Circuit Analysis
15EECC201
Unit: II
Chapter No: 05
First Order circuits
Dr Sujata Sanjay Kotabagi
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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.
𝟏
𝑿𝑪 = ; 𝒂𝒕 𝒇 = 𝟎 𝒍𝒐𝒘 , 𝑿𝑪 = ∞; 𝒐𝒑𝒆𝒏 𝒄𝒊𝒓𝒄𝒖𝒊𝒕; 𝑽𝒐𝒖𝒕 = 𝑽𝒊𝒏
𝟐𝝅𝒇𝑪
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.
𝟏
𝑿𝑪 = ; 𝒂𝒕 𝒇 = ∞ 𝒉𝒊𝒈𝒉 , 𝑿𝑪 = 𝟎; 𝒔𝒉𝒐𝒓𝒕 𝒄𝒊𝒓𝒄𝒖𝒊𝒕; 𝑽𝒐𝒖𝒕 = 𝟎
𝟐𝝅𝒇𝑪
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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.
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The Low-Pass Filter
Frequency Response of a 1st-order Low Pass Filter
Ideal Filter Response Curves
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The Low-Pass Filter
Frequency Response
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Pole-Zero Plot
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Examples
1. Design a low-pass RC filter to have a cutoff frequency
of 30 krad/s. Use a 0.01-mF capacitor.
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Examples
2a. Write the transfer function for the circuit.
2b. Sketch the frequency response.
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Examples
3. A Low Pass Filter circuit consisting of a resistor
Voltage Output at a Frequency of
of 4k7Ω in series with a capacitor of 47nF is connected
10,000Hz (10kHz).
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.
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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.
𝒕
𝟏
𝒗𝒐𝒖𝒕 = 𝒊 𝒕 𝒅𝒕
𝑪 𝟎
𝒕
𝟏
𝒗𝒐𝒖𝒕 = 𝒗𝒊𝒏 𝒕 𝒅𝒕
𝝉≫𝑻 𝑹𝑪 𝟎
𝐑𝐂 ≫ 𝑻 𝒕
𝒗𝒐𝒖𝒕 ∝ 𝒗𝒊𝒏 𝒕 𝒅𝒕
𝟎
𝒗𝒊𝒏
𝒊 𝒕 =
𝑹
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.
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The RC Integrator
Fixed RC Integrator Time Constant
RC Integrator Charging/Discharging Curves
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.
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The RC Integrator
RC Integrator as a Sine Wave Generator
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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.
𝟏
𝑿𝑪 = ; 𝒂𝒕 𝒇 = 𝟎 𝒍𝒐𝒘 , 𝑿𝑪 = ∞; 𝒐𝒑𝒆𝒏 𝒄𝒊𝒓𝒄𝒖𝒊𝒕; 𝑽𝒐𝒖𝒕 = 𝟎
𝟐𝝅𝒇𝑪
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.
𝟏
𝑿𝑪 = ; 𝒂𝒕 𝒇 = ∞ 𝒉𝒊𝒈𝒉 , 𝑿𝑪 = 𝟎; 𝒔𝒉𝒐𝒓𝒕 𝒄𝒊𝒓𝒄𝒖𝒊𝒕; 𝑽𝒐𝒖𝒕 = 𝑽𝒊𝒏
𝟐𝝅𝒇𝑪
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The High-Pass Filter
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The High-Pass Filter
Frequency Response of a 1st-order High Pass Filter
Ideal Filter Response Curves
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The RC Differentiator
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.
𝒗𝒐𝒖𝒕 = 𝒊 𝒕 𝑹
𝒅𝒗𝒊𝒏 (𝒕)
𝒗𝒐𝒖𝒕 = 𝑹𝑪
𝒅𝒕
𝝉≪𝑻
𝐑𝐂 ≪ 𝑻
𝒅𝒗𝒊𝒏 (𝒕)
𝒗𝒐𝒖𝒕 ∝
𝒅𝒗𝒊𝒏 (𝒕) 𝒅𝒕
𝒊 𝒕 =𝑪
𝒅𝒕
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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.
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RC Differentiator Output Waveforms
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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
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