EE3101 Experiment 3

EE3101 LAB REPORT EXP#3 26 OCT 2006

Non-ideal Behavior of Electronic Components at High Frequencies and Associated Measurement Problems

Matt xxxxx

Student ID : xxxxxxxx

10/2/06 – 10/16/06

Abstract

Throughout this experiment we take the input and output measurements of given circuits at

various frequency rates. This is to demonstrate the frequency response of these circuits. In

other words, the circuits behave differently at different frequencies. At high frequencies we

can see the effects of shunt capacitance of the measurement terminals and interconnection

cables, resonance of the circuits, and the non-idealistic frequency behavior of passive

components. This experiment is designed to explore the response of circuits at high

frequencies and to modify the circuits to have the proper responses that are would be required

in electrical circuit design.

1


Introduction

AS opposed to dc or low frequencies, the characteristic of the circuits changes with each

component. This experiment focuses on the aspects of high frequency measurement divided

into three basic parts. We start with experiments to understand the characteristic of shunt

capacitance of the interconnection cables and the measuring instruments. Next we study about

the resonance in RLC circuits. Finally, we look at the non-ideal frequency behavior of passive

circuit components. All of the measurements throughout these three basic parts use similar

procedures. We vary the input frequency from low to high to gather the most important value,

either the resonant frequency or the break frequency of the circuit. These two characters are

results of the capacitance in the circuit that works as a short circuit at high frequencies.

,

The break frequency is when the output is the -3dB of the max value, and the resonant

frequency is the when the phase shift of the input to the output signal is zero. Throughout the

report you are able to see other characters as Q – factors. Through the following 7

experiments we are able to gather a broad understanding on high frequency responses

Body

Part 1 – Shunt Capacitance and the RC Compensator

Experiment 3.1

In the first experiment we measure the transfer function, which would be the gain, or | Vo

/ Vi | of the circuit. We take our measurements from low to high frequencies. We do this to see

the effects of the shunt capacitance that the oscilloscope and the interconnecting cables that

occur. We start at the low frequency of 1 kHz and go up to 1MHz. We construct circuit Figure

1 to measure the shunt capacitance. We could expect that, because of the shunt capacitance,

Vout will have a smaller value as the input voltage frequency increases.

2


Fig 3.1.1

1. Data & Results

f(hz) Vin Vout | Vo / Vi |

1000 5.000 2.380 0.476

5000 5.000 2.380 0.476

10000 4.940 2.280 0.462

28160 5.000 1.767 0.353

50000 5.000 1.300 0.260

100000 5.000 0.750 0.150

500000 4.820 0.167 0.035

1000000 4.820 0.099 0.021

Fig 3.1.2

Gain vs Frequency

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0 200000 400000 600000 800000 1000000 1200000

Frequency (Hz)

Gain

(V

ou

t/V

in)

Gain vs Frequency

3


Fig 3.1.3

As expected, we can see a decrease of gain that is result of the decrease in the Vout. This is

the effect of the shunt capacitance as it works as a short circuit.

Now we calculate the exact value of the shunt capacitance we need further calculations.

2. Shunt Capacitance

In order to calculate the shunt capacitance we look for the break frequency, which has the

information of the capacitor in the circuit:

, where R = R1||R2 = 50KΩ

Then we pick the closest value from our measurements, which would be:

So the shunt capacitance of the oscilloscope and the interconnecting wires are 113pF by

measurement. As you can see the shunt capacitance has a very small value. This is why we

can see the effects of the shunt capacitance only at the high frequency rates that make the

shunt capacitance work as a short circuit. This effect needs to be considered when we are

making measurements of circuits at high frequencies as mentioned.

Part 2 – Resonance in RLC Circuits

Experiment 3.2

4


We now build our second circuit (Fig 3.2.1) to see the effects of the resonance in the RLC

circuits that has its resonant frequency at 2 kHz.

Fig 3.2.1

Our two goals here are to:

1. Determine the resonant frequency and Q-factor.

2. Determine the impedance at the resonant frequency.

We design our circuit to have an R relatively very small R<<1kΩ in order to make the circuit

dependent more on the inductor and capacitor. So we put our R as 10Ω. For L we use a

100mH component from our circuit kit. To determine our capacitor value, we need further

calculations to make the resonant frequency at approximately 2 kHz:

For our resonant frequency fr = 2 kHz, and L = 100mH,

However, in our lab kit the closest value of capacitor we have was 100nF.

With a modified capacitance:

5


If we would have used a series of 100nF capacitance we could have gathered a resonant

frequency closer to 2 kHz. However with the resonant frequency at around 1.6 kHz, we are

still able to see the effect of the resonant frequency at high rates frequencies. In this

experiment we start with the low frequency at the input and increase the frequency to see the

effects.

We cannot rely on correct gain values in this situation because of the parasitic resistance

though the inductor and capacitor. Instead, we will look at the phase shift between Vin and

Vout. In this case, we know that out break frequencies will occur at phase shifts of +45° and -

45° and our resonant frequency will occur at a phase shift of 0°.

1. Data & Results

f(hz) Phase Vo -> Vin

1552 -45°

1796 45°

1670 0°

Fig 3.2.2

* Determine the resonant frequency and Q-factor.

By measurements circuit Figure 3.2.2 has its resonant frequency at 1.67 kHz because that is

where we have a phase shift of 0°.

If we compare our value with our theoretical value 1591.5 Hz, we can see we have evaluated a

reasonable result.

To determine the circuit’s Q-factor:

* Determine the impedance at the resonant frequency.

To determine the impedance of our circuit (Figure 3.2.1), we can use another formula for the

Q-factor and solve for Zin.

6


Since our Zin is R plus the parasitic resistances of the inductor and capacitor we can calculate

what the parasitic resistance is by subtracting 10 from it.

The calculation will be the following:

Experiment 3.3

We now modify our circuit (Figure 3.2.1) to have a Q-factor = 5. Then we drive the resultant

circuit with a 2 kHz square wave to compare the spectra of the input voltage and the current

waveforms.

1. To modify the circuit to have a Q-factor at 5:

with Q = 5, fr = 1.6kHz, and L = 100mH

If we look back to our circuit we see the parasitic resistance of the circuit and the original R.

So we should put a resistance of value of

We can do this by putting two 100 resistors in parallel to give 50 and then put a 10

resistor in series to make the equivalent resistance 60

We now modify the circuit to have a Q factor of 5 by making our circuit look like the

following:

7


Fig 3.3.1

2. Gather spectra of the input voltage and current waveforms.

Figure

3.3.2

8


Figure 3.3.3

Figure 3.3.2 would be the FFT of the input voltage, and Figure 3.3.3 would be the FFT of the

output voltage. Because the output voltage is a part of the input current we used the output

voltage of the circuit. If we compare the differences of the two spectra of the waveforms we

could see that while the input voltage (Figure 3.3.2) is a combination of several harmonics,

the current (Figure 3.3.3) has only one harmonic as an effect. The other harmonics could be

assumed to be cut off by the circuit. The combination of the input harmonics makes the input

to be a square wave, and the only harmonic on the output makes the wave to look as a

sinusoidal waveform.

Note: In order to gather a clear waveform, we turned on the Noise Rej., and put the

center of the FFT at 2 kHz with a Span of 50 kHz.

Experiment 3.4

We now determine the resonant frequencies of the circuit of Figure 3.3.1 for C values from

0.0001uF to 0.1uF to see the relationship between the capacitance of the circuit with its

resonant frequency. The circuit used in this experiment is identical to Experiment 3.2.

To make the experiment easy, I observed the circuit phase shift and took the 0° frequency with

different values of C.

Where fr calculated is:

9


C(uf) fr calculated (hz) fr measured (hz) % error

0.001 15916 16130 1.34%

0.01 5033 5066 0.66%

0.1 1592 1776 11.56%

Fig 3.4.1

Frequency vs Capacitance

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 0.02 0.04 0.06 0.08 0.1 0.12

Capacitance (uF)

Fre

qu

ency

(H

z)

fr calculated (hz)

fr measured (hz)

Fig 3.4.2

From the results in Fig 3.4.1, Fig 3.4.2, we can see that the resonant frequency is inversely

proportional to the square of the capacitance:

Experiment 3.5

Goals for experiment 5.

1. Design a parallel resonant circuit (with R = infinity) with a resonant frequency of 2

kHz.

2. Determine the Q of the circuit.

10


3. Modify circuit to make Q = 5.

4. Measure the modified circuit and gather the resonant frequency and Q of the circuit

5. Determine the admittance of the circuit at the resonant frequency.

The procedures of this experiment are similar to the earlier experiments.

Fig 3.5.1

Rs is used to derive a current source for the circuit. (L=100mH, without R)

Is = Vs / Rs. We will use 100 kΩ.

1. To make resonant frequency at 2 kHz with L = 100 mH.

However, in our lab kit the closest value of capacitor we have was 100nF so we will use two

100nF in series to create an equivalent capacitance of 50nF.

With a modified capacitance:

11


Results (by using the same experimental procedure as outlined in experiment 3.2):


2205 -45°

2503 45°

2361 0°

Fig 3.5.2

2. So the cutoff frequencies would be at 2.205kHz and 2.503 kHz.

3. Now to modify Q to be 5.

We know:

with the measured Q = 7.923

So we connect an additional Rextra as 20K that is a series of two 10K resistors. We repeat

the experiment to do the measurements to see if the modifying worked out.

4. Now, using the same process as outlined previously, the results show:

12



2123 -45°

2593 45°

2356 0°

Fig 3.5.3

Analysis of Figure 3.5.3:

This could be considered as a good result as it is very close to 5.

5. Determine the admittance of the circuit at the resonant frequency.

To do this, we calculate the equivalent resistance of the resistors in parallel, and then put them

in parallel with the equivalent impedance of the capacitor and inductor. Then we can put that

impedance in series with Ri. Once we know the total equivalent impedance of the circuit, we

can take the reciprocal of that to find the admittance.

In phasor form, Zinductor = jwL, and Zcap = -1/(jwC)

This answer makes sense because resonance is where the impedances of the circuit

components all cancel out

From part 2 we gathered the relationship between L, C, and R components of circuits at high

frequencies. We take our experiment further in part 3.

Part 3 – Non-ideal Frequency Behavior of Passive Components

13


Experiment 3.6

We repeat the measurement of experiment 3.1, however as R1 = 1M Ω and for R2 = 5 KΩ to

see the effect of the shunt capacitance at R1 at high frequencies. Note that we are actually

using two 10KΩ resistors in parallel to represent the 5KΩ R2.

Fig 3.6.1

Note: To minimize the shunt capacitance of the oscilloscope we must use a 10x probe.

Since the circuit acts as a high-pass voltage divider, we need to normalize the gain relative to

the high pass-band gain (~.075) so we will multiply our gains by (1/.075) to normalize them.

The .707 normalized gain will be our -3dB point. Aside from that, the process is the same as

stated in experiment 1.

The results were as follows.

f(hz) Vin Vout | Vo / Vi | Normalized Gain

10 10.130 0.058 0.006 0.0765

50 10.190 0.059 0.006 0.0768

100 10.190 0.058 0.006 0.0760

500 10.190 0.058 0.006 0.0760

1000 10.190 0.058 0.006 0.0760

5000 10.190 0.058 0.006 0.0760

10000 10.190 0.056 0.005 0.0731

14


50000 10.190 0.058 0.006 0.0760

100000 10.190 0.069 0.007 0.0900

500000 10.000 0.191 0.019 0.2547

1000000 10.060 0.338 0.034 0.4480

1897000 10.000 0.530 0.053 0.7067

5000000 11.300 0.830 0.073 0.9794

10000000 13.800 1.030 0.075 0.9952

20000000 17.500 1.250 0.071 0.9524

Fig 3.6.2

Gain vs Frequency

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

0 5000000

1E+07 1.5E+07

2E+07 2.5E+07

Frequency (Hz)

No

rmal

ized

Gai

n

Gain vs Frequency

Fig 3.6.3

We see from our table (Figure 3.6.2) that the fB is approximately 1.897MHz

fb = 1.897MHz

15


Where

Note: We treat our resistors in parallel because they are seen in parallel by the probes. The

capacitors are treated in parallel too for the same reason.

Since we know Cprobe = 15pF (as stated on the device), we can plug in all of our values and

solve for Cresistor.

Experiment 3.7

Use the circuit shown below to determine |Z(jw)| of the impedance of the inductor from 100

Hz to 1 MHz.

Fig 3.7.1

For the inductor, (Figure 3.7.2) with the parasitic capacitance and resistance (Figure 3.7.3)

=>

Fig 3.7.2 Fig 3.7.3

To obtain the Rw we use dc voltage across the inductor from Figure 3.7.3.

Rw = 97.89Ω

16


Now we do another measurement to get the impedance of the circuit:

f(hz) Vin Vout | Vo / Vi | Phase (°)

100 10.000 9.200 0.920 5

500 10.000 8.800 0.880 180

1000 10.000 8.000 0.800 30

5000 10.300 3.400 0.330 74

10000 10.300 1.900 0.184 90

50000 10.300 0.340 0.033 89

100000 10.300 0.138 0.013 90

136000 10.300 0.006 0.001 unreadable

500000 10.300 0.425 0.041 -85

1000000 10.300 0.920 0.089 -76

Fig 3.7.4

Gain vs Frequency

-0.200

0.000

0.200

0.400

0.600

0.800

1.000

0 200000 400000 600000 800000 1000000 1200000

Frequency (Hz)

Gai

n

Gain vsFrequency

Fig 3.7.4

17


We could see from the measurements that the resonant frequency is where gain is its

minimum value. That is 136 KHz (Figure 3.7.4 & 3.7.5).

where L = 100mH, fr = 136 KHz

Now we could characterize the 100mH inductor that we used through out the experiments

that has 97.89Ω parasitic resistance and 13.7pF parasitic capacitance (Figure 3.7.3).

We can calculate |Z(jw)| by plugging these values into a phasor form equivalent equation.

Conclusion

First we looked at the effects of the shunt capacitance of the interconnecting cables and

oscilloscope. By measuring the transfer function of a basic series connection of resistance we

were able to obtain a shunt capacitance of 113pF, which would be able to be a reasonable

value of the cables. The break frequency was gathered at the 3dB (70.7%) point of the circuit.

Next we checked the resonance of the RLC circuits and we were able to see that the resonant

frequency is inversely proportional to the square root of C. We also saw the input and output

spectra differences from the RLC circuit by using the FFT math function on the oscilloscope.

We found out that through the circuit we are only able to gather one harmonic that results a

sinusoidal waveform at the output. Further we made an understanding that in order to modify

a circuit to give an expected or proper response at high frequencies, we need to consider the

more specific components of the circuit, such as the resistance of the function generator or the

impedance of the components, and so forth.

Finally, we obtained the knowledge on how to gather the exact characteristics of passive

circuit components that are used for circuit designing. Typically a resistor has extra shunt

capacitance. For the component we used (R=1M ), we were able to see a parasitic

capacitance of 1.86pF. An inductor has extra parasitic capacitance and resistance contained

within the component. And for the component given in our lab kit (L=100mH) we were able

to see the 98.89 resistance and the 13.7pF capacitance.

In this experiment, we have successfully practiced the process of interpreting the high

18


frequency response of a circuit. This information should be used on further circuit analysis or

designing projects to gather more accurate results.

References

1) Sedra/Smith, “Microelectronic Circuits”, Fifth Edition

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EE3101 Experiment 3

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