# AC circuits

Rather than having constant voltages and currents as in DC (direct current) circuits, AC (alternating current) circuits involve sinusoidally varying voltages and currents. It may seem strange at first to be using sinusoidal quantities until one realizes that devices rotating in some stationary environment generate most electrical power in the world. In a natural way this results in the voltages and currents being sinusoidal.

The complicating factor in AC circuits is that inductors and capacitors introduce phase shifts between the voltages across the components and the currents flowing through them. Thus if one looks solely at the magnitudes of the voltages across the components in a circuit containing inductors and capacitors, the algebraic sum of the voltages around the circuit will not in general be zero, an apparent violation of Kirchhoff’s Voltage or Loop Rule. This is equivalent to saying that the voltage from the power supply in the circuit is unequal to the sum of the voltages across the components making up the circuit. In this experiment you will explore the relationships between voltages and currents for inductors, capacitors, and resistors. This will include phase relationships and frequency dependencies. For this study a simple circuit consisting of a resistor, capacitor, and inductor connected in series and driven by a sinusoidal voltage source is used.

Theory (A Brief Review)

Capacitors are essentially two conducting sheets or plates separated by some insulating material that may include air or a vacuum. When a voltage is applied between the two plates of

the capacitor, charge is transferred from one plate to the other. Thus a current flows through the voltage source and the connecting wires. As the voltage increases and more charge collects on the plates, adding more charges becomes harder and harder remembering the fact that like charges repel one another. Because of this the current flowing into the capacitor is greatest when the plates initially begin charging. Then the current goes to zero when the charge build-up reaches a maximum. If a sinusoidally varying voltage source, one that oscillates positively and negatively in time with the shape of a sine wave, is connected across the capacitor, it can be shown that the voltage across the capacitor “lags” the current by 90o in phase, meaning that the voltage peaks occur ¼ of an oscillation period later in time relative to the current peaks. This is designated as a –90o phase shift for the voltage with the current taken as the zero reference phase.

An inductor is usually in the form of multiple loops of wire called a coil. When a time-varying electrical current passes through the loops, the resulting time-varying magnetic field induces a voltage in the coil, and according to Lenz’s law (and energy conservation) this induced voltage opposes the source voltage, making the current small. When sinusoidally driven, the voltage can be shown to “lead” the current by 90o in phase, meaning that the voltage peaks occur ¼ of an oscillation period earlier in time relative to the current peaks. This is designated as a +90o phase shift.

For ideal inductors and capacitors like those just described that are connected in series with a resistor, the individual voltages across the three components can be added using the rules of vector addition assuming that the voltage across the resistor is in phase with the current and thus can be represented as lying along the +x-axis, the voltage across the inductor as along the +y-axis (+900), and the voltage across the capacitor as along the –y-axis (-900). For non-ideal components the actual phase differences between the voltages and the current can be measured. Then using the actual measured voltage phase angles for the inductor and capacitor, the voltage “vectors” can be drawn on a diagram similar to that just described and added using the normal rules of vector addition with components, etc.

Root mean square (abbreviated as rms) values of voltages and currents are also useful. In DC circuits the product of the current and voltage gives the power. In an AC circuit with the current and voltage in phase it is convenient to use the same formula to calculate the power averaged over time, namely current times voltage, but what current and voltage should be used since they are varying positively and negatively with time? If we use the product of the zero to peak voltage and zero to peak current, the answer is twice as large as the actual average power. Thus we take the rms voltage, which is the peak voltage divided by the √2, and the rms current, which is the peak current divided by the √2. By using the rms voltage and current we can keep the same formulation for power in both AC and DC circuits. Note that most AC voltmeters and ammeters read out values in rms. The voltage at a wall plug in the United States is 120 V rms, whereas the actual peak voltage is 170 V.

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