First Order Circuits_Low PF and High PF.
- 1. Unit: II Chapter No: 05 First Order circuits Dr Sujata Sanjay Kotabagi Circuit Analysis 15EECC201 11/5/2020 School Of ECE 1
- 2. 11/5/2020 School Of ECE 2 The Low-Pass Filter Low-pass RC filter circuit: Permits low-frequency signals to pass from the input to the output while attenuating high frequency signals. At low frequencies, the capacitor has a very large reactance. Consequently, at low frequencies the capacitor is essentially an open circuit resulting in the voltage across the capacitor, Vout, to be essentially equal to the applied voltage, Vin. At high frequencies, the capacitor has a very small reactance, which essentially short circuits the output terminals. The voltage at the output will therefore approach zero as the frequency increases. 𝑿𝑪 = 𝟏 𝟐𝝅𝒇𝑪 ; 𝒂𝒕 𝒇 = 𝟎 𝒍𝒐𝒘 , 𝑿𝑪 = ∞; 𝒐𝒑𝒆𝒏 𝒄𝒊𝒓𝒄𝒖𝒊𝒕; 𝑽𝒐𝒖𝒕 = 𝑽𝒊𝒏 𝑿𝑪 = 𝟏 𝟐𝝅𝒇𝑪 ; 𝒂𝒕 𝒇 = ∞ 𝒉𝒊𝒈𝒉 , 𝑿𝑪 = 𝟎; 𝒔𝒉𝒐𝒓𝒕 𝒄𝒊𝒓𝒄𝒖𝒊𝒕; 𝑽𝒐𝒖𝒕 = 𝟎
- 3. 11/5/2020 School Of ECE 3 The Low-Pass Filter The cutoff frequency, 𝝎𝒄, as the frequency at which the output power is equal to half of the maximum output power (3 dB down from the maximum). This frequency occurs when the output voltage has an amplitude which is 0.7071 of the input voltage.
- 4. 11/5/2020 School Of ECE 4 The Low-Pass Filter Ideal Filter Response Curves Frequency Response of a 1st-order Low Pass Filter
- 5. 11/5/2020 School Of ECE 5 The Low-Pass Filter Frequency Response
- 6. 11/5/2020 School Of ECE-Prakalp18-22 6 Pole-Zero Plot
- 7. 11/5/2020 School Of ECE 7 1. Design a low-pass RC filter to have a cutoff frequency of 30 krad/s. Use a 0.01-mF capacitor. Examples
- 8. 11/5/2020 School Of ECE 8 2a. Write the transfer function for the circuit. 2b. Sketch the frequency response. Examples
- 9. 11/5/2020 School Of ECE 9 3. A Low Pass Filter circuit consisting of a resistor of 4k7Ω in series with a capacitor of 47nF is connected across a 10v sinusoidal supply. Calculate the output voltage ( VOUT ) at a frequency of 100Hz and again at frequency of 10,000Hz or 10kHz. Voltage Output at a Frequency of 100Hz. Voltage Output at a Frequency of 10,000Hz (10kHz). Examples
- 10. 11/5/2020 School Of ECE 10 The RC Integrator The Integrator is basically a low pass filter circuit operating in the time domain that converts a square wave “step” response input signal into a triangular shaped waveform output as the capacitor charges and discharges. A Triangular waveform consists of alternate but equal, positive and negative ramps. The RC time constant is long compared to the time period of the input waveform, the resultant output waveform will be triangular in shape and the higher the input frequency the lower will be the output amplitude compared to that of the input. 𝐑𝐂 ≫ 𝑻 𝝉 ≫ 𝑻 𝒗𝒐𝒖𝒕 = 𝟏 𝑪 𝟎 𝒕 𝒊 𝒕 𝒅𝒕 𝒊 𝒕 = 𝒗𝒊𝒏 𝑹 𝒗𝒐𝒖𝒕 = 𝟏 𝑹𝑪 𝟎 𝒕 𝒗𝒊𝒏 𝒕 𝒅𝒕 𝒗𝒐𝒖𝒕 ∝ 𝟎 𝒕 𝒗𝒊𝒏 𝒕 𝒅𝒕
- 11. 11/5/2020 School Of ECE 11 The RC Integrator RC Integrator Charging/Discharging Curves Fixed RC Integrator Time Constant An input pulse equal to one time constant, that is 1RC, the capacitor will charge and discharge not between 0 volts and 10 volts but between 63.2% and 38.7% of the voltage across the capacitor at the time of change.
- 12. 11/5/2020 School Of ECE 12 RC Integrator as a Sine Wave Generator The RC Integrator
- 13. 11/5/2020 School Of ECE 13 The High-Pass Filter The high-pass filter is a circuit which allows high-frequency signals to pass from the input to the output of the circuit while attenuating low-frequency signals. At low frequencies, the reactance of the capacitor will be very large, effectively preventing any input signal from passing through to the output. At high frequencies, the capacitive reactance will approach a short-circuit condition, providing a very low impedance path for the signal from the input to the output. 𝑿𝑪 = 𝟏 𝟐𝝅𝒇𝑪 ; 𝒂𝒕 𝒇 = 𝟎 𝒍𝒐𝒘 , 𝑿𝑪 = ∞; 𝒐𝒑𝒆𝒏 𝒄𝒊𝒓𝒄𝒖𝒊𝒕; 𝑽𝒐𝒖𝒕 = 𝟎 𝑿𝑪 = 𝟏 𝟐𝝅𝒇𝑪 ; 𝒂𝒕 𝒇 = ∞ 𝒉𝒊𝒈𝒉 , 𝑿𝑪 = 𝟎; 𝒔𝒉𝒐𝒓𝒕 𝒄𝒊𝒓𝒄𝒖𝒊𝒕; 𝑽𝒐𝒖𝒕 = 𝑽𝒊𝒏
- 14. 11/5/2020 School Of ECE 14 The High-Pass Filter
- 15. 11/5/2020 School Of ECE 15 The High-Pass Filter Ideal Filter Response Curves Frequency Response of a 1st-order High Pass Filter
- 16. 11/5/2020 School Of ECE 16 The input waveform to the filter has been assumed to be sinusoidal or that of a sine wave consisting of a fundamental signal and some harmonics operating in the frequency domain giving us a frequency domain response for the filter. However, if we feed the High Pass Filter with a Square Wave signal operating in the time domain giving an impulse or step response input, the output waveform will consist of short duration pulse or spikes as shown. The RC Differentiator 𝐑𝐂 ≪ 𝑻 𝝉 ≪ 𝑻 𝒗𝒐𝒖𝒕 = 𝒊 𝒕 𝑹 𝒊 𝒕 = 𝑪 𝒅𝒗𝒊𝒏(𝒕) 𝒅𝒕 𝒗𝒐𝒖𝒕 = 𝑹𝑪 𝒅𝒗𝒊𝒏(𝒕) 𝒅𝒕 𝒗𝒐𝒖𝒕 ∝ 𝒅𝒗𝒊𝒏(𝒕) 𝒅𝒕
- 17. 11/5/2020 School Of ECE 17 The RC Differentiator Each cycle of the square wave input waveform produces two spikes at the output, one positive and one negative and whose amplitude is equal to that of the input. The rate of decay of the spikes depends upon the time constant, ( RC ) value of both components, ( τ = R x C ) and the value of the input frequency. The output pulses resemble more and more the shape of the input signal as the frequency increases.
- 18. 11/5/2020 School Of ECE 18 RC Differentiator Output Waveforms
- 19. 11/5/2020 School Of ECE 19 1. A square pulse of 3 volt amplitude is applied to C-R circuit shown in the figure. The capacitor is initially uncharged. Calculate the output voltage at time t = 2 sec. For a R-C network, time constant, 𝝉 = 𝑹𝑪 = 𝟏𝟎𝟎𝟎 × 𝟎. 𝟏 × 𝟏𝟎−𝟔 = 𝟏𝟎𝟎 𝝁S Time duration of pulse, 𝑻𝒑 = 𝟐 𝑺 Settling time is given by, 𝒕𝒔 = 𝟓𝝉 = 𝟓 × 𝟏𝟎𝟎𝝁 = 𝟓𝟎𝟎𝝁𝑺 at time t = 2 sec. 𝑽𝒐 = −𝟑𝑽
- 20. 11/5/2020 School Of ECE 20