Network Analysis Laboratory

EEE 117L Network Analysis Laboratory Lab 2

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EEE 117L Network Analysis Laboratory Lab 2

Thevenin/Norton Equivalent and LCR Resonant Circuits

Lab Overview

The objective of Lab 2 is to familiarize students with the use of both Thevenin/Norton equivalent and LCR resonant circuits. The performance of these are determined using both Spice simulations and the Digilent Analog Discovery 2 on the circuits constructed.

Prelab

Before coming to lab, students need to complete the following items for each of the circuits studied in this lab : • Any hand calculations needed to determine the values of components used in

the circuits such as resistors and capacitors, or specifications such as frequencies. • A Spice simulation of each circuit to get familiar with how it works, and determine

what to expect when the circuit is built and its performance is measured.

Circuits to be studied

When choosing resistor and capacitor values use standard values available to you, and keep all resistor values between 100 W and 100 kW. 1. Thevenin and Norton Equivalent Circuits

Often the analysis of a circuit can be greatly simplified through the use of a Thevenin or Norton’s equivalent circuit. For example, determining the pole frequency for the low pass filter shown in Figure 1.a is much simpler if the resistor network is first replaced with the Thevenin’s equivalent circuit shown in Figure 1.b. A simple way to measure the Thevenin’s equivalent resistance for a circuit, shown as Rth in these figures, is to replace the capacitor C with a load resistor RL and measure the output voltage. When the output voltage is ½ of the value seen when the output is open circuited, the value of the load resistor is equal to the Thevenin’s equivalent resistance.

Figure 1.b Figure 1.a

EEE 117L Network Analysis Laboratory Lab 2

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If it is not possible to add a load resistance that is exactly equal to Rth, then the value of this Thevenin’s equivalent resistance can still be determined using voltage division :

"# "$%

= ' () ()*($%

+ where: Vth = the Thevenin equivalent (open circuit) voltage Tip: The Thevenin’s equivalent resistance and the Norton’s equivalent resistance are the same value. This is often called the “dead circuit resistance” since it is the resistance seen when all of the independent voltage and current sources are turned off, by setting V = 0 for the independent voltage sources (replace them with short circuits) and I = 0 for the independent current sources (replace them with open circuits). Determine the Thevenin equivalent resistance and pole frequency for the circuit in Figure 1.a Choose any 4 resistors between 1 kW and 10 kW for R1 through R4 in Figure 1.a, and use C = 1 µF. Calculate the Thevenin equivalent resistance Rth and the pole frequency for this circuit. Verify Rth in DC Spice simulations using the voltage division method described above, and verify the pole frequency using AC Spice simulations. Then build the circuit on your breadboard and make use the Digilent Analog Discovery 2 to measure the actual values of Rth and the pole frequency achieved. Compare the measured results to both your calculated and simulated values. 2. LCR Parallel Resonant “Tank” circuits

Another way to create a bandpass filter is to use an inductor and capacitor in parallel to create a resonant “tank” circuit, as shown in Figures 2.a and 2.b. The impedance Zin of the circuit in Figure 2.a is given by :

𝑍-. = / 0

123* 0 1245 6 = 7

0 18239 0 245 :

; = ∞ 𝑎𝑡 ω = 𝜔A = 0 √43

Figure 2.a Figure 2.b

EEE 117L Network Analysis Laboratory Lab 2

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Figure 3.a

As this shows, at the resonant frequency wo the admittances of the inductor and capacitor exactly cancel, resulting in an open circuit. If a resistor is added in parallel as shown in Figure 2.b then the impedance Zin of the circuit will be just R at w = wo . At frequencies below wo the impedance Zin will be much smaller and look inductive with a positive phase, and at frequencies above wo the impedance Zin will be much smaller and look capacitive with a negative phase. At wo the impedance Zin = R with 0° phase. This resonant LCR circuit can be used to create a bandpass filter, as shown in Figure 2.c. At w = wo the voltage gain is given by :

"# "C = ' (

(*(C +

This type of bandpass filter typically only passes a narrow band of frequencies, which can be useful when trying to select a single signal such as a radio or television channel. How selective the filter is can be characterized using the quality factor or Q of the filter, which is given by the resonant frequency divided by the -3dB bandwidth :

𝑄 = ' E# EF 9 E)

+ where fH and fL are the upper and lower -3dB frequencies in Hertz Build a bandpass filter using the circuit shown in Figure 2.c Use R = Rs = 1 kW, L = 1 mH, and C = 1 µF. Calculate the resonant frequency in Hertz and the voltage gain at this frequency. Then use an AC simulation in Spice to create a Bode plot for this circuit, and verify that the resonant frequency and voltage gain agree with your hand calculations. Plot both the magnitude of the voltage gain in dB and also the phase in degrees. Also measure the Q of the filter. Finally construct this circuit on a breadboard and measure the resonant frequency, the voltage gain at this frequency, and the Q of the filter. 3. LCR Series Resonant circuits Resonance also occurs when an inductor and capacitor are connected in series, as shown in Figure 3.a. The impedance Zin of this circuit is given by :

𝑍-. = '𝑗𝜔𝐿 + 1 𝑗𝜔𝐶5 + = 𝑗8𝜔𝐿 − 1 𝜔𝐶5 :

𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0 𝑎𝑡 ω = 𝜔A = 0 √43

Figure 2.c

EEE 117L Network Analysis Laboratory Lab 2

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As this shows, at the resonant frequency wo the impedances of the inductor and capacitor exactly cancel, resulting in a short circuit. If a resistor is added in series as shown in Figure 3.b then at w = wo the impedance Zin of the circuit will be just R. At frequencies below wo the impedance Zin will be much larger and look capacitive with a negative phase, and at frequencies above wo the impedance Zin will be much larger and look inductive with a positive phase. At wo the impedance Zin = R with 0° phase. This resonant LCR circuit can be used to create a band reject or “notch” filter, as shown in Figure 3.c. At w = wo the voltage gain is given by :

"# "C = ' (

(*(C +

At frequencies either well below or well above wo the gain will approach 1 with a phase of 0° as either the impedance of the capacitor (at low frequencies) or the inductor (at high frequencies) becomes very large. Build a band reject or “notch” filter using the circuit shown in Figure 3.c Use Rs = 100 W, R = 1 W, L = 1 mH, and C = 1 µF. Calculate the resonant frequency in Hertz and the voltage gain at this frequency. Then use an AC simulation in Spice to create a Bode plot for this circuit, and verify that the resonant frequency and voltage gain agree with your hand calculations. Plot both the magnitude of the voltage gain in dB and also the phase in degrees. Finally construct this circuit on a breadboard and measure the resonant frequency and the voltage gain at this frequency. 4. Non-ideal effects in real inductors and capacitors due to parasitics Unfortunately it is not possible to build ideal inductors and capacitors. Real manufactured inductors and capacitors have additional unwanted “parasitic” elements that cannot be avoided. For example, an inductor created by winding wire around a core has series resistance due to the resistance of the metal wire used, and parallel capacitance due to the capacitance between the wire coils that are packed close together. A real capacitor has both resistance and inductance in series due to the metal wire used to connect to the parallel plates of the capacitor. So actual manufactured devices are really combinations of R, L, and C elements connected in either series or parallel, which are designed to have one of these elements (either R, L or C) dominate.

Figure 3.b

Figure 3.c

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Typical circuit models for actual manufactured inductors are shown in Figures 4.a and 4.b and a typical circuit model for an actual manufactured capacitor is shown in Figure 4.c. The inductor model in Figure 4.b is an approximate version of the inductor model shown in Figure 4.a that is valid near the resonant frequency, with :

𝑅Y = 𝑄Z𝑅[ where the Q of the inductor is given by : 𝑄 = 24 (C

Note that the Q of the inductor is essentially a measure of the quality of the inductor, using the ratio of the desired part of the inductor impedance wL to the undesired part due to the parasitic resistor RS . An ideal inductor has RS = 0 so the Q = ¥ . This second version of the inductor model is useful when analyzing parallel resonant tank circuits like those shown in Figures 2.b and 2.c. Since the parasitic resistance of the inductor RP appears in parallel with the intended resistance R, it lowers the gain of the bandpass circuit shown in Figure 2.c at the resonant frequency wo . An important consequence of these undesired parasitic elements in real manufactured inductors and capacitors is that they have a limited range of frequencies where they look like the intended element. Above the “self-resonant frequency” of an inductor the parallel capacitance CP dominates, and the impedance looks like a capacitor instead of an inductor. Above the “self-resonant frequency” of a capacitor the series inductance LS dominates, and the impedance looks like an inductor instead of a capacitor. Tip: This makes it critical for engineers to select the correct type of inductor or capacitor for different applications, since these self-resonant frequencies vary greatly depending on how the device is manufactured and the materials used. (e.g., electrolytic capacitors have much lower self-resonance frequencies than ceramic capacitors.) Review the datasheets for the inductors and capacitors used in this lab to determine the self-resonance frequencies of these components.

Figure 4.a Figure 4.b

Þ

Figure 4.c