Alternating Current Circuits

Alternating currents dominate the present day industrial and consumer world. Examples are the overwhelming majority of the electrical power generation plants, the aerial radio and television broadcast and communication systems, the electric motors in all sorts of manufacturing machinery and home appliances, etc. A fundamental advantage of the alternating current is that it can be transported efficiently on very large distances from the generating facility. The latter can therefore, be optimized based on the power source, scale and location, as it is the case in the recent decades. The transportation advantage of the alternating current arises from the ease to produce various voltages using transformers. Very high voltages (hundreds of thousands of volts) combined with low currents are used through the electricity transportation grids, resulting in low resistive energy dissipation. In contrast, direct current carried by copper cable on a distance of even few hundreds of meters has prohibitively large losses and cost of the transmission line.

Reactance and Resonance In alternating current circuits the voltage and current are related in a more complex way than in the DC circuits. The presence of frequency-sensitive elements such as inductors and capacitors leads to phase shift between the generator voltage and the circuit current. Suppose a resistor, R, inductor, L and capacitor, C, are connected in series (Figure 1, consider the dashed line with switch). If we close the resulting series RLC circuit after the capacitor is charged, it will operate in a free oscillation mode, provided no energy is further supplied. For low resistor values the free mode is a damped oscillation under an exponentially decreasing envelope, with frequency

1 1 R2 1 1 fR =  ≈ = f0 (1) capacitor C 2 LC 4L2 2 LC

where f0 is the free mode oscillation frequency (natural r

frequency) of the idealized circuit with the same L and C, inductor L but R=0 (called LC circuit). ~ If the circuit is connected to an AC generator (Figure 1, dashed connection line now ignored), it operates in a forced AC generato oscillation mode. In this mode, the quantity relating the

generator voltage and the circuit current is called impedance resistor R Z. The impedance includes the circuit resistance R (the same Oscilloscope as in the DC circuits) and circuit reactance X. The reactance consists of inductive reactance XL, due to the inductors with Figure 1. total inductance L, and capacitive reactance XC, due to the capacitors with total capacity C. It can be shown that the circuit impedance satisfies the equation: 2 2 2 2 2 Z = R + X = R + (X L – X C ) (2) The reactance is frequency dependent. For an AC generator of sinusoidal voltage with frequency f

XL = 2πfL, XC = 1 / (2πfC) (3)

The situation of X L =XC (reactance X=0) can be achieved by appropriate generator frequency. Equating the right-hand sides of Eq. 3 gives the condition 2πfL=1/(2πfC), and therefore:

f = 1/(2π LC )= f0 (4) At X=0 the impedance has minimum, the circuit current is at maximum and the phase shift between voltage and current is zero. We say that the AC generator is in resonance with the RLC circuit at the resonance frequency f0. The generator compensates for the resistive energy dissipation in the circuit and in resonance drives it to oscillate like a “no-resistance” LC circuit, with the frequency f0 instead of fR.

2 Alternating Current Circuits

The unit for Z, X, XL and XC, as evident from Eq. 2, is Ohm (Ω). The units for f, L and C are Hertz (Hz), Henry (H) and Farad (F) respectively.

The Experiment The series RLC circuit shown in Figure 1 should be built using the provided components and wires. The frequencies fR and f0 should be calculated from Eq. 1 and Ed. 4 with values of the R, L and C obtained either from the marks on the respective components, or from data given by the instructor. First, the free oscillation mode will be investigated. The charge of the capacitor and closing of the circuit required for this mode are achieved by using a square wave signal ( ) from the digital function generator. The square wave front raises almost instantly (≤120 ns) and produces the initial voltage on one of the capacitor terminals. Past the signal front the voltage level remains constant (high or low) for each half period, with the generator’s own impedance (mostly resistive, 50 Ω by specification) in series with the RLC circuit and contributing negligibly to its resistance. Therefore, the circuit is in free oscillation mode during both half-periods of the square wave signal, with raised voltage level in the first half period and a lowered one - in the second. The oscilloscope is operated in dual channel mode. One of the channels should monitor the square wave signal and provide the time synchronization of the oscilloscope. The channel monitoring the free oscillation of the circuit measures the voltage drop over the resistor (Figure 1, right). With appropriate vertical (voltage) resolution, it should display the characteristic picture of the free damped oscillations. Recommended square wave frequency for good observation is ≈f 0 /18. The elapsed time for five or more full periods of damped oscillation should be measured in order to achieve better accuracy. Pay attention that true readings are taken only when the variable regulation knob of the horizontal sweep is in locked position. In the second part of the experiment you drive the RLC circuit with a sine wave ( ~ ), and obtain the frequency dependence of the circuit current. The measurements consist of recording the amplitude of the voltage drop over the resistor at a number of frequencies below and above the calculated resonance frequency. The relationship “current-voltage drop” is analogous to that of DC circuits. It is helpful if the frequency points are chosen ≈2 kHz apart in an interval ≈(0.8f 0 –1.2f 0 ). Data processing is facilitated by a tabulated record “frequency (kHz) – voltage drop amplitude (V) – current amplitude (mA)”.

Analysis Once the data table is completed, a graph “current vs. frequency” has to be plotted, the points being exp connected with a smooth line. The experimental value of the resonance frequency f0 corresponds to the current maximum on the graph. Now you can compare the experimental value of fR with that calculated exp from Eq. 1. You also compare f0 with f0 calculated from Eq. 4. Propose explanation for the eventual difference between them. Take into account that the values of the electronic components are usually with tolerance of 5%, and the inductor has a wire resistance of ≈70Ω. Recall that the experimental fR was obtained using the oscilloscope’s timing scale. Does the precision of the experimental fR allow exp distinguishing it from f0 ? Can you compare the timings of the oscilloscope and the function generator and make corrections for the eventual difference?

Apparatus on each bench: 1. Oscilloscope 3. Components (resistor 1 kΩ, inductor 0.025 H, capacitor 200 pF) 2. Function generator 4. Wires (1 coaxial – signal, 5 – connecting)