Full Report for Experiment Nr. 2: REACTIVE COMPONENTS, RESONANCE AND FILTERS

SummaryFilters are very important circuits used in signal processing. This experiment shows what properties on the signal has L-R low pass filter and the analyzed behavior on signals with different frequencies. The prediction was that it should not change low frequency signals and block those with high frequency. On the contrary, the C-R circuit should block all low frequencies and let the high ones pass. The third circuit connected with L-R-C components should let to go through some small range of frequencies.

Contents

1. Introduction

2. The C-R Circuit analysis 2.1 Analysed C-R circuit and the transfer functions2.2 Representation of gathered data2.3 Cutoff frequency3. The L-R Circuit analysis3.1 Analysed L-R circuit and the transfer functions3.2 Representation of gathered data3.3 Cutoff frequency4. The L-R-C Circuit analysis4.1 Analysed L-R-C circuit and the transfer functions4.2 Representation of gathered data4.3 Cutoff frequency

5. Conclusions

6. Theory

7. References

Introduction

This experiment purpose was to analyze the behavior of L-R circuit (low pass filter) and to find out what effect it has on signals of different frequency. During experiment the measurements were taken using oscilloscope, input signal were supplied by waveform generator and all the components that were used in the circuit were tested with LCR Bridge to find out their real resistance or inductance values.The C-R Circuit analysis

Figure 1: The C-R circuit

The circuit was set up with 1466 resistor and 14.08 nF capacitor. Input Peak-to-Peak voltage value was kept around 4 V and measurements of output voltage were taken while varying the frequency from 200 Hz to 20 kHz. The total impedance of the circuit is: Derived transfer function for the L-R circuit in the frequency domain: Derived transfer function for the L-R circuit in the Laplace domain: = = Equation for phase of a transfer function:

Graph of the phase shift of a transfer function. The blue line in the graph shows theoretical curve and the red line depicts measured values:

Figure 2: The C-R circuit phase of H(f) graph

Graph of the amplitude of a transfer function. The blue line in the graph shows theoretical curve and the red line depicts measured values:

Figure 3: The L-R circuit module of H(f) graph

As it can be seen from the graph only low frequency signals are at almost the same voltage level as the input and the amplitude of transfer function is close to one.

Cutoff frequencyCutoff frequency is given by

This frequency is shown by a red circle on the graphs.For our circuit this was: For our circuit the measured value was: 8.2523 KHzTheL-R Circuit analysis

Figure 4: The L-R circuit graph

The circuit was set up with 98.49 resistance and 0.9481 mH. Input Peak-to-Peak voltage value was kept around 4 V and measurements of output voltage were taken while varying the frequency from 50 Hz to 50 kHz. The total impedance of the circuit is: Derived transfer function for the L-R circuit in frequency domain: Derived transfer function in Laplace domain:

Equation for phase of a transfer function:

Graph of the phase shift of a transfer function. The blue line in the graph shows theoretical curve and the red line depicts measured values:

Figure 5: The L-R circuit phase of H(f) graph

Graph of the amplitude of a transfer function. The blue line in the graph shows theoretical curve and the red line depicts measured values:

Figure 6: The L-R circuit module of H(f) graph

As it can be seen from the graph only low frequency signals are at almost the same voltage level as the input and the amplitude of transfer function is close to one.

Cutoff frequency

Inspection of transfer function shows that, when 1

And the cutoff frequency is given by

This frequency is shown by a red circle on the graphs.For our circuit this was expected to be And the measured value was

The L-R-C Circuit analysis

Figure 7: The L-R-C circuit

The circuit was set up with 54.88 resistance and 0.968 mH and 44.05 nF. Input Peak-to-Peak voltage value was kept around 4 V and measurements of output voltage were taken while varying the frequency from 5k Hz to 60 kHz. The total impedance of the circuit is: Derived transfer function for the L-R-C circuit in frequency domain: Derived transfer function for the L-R-C circuit in the Laplace domain: H= = =

Equation for phase of a transfer function:

Graph of the phase shift of a transfer function. The blue line in the graph shows theoretical curve and the red line depicts measured values:

Figure 8: The L-R-C circuit phase of H(f) graph

Graph of the amplitude of a transfer function. The blue line in the graph shows theoretical curve and the red line depicts measured values:

Figure 9 : The L-R-C circuit module of H(f) graph

As it can be seen from the graph only low frequency signals are at almost the same voltage level as the input and the amplitude of transfer function is close to one.

Cutoff frequency

And the cutoff frequency is given by

This frequency is shown by a red circle on the graphs.The expected cut off frequency for our circuit was:= 23.215 kHzAnd the measured value was:= 25.012 kHz

ConclusionAs it can be seen from the graphs the theoretical curve and the one obtained by measurements are very similar. Low pass filter acts as predicted, it passes through low frequency signals without significantly changing their amplitude, controversially high frequency signals are diminished and they cannot pass through this L-R circuit. All the theory also can be seen to hold when analyzing R-C circuit, only the high frequency signals can pass and the low frequency signals cannot get through. The L-R-C circuit a.k.a. Band pass filter only approximately 20-25 kHz signals.Theory

To analyze our circuits we used phasors. In physics and engineering, a phasor, is a complex number representing a sinusoidal function whose amplitude (A), frequency (), and phase () are time-invariant. It is a special case of a more general concept called analytic representation. Phasors separate the dependencies on A, , and into three independent factors. This can be particularly useful because the frequency factor (which includes the time-dependence of the sinusoid) is often common to all the components of a linear combination of sinusoids. In those situations, phasors allow this common feature to be factored out, leaving just the A and features. A phasor may also be called a complex amplitude andin older textsa phasor is also called a sinor or even complexor.

With phasors, the techniques for solving DC circuits can be applied to solve AC circuits. A list of the basic laws is given below. Ohm's law for resistors: a resistor has no time delays and therefore doesn't change the phase of a signal therefore V=IR remains valid. Ohm's law for resistors, inductors, and capacitors: V = IZ where Z is the complex impedance.ReferencesThis report used a lot of information from Electrical and electronic Engineering Linear Circuits laboratory handbook