RC Circuit Transfer Functions with Bode Diagrams

RC Circuit Transfer Functions with Bode Diagram Magnitude and Phase Diagrams simulated in National Instruments (NI) Multisim

1. Katrina Little
Matt
Station :3B

Experiment #8:
Transfer Functions

2. Objective:
The objective of the experiment was to study the transfer functions of various RC networks and find the
frequency of the output voltage at certain phase angles.


Equipment:
 Oscilloscope
 Function Generator
 Frequency counter
 Resistors
 Capacitors


Background:
A two- port linear network can be in general described by a transfer function :

given a sinusoidal input voltage:
,
the output voltage will be :

where |H(jwo)| and ϴ(wo) are the magnitude and phase of the transfer function at frequency
wo.



Preparation:
Selected Circuit Values:
 C = 0.1µF
 R = 1000 Ω

A. Refer to the circuits in figures 1.a and 1.b

3. 1) For each circuit, find the sinusoidal steady- state transfer function
2) Assume that the input is a sinusoid. Find the frequency where the output voltage will be 45® out of
phase than the input voltage. Find the amplitude of the output voltage at that frequency.
3) Find the frequency where the amplitude of the voltage is ¼ of the amplitude of the input voltage.

B. Refer to the circuits in figures 2 and 3.







1) Verify that the transfer function is the one given in each figure
2) Assume sinusoidal input. Find the frequencies where the phase difference of input and output voltages is
either +90° or -90°. Find the amplitude of the output voltage at these frequencies.
3) Repeat part (2) for 0° and 180° phase shift.

4. Simulation:

Run simulations to verify the theoretical results found in the preparation. The amplitude of the sinusoidal
input is 10V.

FIGURE 1A:

5. FIGURE 1.B:

6. FIGURE 2A:

7. FIGURE 2B:

8. FIGURE 3:

9. Experiment:
a) Connect the circuits in figures 1.a and 1.b. Use component values as in the preparation and
simulations
1.a:


1.b:

10. b) Experimentally verify the results you have calculated in parts A.2 and A.3 of the prep. What is the
maximum phase shift that can be achieved with these circuits? Are there any limitations in
achieving this phase-shift?
c) Connect the circuits of figures 2 and 3, with the component values as in the prep.
2.a:

2.b:

11. Fig 3:


d) Experimentally verify the results you have calculated in parts B.2 and B.3 of the prep.
Results:
Simulated Measured Calculated Phase- Shift
Figure 1.a
Frequency (Hz) 1607 1465 1592 45°
Output Voltage (V) 7.03 7.36 7.07
1/4 Amplitude Frequency 6201 6355 6164
Figure 1.b
Frequency (Hz) 1554 1325 1592 45°
Output Voltage (V) 6.97 6.32 7.07
1/4 Amplitude Frequency 408 422 411.4
Figure 2.a
Frequency (Hz) 1628 1605 1591 -90°
Output Voltage (V) 3.25 2.96 2.92
Frequency (Hz) 0° as f→ 0 Hz -180° as f→∞ Hz
Figure 2.b
Frequency (Hz) 1554 1405 1450 90°
Output Voltage (V) 2.92 2.8 2.92
Frequency (Hz) 180° as f→ 0 Hz 0° as f→∞ Hz
Figure 3
Frequency (Hz) 3529 3295 3300 90°
Output Voltage (V) 3.57 3.36 3.57
Frequency (Hz) -90° as f→ 0 Hz 0° as f→∞ Hz

12. Conclusions:

 Referring to the results section, the experiment was conducted accordingly because the values very ,
closely match up.

 For Figures 2.a, 2.b , and 3 it is important to note that for part d) of the experiment : “Find the
frequencies where the phase shift is either 0 or 180°” that it is not possible to get a definitive value for
a frequency at those particular phase angles. That is because the frequency approaches a value (see
the results table for clarification).

The amplitude of the output voltage is greatest across the circuit in Figure 3.


 For the circuits in Figures 1.a & 1.b:

“What is the maximum phase shift that can be achieved with these circuits?”
The maximum phase shift these circuits can reach is 90° because as the phase shift approaches 90° the
output voltage gets very small. So anything past 90° is unattainable.

When the output voltage is connected across the capacitor (as compared to across the resistor) of
these two series RC circuits, the amplitude is slightly larger.

Also, the frequencies where the amplitude of the output voltage is ¼ the amplitude of the input
voltage measurements varies greatly.

 The experiment further expanded our knowledge of transfer functions in parallel and series RC circuits
by measuring the amplitude of the output voltage and frequencies at different phase shifts between
the input/output voltages.