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:

1. CRO (Dual Channel  0-20 MHz)                     -           1 No.
2. Signal Generator ( 1Hz to 1 MHz)                  -           1No.
3. Decade capacitance box                              -           1 No.
4.  Resistor (100  K)                                      -           1 No.
5. Connecting wires        
6. Bread board

Circuit Diagrams:

Fig 1.1 High Pass RC circuit	Fig 1.2 Low Pass RC circuit

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., 

i) High Pass RC circuit   ii) 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 and T is time period of input signal.

Let RC = 10 T,

Choose R = 100k, f = 1kHz.

C = 10 / (103X 100X103 ) = 0.1µf

2. Medium time constant: RC = T

C = T/R = 1/ (103X100X103 ) = 0.01µf

3. Short time constant: RC << T

RC = T/10 => C = T/10R = 1/(10X103X100X103)  = 0.001 µf.

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

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

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. RC = T

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

3. 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 V₀₁, V_O2 during T₁ and T₂ is given by following equations.

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

1. RC = T

2. RC >> T

3. 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 V_P-P, f = 1KHz)
3. Observe the output waveform for (a) RC = T, (b) RC<<T, (c) RC>>T
4. Verify the values with theoretical calculations.

Observations:

Result:

1. The response of Low pass and High pass RC circuits have been verified for non-sinusoidal inputs.
2. Verified the theoretical and practical values of %P.

Viva Questions:

1. What is linear wave shaping?
2. Define Time constant.
3. Define %tilt and rise time.
4. When High pass RC circuit is used as Differentiator?
5. When Low pass RC circuit is used as Integrator?
6. What is the Difference between Low pass and High pass RC circuits.
7. Why High pass circuit blocks the DC signal?
8. What is rise time and what is its value.
9. Explain 3 dB value for a LP and HP circuit.
10. A differentiator converts a square wave into what form? An integrator converts a square wave into what form?
11. What is the formulae for charging a capacitor from an initial voltage of Vi to a final voltage of Vo.
12. Instead of using RC components for a  low pass or high pass, how the circuit changes , if we want to use RL components?
13. What is double differentiation?
14. What are units of time constant?

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.