Linear Wave Shaping

Prior to Lab session:

1. Study the working principle of high pass and low pass RC circuits for non-sinusoidal signal inputs.
2. Study the definitions of % tilt, time constant, cut-off frequencies and rise time of RC circuits.
3. Study the procedure for conducting the experiment in the lab.

Objectives:

1. To design High pass and Low pass RC circuits for different time constants and verify their responses for a square wave input of given frequency.
2. To find the % tilt of high pass RC circuit for long time constant.
3. To study the operation of high pass RC circuit as a differentiator and low  pass circuit as an integrator.

Apparatus:

Circuit Diagrams:

Theory:

Resistors and Capacitors can be connected in series or parallel in various combinations. The RC circuits can be configured in two ways as shown above circuit diagrams. i.e., 

1. High Pass RC circuit
2. Low Pass RC circuit

High Pass RC circuit:

The reactance of the capacitor depends upon the frequency of operation. At very high frequencies, the reactance of the capacitor is very low. Hence the capacitor in fig.1.1 acts as short circuit for high frequencies. As a result the almost entire input appears at the output across the resistor.

At low frequencies, the reactance of the capacitor is very high. So the capacitor acts as almost open circuit. Hence the output is very low. Since the circuit allows only high frequencies, it is called as high pass RC circuit.

High - pass RC circuit as a differentiator:

In high pass RC circuit, if the time constant is very  small in comparison with the time required for the input signal to make an appreciable change, the circuit is called  a “Differentiator”. Under these circumstances the voltage  drop across R will be very small in comparison with the drop across C. Hence we may consider that the total  input Vi appears across C. So that the current is determined entirely by the capacitor. 

i = C dVi/dt.

The output voltage across R is, Vo  = RC (dVi/dt).

i.e., The output voltage is proportional to the differential of the input. Hence the high pass RC circuit acts as a differentiator when RC <<  T.

Low Pass RC circuit:

The reactance of the capacitor depends upon the frequency of operation. At very high frequencies, the reactance of the capacitor is almost zero. Hence the capacitor in fig.1.2 acts as short circuit. As a result, the output will fall to zero.

At low frequencies, the reactance of the capacitor is infinite. So the capacitor acts as open circuit. As a result the entire input appears at the output. Since the circuit allows only low frequencies, it is called as low pass RC circuit.

Low - Pass RC circuit as an integrator:

In low pass circuit, if the time constant is very large in comparison with the time required for the input signal to make an appreciable change, the circuit is called an  “integrator”. Under these circumstances the voltage drop across C will be very small in comparison to the drop across R and almost the total input Vi appears across R .i.e., i = Vi/R.

?The output signal across C is  

i.e., The output is proportional to the integral of the input. Hence the low pass RC circuit acts as a integrator for RC >>  T.

Design:

RC high pass circuit:

1. Large time constant: RC >> T;  Where RC is time constant

Let RC = 10 T, T is time period of input signal Choose R = 100kohms, f = 1kHz.

C = 10 / (103 X 100 X 103) = 0.1 microfarads

2. Medium time constant: RC = T

C = T/R = 1/ (103 X 100 X 103) = 0.01 microfarads

3. Short time constant: RC << T

RC = T/10 => C = T/10R = 1/(10 X 103 X 100 X 103)  = 0.001 micro farads

RC low pass circuit:   (Design procedure is same as RC high pass circuit)

1. Long time constant         :   RC >> T, C = 0.1 micro farads
2. Medium time constant    :   RC = T, C = 0.01 micro farads
3. Short time constant        :   RC = T/10, C = 0.001 micro farads

Expected output wave forms of High pass RC circuit for square wave input:

Consider the input at V1 during T1 and V11 during  T2 then the voltages V1, V11, V2, V2 1 are given by following equations.

	V11- V2 = V
	V1-V21 = V

For a symmetrical square wave

and because of symmetry V1= -V2  and V11= -V21 

The percentage tilt ‘P’ is defined by P= (V1-V11) / (V/2) X 100   ------------ (1.1)

Input wave Form

1. RC = T

1. RC >> T ( RC = 10T)

1. RC << T (RC = 0.1T)

Expected output wave forms of Low pass RC circuit for square wave input:

Consider the input at V1 during T1 and V11 during T2 then the voltages V1, V11, V2 , V2 1 are given by following equations.

For a symmetrical square wave V2= V/2(tanhx) and V1= -V2 where   x = T/(4RC)

1. RC = T

1. RC >> T

1. RC << T

Procedure:

1. Connect the circuit as shown in figure (fig.1.1 and fig 1.2).
2. Apply the Square wave input to this circuit (Vi = 2 VP-P, f = 1KHz)
3. Observe the output waveform for
   1. RC = T
   2. RC << T
   3. RC >> T
4. Verify the values with theoretical calculations.

Observations:

Result:

1. The responses of Low pass and High pass RC circuits have been verified for square wave inputs for different time constants.
2. Verified the theoretical and practical values of %P.
3. Observed the operation of differentiator and integrator circuits.

Viva Questions:

1. What is a linear network? What is linear wave shaping?

Ans: A linear network is a network made up of linear elements only.The principle of superposition and the principle of homogeneity hold good for linear networks.

Linear wave shaping: it is the process whereby the form of non sinusoidal signal is altered by transmission through a linear network.

2. Define Time constant. What is its formula?

Ans: The time constant of  a circuit is defined as the time taken but the output to rise to 63.2% of the amplitude of the input step T = RC.

3. Define %Tilt and Rise Time. Write the expressions for the same.

Ans: %Tilt: %Tilt is defined as decay in the amplitude of the outut voltage wave due to the inut voltage maintaining constant level.

%Tilt = V1 –V11/(V/2) ×100

Rise Time: The rise time t

 is defined as the time taken by the output voltage waveform of a low pass circuit excited by a step input to the rise from 10% to 90% of its final value.

The rise time of the output of a low pass circuit excited by a step input is given by 

t

 = 2.2RC = 0.35/ f2 = 0.35 / BW 

4. When High pass RC circuit is used as Differentiator? What is the for mula for the output, when operated as differentiator?

Ans: The high pass RC circuit acts as a differentiator provided the RC time constant of the circuit is very small in comparison with the time required for the input signals to make an appreciable change.

Vo (t) = RC  dvi(t) / dt

5. When Low pass RC circuit is used as Integrator? What is the formula for the output, when the circuit is operated as Integrator?

Ans: If the time constant of an RC low pass circuit is very large in comparison with the time required for the input signal to make an appreciable change, the circuit acts as an integrator.

Vo (t) = 

t2 / 2RC

6. What is the Difference between Low pass and High pass RC circuits.

Ans: Low Pass: A Low pass circuit is a circuit which transmits only low frequency signals and attenuates or stops high frequency signals.

High Pass: A High pass circuit is a circuit which transmits only high frequency signals and attenuates or stops low frequency signals.

7. A Capacitor blocks __ signal and passes __ signal. The voltage across the ___ will not change suddenly.

Ans: A Capacitor blocks DC signal and passes AC signal. The voltage across the capacitor will not change suddenly. 

8. Explain 3 dB values for a LP and HP circuit.

Ans: 

High Pass 

Low Pass 

9. A differentiator converts a square wave into what form? An integrator converts a square wave into what form?

Ans: A differentiator converts a square wave into spikes waveform An integrator converts a square wave into triangle waveform.

10. What are the formulae for charging a capacitor from an initial voltage of Vi to a final voltage of Vo.

Ans: Vo = Vf + (Vi - Vf) e-T/RC

11. Instead of using RC components for a low pass or high pass, how the circuit changes , if we want to use RL components? What are the values for the Time constant for RL circuits?

Ans: T = L/R

12. When a capacitor in a low pass circuit charges to 99.3 % ( treated as fully charged) for a step input to a Low pass filter?

Ans: V0 = Vi

13. What is a peaking circuit?

Ans: The process of converting pulse into pips by means of a circuit of very short time constant is called peaking.

14. What is a ringing circuit?

Ans: A ringing circuit is a circuit which can proved as nearly and un-damped oscillations as possible.

15. Why resistive attenuators are to be compensated?

Ans: A resistive alternator needs to be compensated in order to reduce or eliminate the rise time of the output waveform.

Design Problems:

1. Design RC Differentiator circuit for frequency of 2kHz.
2. Design RC high circuit for a square wave input signal of frequency 2.5KHz for
   1. RC = 10T
   2. RC = T
   3. RC = T/10
3. Design low pass circuit for a square wave signal of 3KHz for
   1. RC = 5T
   2. RC = T
   3. RC = T/5
4. Verify the output of  circuits given in Fig1.1 and Fig 1.2 for input square wave of frequencies 10KHz and 500Hz.
5. Verify the RC high pass circuit output for sinusoidal input.

Outcomes: 

After finishing this experiment the students are able to

1. Design High pass and Low pass circuits with different time constants.
2. Find % Tilt
3. Observe the output waveforms for a given square wave.