AC Circuit Essay: Voltage And Impedance Response

This Experiment is the first of two in which AC circuits will be more explicitly introduced. As a result of using an oscilloscope, we may determine the voltage response and impedance response to the frequency of an AC signal that has passed through a series circuit consisting of a resistor and capacitor (RC), and a resistor & inductor (RL).

The amount and direction of the current fluctuate with time in AC circuits. The magnitude and polarity of voltage decreases across elements change with time. During the late 19th century, Nikola Tesla established that AC's time-varying current and voltage is the most effective method of delivering electricity.  This can be illustrated as shown in the figure below; (Lab06Manual)

Figure 1: Graph of Amplitude Vs Time

The sine waveform, as depicted in the image above, is the most commonly encountered time-varying AC signal. There is a distinct peak voltage  in the graph, which rises to zero before returning to zero. The voltage then reverses its polarity, starting at zero, falling to a negative maximum value, and then returning to zero.  Similarly, when the polarity of the voltage changes from positive to negative, so does the current (switching directions, forward then backward). Charges move in a wavelike pattern via the wires. This lab will study a few of the effects that are only seen in AC circuits because of the fluctuating current and voltage. (Lab06Manual) (Bhargava & Kulshreshtha)

From the above graph shown in figure 1, the following terms may be defined as shown below; (Lab06Manual).

This is the absolute maximum displacement of current or voltage on either side of the mean position. For instance, from the above graph, the amplitude is 10 Volts.

This is the difference in voltage or current between the highest and lowest values. From the above graph, this value is 20 Volts.

This is the time taken by a waveform to form one complete oscillation. The horizontal displacement in time units can be measured between any two corresponding positions, from peak to peak, trough to trough, or any other two corresponding places.

This is generally the number of complete cycles or oscillations forms in one second. Mathematically, it is the reciprocal of the period. 

AC voltage peak-to-peak measurements are easy with an oscilloscope . The voltage difference between the signal's peak and trough is what we're interested in. If the signal's amplitude is doubled, the peak-to-peak voltages will be twice as high. We can eliminate this two-fold error if we measure all voltages from the highest to the lowest point.  

It's possible to store electrical energy in the form of a charge using a capacitor. Separated by a non-conducting substance are two or more conducting electrodes (the dielectric). A capacitor doesn't conduct any current. When a voltage is supplied across the electrodes, charges build up (or down) on the plates. The capacitance of a capacitor determines the maximum charge it can carry at any given potential. (C). mathematically; 

Capacitance is measured in Farads, which is equivalent to seconds per ohm, .  (Bhargava & Kulshreshtha) (Lab06Manual)

Definition of Terms

The charge accumulated at the electrodes of the capacitor determines the voltage across it at any given moment t. There is a long buildup period before that charge may be felt (or down). Voltage and current reverse direction too quickly when driven by an AC signal with a high frequency, causing the capacitor to begin discharging before it has a chance to fully charge. Therefore, the voltage across a capacitor is inversely proportional to the frequency of the AC signal and reduces at higher frequencies, and increases at lower frequencies.

As a result, the voltage across the capacitor and the current flowing through the circuit have a lag period. To put it another way, when it comes to voltage, the current is out of sync with the voltage. A quarter of a cycle separates the voltage of the capacitor (Lab06Manual)or from the current. The current and voltage across the capacitor are out of phase by 90° if we consider 360° as one full cycle.

The coil of wire that makes up an inductor might have or not have a core. To counteract the EMF of the time-varying voltage applied to the coil, a back-EMF is generated when a time-varying current is passed through it. Depending on the frequency of the AC signal, the back-EMF can be large or small; at higher frequencies, the effect is more pronounced, whereas, at lower frequencies, the effect is less pronounced. A direct current inductor is essentially a piece of wire at zero frequency, therefore there is no effect. Inductance determines the greatest amount of back-EMF that may be created by an inductor at any given changing current (L). Mathematically; 

The inductance is measured in Henry (H), which is equivalent to .

The voltage across the inductor and the current across the circuit are in phase with each other, just like the capacitor is. When it comes to the inductor, however, the current lags behind the voltage by . A resistor does not affect the phase relationship between voltage and current because resistance does not affect frequency. In other words, current and voltage in an AC circuit are always in phase. (Bhargava & Kulshreshtha) (Lab06Manual)

Phase shift is calculated by comparing the times of two corresponding peaks or valleys (any two adjacent corresponding points suffice as well). It is possible to compute the phase shift (in degrees) of an AC generator.

Like resistors, inductors and capacitors obstruct the flow of current in an AC circuit when they are operating in this mode. Their impedance, however, differs from that of a resistor fundamentally. Inductors and capacitors react to the flow of current, and their resistance to current flow is called reactance, as described earlier. The unit of reaction is the ohm (?), and the symbol for it is 

where is the frequency of the AC signal and C is the capacitance of the capacitor, and  is the symbol for its reactance.

It's important to note that the reactance is larger at lower frequencies, while the reactance decreases as the frequency increases. Taking the DC limit into account, the reactance is infinite as the frequency approaches zero. Because the capacitance gap is infinitely resistive, the capacitor is effectively a barrier to the flow of current. At its most extreme, the reaction has no value because the frequency is approaching infinity. Due to the rapidity with which the current oscillates at high frequencies, the gap is mostly irrelevant, and the resistance is minimal.

Since both  and  are functions of time, they are analogous to Ohm's Law, which describes the resistance of a DC circuit.

At low frequencies, the reactance is small, whereas, at high frequencies, the reactance is large. This behavior is consistent. Once again, as the frequency approaches 0 (the DC limit), the reactance is zero, which is exactly what one would expect. All an inductor is is a wire through which no resistance is encountered by the current it carries. Reactance becomes limitless when frequency approaches infinity. The inductor creates enough back-EMF to halt the current while operating at high frequencies because the current oscillates so rapidly. The drop in voltage across the inductor is given by the following equation: (Lab06Manual) 

The voltage drop across the resistor is given by the following equation: 

There is no frequency response in resistors; they nevertheless obey Ohm's law.

Parts A and B are involved in the experiment. It was determined that the peak-to-peak voltages for two pairs of series AC circuits containing a resistor and capacitor parts were tested. Between the resistor and capacitor, the phase shifts were detected and measured. In addition, the frequency at which the voltage decreases across the elements is equal was established. Peak-to-peak voltages and currents, as well as the AC impedance of the circuit, were calculated using the resulting data.

A decade resistance box was used to create resistances. The equipment kit is as shown below;

In Kit-return to kit

• Red & Black Banana leads
• Yellow & Blue Banana leads
• Set 1 1x capacitor
• Set 2 1x capacitor
• 2x Dongles
• Decade Resistance Box

In Drawer-Return Drawer

• Ruler
• Protractor

Available in the Lab

• Green LCR Meter
• Signal Generator
• Oscilloscopes

Setup the AC Generator and the Oscilloscope

The AC Generator and the Oscilloscope were set using the steps and instructions of the Machine manual found in the laboratory. This was then followed with machine setups to acquire the average settings and the peak-to-peak voltage measurements settings. The experiment was then carried out for PART A and PART B as illustrated below;

This was set as AC circuit set 1. It was done by using the components of set 1: 

The test frequency  was set to .

• The decade resistance box was set as the resistor and the required adjustments s per the lab manual were done appropriately.
• The values of the were then measured using the .
• The value of the test frequency was then measured using the AC generator.
• The peak-to-peak voltage for the resistor and capacitor was then recorded.
• The defined voltage in the A.C generator was then labeled as .
• Other adjustments were then made and all the data recorded.

• This was set as AC circuits set 2. The components of AC circuit components were as follows: 

• The frequency was then set to
• All the procedures and adjustments used in PART A as per the lab manual were then repeated using the components of the resistance values for set 2.
• The A.C generator and the Oscilloscopes were then turned OFF and the circuit disassemble.  

This is a positive shift since the calculated is leading with a positive deviation of .     

Conclusion

The objective of the experiment to examine the concepts of the alternating circuits was successfully achieved. Both the voltage response and the impedance response were effectively examined to the frequency of an alternating signal that was sent to the Resistor/capacitor series circuits and Resistor/inductor series circuits.

References

Bhargava, N., & Kulshreshtha, D. (n.d.). Basic Electronics & Linear Circuits. Tata McGraw-Hill Education.

Lab06Manual. (n.d.). AC Circuits lab manual.