvuzNrcDYAIwnBgC9ddjD.html Analog Filters playlist.

@@behzadabf Thank you. Glad that you liked this ECG amplifier video. Here are a few more examples, Digital Stethoscope Microphone Amplifier Explained th-cam.com/video/ez5KtkPbsHg/w-d-xo.html Thermometer Sensor Circuit Explained th-cam.com/video/5jmbZ9ak6EI/w-d-xo.html Strain Gauge Wheatstone Bridge Instrumentation Amplifier Explained th-cam.com/video/io1yBcCsP-Y/w-d-xo.html Thermometer Design with Op Amp and BJT transistor th-cam.com/video/55YsraFE0rg/w-d-xo.html Hope that these videos are interesting as well.

Thank you for sharing your thoughts on this circuit and op-amp. Please note that the op-amp's maximum allowed supply voltage is not violated in this design. The additional voltage drop across the series resistor in the supply rail will accommodate the extra voltage drop. On a separate note, there are indeed high-voltage operational amplifiers with push-pull output stages. An example is the Apex Microtechnology PA198 High Voltage Power Operational Amplifier, which delivers output voltages ranging from ±215V with a dual supply to +440V with a single supply configuration. For more information, see the APEX PA198 datasheet at this link www.mouser.com/pdfDocs/pa198u.pdf Additionally, you can find more examples of power amplifiers in the th-cam.com/play/PLrwXF7N522y4vWr-8XRBqxpi5idFDE9BV.html Power Amplifier and Voltage Regulator Video Playlist. I hope this is helpful!

I suggest a low noise Zero-Drift CMOS op amp from Texas Instruments or Analog Devices. TI's OPA4189 is a Quad OpAmp Chip that should be a good choice for this circuit. Additional choices are OPA4188 or Analog Devices LTC1151 www.ti.com/product/OPA4189 www.ti.com/product/OPA4188 www.analog.com/media/en/technical-documentation/data-sheets/1151fa.pdf I hope this is helpful.

Thank you for your interest in this ECG video. Regarding the Op Amp, I highly recommend using zero drift ultra low noise CMOS Op Amp from Texas Instrument or Analog Devices. For example AD8630 Zero Drift Quad Op Amp from Analog Devices or Texas Instrument's TLV4387 Quad, ultra-high-precision (10 μV) zero-drift (0.01 μV/°C) low-input-bias-current op amp.

Thank you! Glad that you liked my video. A few more examples that worths checking: Push-Pull Power Amplifier Design with Op Amp th-cam.com/video/8BFzsi7-Vbs/w-d-xo.html High Voltage Push-Pull Power Amplifier with Darlington transistors th-cam.com/video/C9Hse91r5sA/w-d-xo.html I hope these videos are interesting as well.

@@brunoluis2237 Glad that this Multiplier Circuit is useful. Here are related circuit videos: Analog Computer Power Raiser V1^V2 th-cam.com/video/Qe3cY9JSzYg/w-d-xo.html Square Root Analog Computer Circuit th-cam.com/video/Dct5lwQyDuc/w-d-xo.html Analog Voltage Divider th-cam.com/video/2axI5l3cf6c/w-d-xo.html Logarithm Amplifier (Log calculator analog circuit) th-cam.com/video/yMXFzwnk_1w/w-d-xo.html I hope these videos are interesting as well.

You are welcome! Glad that you liked my EKG amplifier video. Here are a few more examples, Digital Stethoscope Microphone Amplifier Explained th-cam.com/video/ez5KtkPbsHg/w-d-xo.html Thermometer Sensor Circuit Explained th-cam.com/video/5jmbZ9ak6EI/w-d-xo.html PhotoDiode Amplifier with Data Compression Explained th-cam.com/video/hqrRx2ufAwg/w-d-xo.html Thermometer Design with Op Amp and BJT transistor th-cam.com/video/55YsraFE0rg/w-d-xo.html Strain Gauge Wheatstone Bridge Instrumentation Amplifier Explained th-cam.com/video/io1yBcCsP-Y/w-d-xo.html Thermometer Current source th-cam.com/video/Ggf0yCaTTiY/w-d-xo.html I hope these videos are interesting as well.

Thanks for watching and sharing your thoughts. Your statement "If R1=0 then Vin=VT" is Not correct. In case helpful, here is another example: th-cam.com/video/kk2c7Gk3nW4/w-d-xo.html which is Anti-Log with Temperature Compensation. It is important to recall that the voltage division realized using R1 and R2 is practically required to scale the input voltage to a range of say +/-100 mV (or less) that can modulate the 0.7v default Vbe of active BJT transistors. With that in mind, whenever (in a given application) Vin is natively guaranteed to be a small enough voltage (within +/-100mV) then we may set R1=0 and that then simply results in the positive input terminal of the first op amp to have the same voltage as Vin meaning V+ = Vin. But that does not mean Vin=VT (note that VT is the thermal voltage of P-N junction that is ~26 mV in room temperature). I hope this explanation is helpful.

@@STEMprof Thanks but that link didn't help me understand since it is not the same circuit. In this circuit, VT is the voltage on the right side of R1, yes? So if R1 is zero then Vin and VT have to be the same, yes? and if your final equation is correct then the result of that final equation becomes a constant, yes? Is that equation not completely correct? What am I missing?

VT is not the voltage at the right side of R1. Input voltage Vin & R1, R2 define what will be at the right side of R1.

@@STEMprof Ahhh!!!! I understand now, its V+, sorry it looked kind of like a VT

Alright, glad that the clarification was helpful.

Glad that you like my videos and circuit design examples. I understand, I had similar limitations back in my early days as well. Regarding your question, this is not a marketing theme. A digital stethoscope requires initial analog signal conditioning (filtering and amplification) before analog to digital conversion for further digital post processing. In case you are interested here are additional practical examples: Electric Guitar Amplifier to XLR Audio Signal th-cam.com/video/X4y8cwZdGEk/w-d-xo.html Bridge Audio Amplifier Explained th-cam.com/video/EDpu6urAtHA/w-d-xo.html Thermocouple Amplifier with Cold Junction Compensation Explained th-cam.com/video/-BsDLBI166U/w-d-xo.html Electronic Gain Control for Op Amp Amplifier th-cam.com/video/NoNgQpbj77Y/w-d-xo.html VCA Electronic Gain Control (Part 1): Voltage-Controlled Attenuator Overview th-cam.com/video/cFzYZsPEtP0/w-d-xo.html I hope these circuit videos are interesting as well.

Your are welcome! Glad that you like this EKG video. Yes, a nominal value of 100k Ohm is fine. I highly recommend checking your Op Amp data sheet regarding input impedance and offset requirements of your operational amplifier. For more sensor videos see the Sensors Amplifiers playlist th-cam.com/play/PLrwXF7N522y7Ut9bm8TXAOhIWqL__FGlj.html

You are welcome. As shown on the circuit schematic in the video, alpha is the ratio of the first Op Amp feedback resistor to the R2 resistor connected between input of op amp and ground.

Sure, it is done in th-cam.com/video/997hfjGK3_w/w-d-xo.html which is Electric Circuit Analogy for Spring-Mass-Damper Mechanical System. See the three methods discussed in the video.

@@nataphongthip You are welcome. Glad that this Push-Pull Power Amplifier video is helpful. Another example is Push-Pull Amplifier with Darlington and Sziklai Transistors th-cam.com/video/866MYibo8yE/w-d-xo.html More analog circuits & Op Amp examples are in the Power Amplifiers & Voltage Regulators Video Playlist th-cam.com/play/PLrwXF7N522y4vWr-8XRBqxpi5idFDE9BV.html I hope these Circuit videos are interesting as well.

@@t1d100 You are welcome! Glad that this video is helpful. Another interesting example is: 1x, 10x, 100x, 1000x Switched-Gain Instrumentation Amplifier th-cam.com/video/9-MLqyewXW8/w-d-xo.html For more videos about Sensors and Instrumentation Amplifiers see the playlist th-cam.com/play/PLrwXF7N522y7Ut9bm8TXAOhIWqL__FGlj.html I hope these videos are interesting as well.

@@animeditstv2628 Glad that you like this three-set proof of the principle of inclusion exclusion. More videos are posted in Computer Science, Math video Playlist th-cam.com/play/PLrwXF7N522y5BUuxzrned72krSgRWhTOh.html

Thank you for your interest! Glad that you liked my videos and happy to hear that they are useful. I have been working on two books. The first one is scheduled to be published early next year. I'll post an announcement in this channel as soon as they are published. Please stay tuned! :)

@@josemanuelromo6832 thanks for your good question. This problem states that the minimum load resistor allowed is 10 Ohm. Hence the max output current is 1.2 Ampere that ONLY happens when output voltage of regulator is 12 Volts (meaning alpha is one). When alpha is set to zero, then output voltage is 6 Volts which results in max output current of 600 mA when minimum allowed load resistor of 10 Ohm is applied. I recommend that you also watch minute 21:10 onward that further explains by showing and plotting the Transistor Power Formula. I hope this is helpful.

Test were surely more rigorous back then focusing on fundamentals and intuition. Glad that this circuit is useful. A few related Circuit Videos to see: Analog Computer Power Raiser V1^V2 th-cam.com/video/Qe3cY9JSzYg/w-d-xo.html Analog Logarithm Computer with Op Amp th-cam.com/video/RpKEq5WyoLg/w-d-xo.html Analog Exponential (Anti-Log) computer th-cam.com/video/kk2c7Gk3nW4/w-d-xo.html I hope these videos are interesting as well.

Thanks for watching and sharing your thoughts. Glad that you liked this Sawtooth Signal Generator video. You have a good practical point. Having a typical 100 Ohm current limiting resistor at the output of operational amplifier is generally a safe and recommended practice. Otherwise Op Amp selection should be carefully done so that there is an internal max current protection included in the design of output stage of the op amp. Here are a few more Oscillator Design and OpAmp Circuit examples: Triangle Oscillator Op Amp circuit th-cam.com/video/JF5Up_cuL9k/w-d-xo.html Sawtooth Oscillator Design th-cam.com/video/2eUsGPfqbW4/w-d-xo.html Complex Sinusoid Oscillator th-cam.com/video/GXRhmwmS5Zk/w-d-xo.html Sawtooth Signal Generator design with OpAmp, JFET, BJT Transistors th-cam.com/video/5zHXTx-Vl20/w-d-xo.html Hope you find these examples interesting as well.

You're welcome! and Glad you liked this circuit video. For more Filter examples see Universal Analog Filter Design with Op Amps th-cam.com/video/2J-0msXZE2o/w-d-xo.html Analog Filter Frequency Response and Poles and Zeros in Bode plot th-cam.com/video/Ga8Lu8sWlAk/w-d-xo.html All-In-One BiQuad Analog Filter: LPF HPF BPF th-cam.com/video/jEIQ2Twvthk/w-d-xo.html Analog Filter with Op Amps: Find Voltage Transfer Function th-cam.com/video/AJEyXOCj8Q4/w-d-xo.html I hope you enjoy these videos as well.

@@STEMprof 🤣🤣 read that again lol, bots at work 🫡

Thank you 🙂

Ok, Opa443 .... But there is a problem with noise in all of them.... ~20 nV/√(Hz) with amplitude from 1V to 60V max all that noise becomes ~1200 nV/√(Hz). In comparison to normal low noise audio OpAmps with noise below ~4..5 nV/√(Hz) Some of them even near only ~1 nV/√(Hz) But all of them have ±18V max. What to do if we need to get low noise 50W power amplifier?

Thanks for your interest in this Push-Pull Power Amplifier Design example. Yes, OPA443 is a high-voltage +/-40V Rail operational amplifier. Careful selection of Op Amp is important in high power amplifier design. For another example see Push-Pull Power Amplifier with Darlington and Sziklai Transistor pairs th-cam.com/video/866MYibo8yE/w-d-xo.html

@@Arwndr Thanks for watching & sharing your thoughts. You have a good point. Noise, drift and input offset are all valid and important concerns in op amp based circuit designs. A good designer need to identify the right operational amplifier that satisfies a practical balance among conflicting set of requirements including cost, power, Noise, drift and input offset. If need be a proper Filtering stage possibly additional feedback might be needed at the input or output of the circuit to further control or remove noise. Here is another example th-cam.com/video/C9Hse91r5sA/w-d-xo.html which is a Push-Pull Power Amplifier design (High Voltage) with Op Amp and Darlington Transistors.

​@@STEMprof Thank you for big open feedback, and links for another solutions. I will study them all. Also, us I understand, to much of feedback filtering for reducing Noise, - also reduce some parts and pieces of original audio. That's why I look for less noisy components where it possible, so to not create an extra problems with Noise so to have less deal with it in post, with some inevitable concomitant quality loss of original audio source. Thanks for your channel! Very helpful for self education 🙌🏻👍🏻👌🏻🤝🏻

You're very welcome! Glad you like my videos. Thank you!

Thanks Matt for your good summary of interesting features of this two-OpAmp Instrumentation Amplifier. Glad that this design is interesting and useful. Here are a few more videos about different features and types of instrumentation amplifiers: 1x, 10x, 100x, 1000x Switched-Gain Instrumentation Amplifier th-cam.com/video/9-MLqyewXW8/w-d-xo.html Variable Gain Instrumentation Amplifier for designing Thermometer Current Source th-cam.com/video/Ggf0yCaTTiY/w-d-xo.html Instrumentation Amplifier Explained th-cam.com/video/Yq767et8BbY/w-d-xo.html Gain & CMRR of Instrumentation Amplifier Explained th-cam.com/video/Bw29QjHHzGo/w-d-xo.html Instrumentation Amplifier design with Differential Instrumentation Amplifier with BJT transistor th-cam.com/video/2xJpqfexsPg/w-d-xo.html I hope these videos are interesting as well. Thank you.

@gwen-ip4qg You're very welcome! Glad that this Excel Tutorial Video is helpful. Best wishes with your Biology Assignment 🙏

You are welcome! And thanks for sharing your observations and insights. Here is another example that you might find interesting, Sallen-Key Filter Design Tutorial: LPF, HPF Frequency Response, Damping Factor th-cam.com/video/KwUnQXbk7gM/w-d-xo.html

That's a good question. One quick way to derive R4/R3>R1/R2 is: For stable circuit we need dominant negative feedback in this circuit. That means the contribution of output voltage Vo to the negative input terminal R4/(R3+R4)*Vo should be larger than its contribution to positive input terminal R1/(R1+R2)*Vo. Therefore R4/(R3+R4)>R1/(R1+R2) which results in R4/R3>R1/R2. I hope this circuit video and explanation are helpful.

@@STEMprof ok, thank you for your answer and wish you good luck.

@@谢泽文-n4x You are welcome, Thank you! Glad that the clarification was helpful.