Introduction to AC Circuit Analysis

Introduction to AC Circuit Analysis

• The overarching objective of this discussion is to introduce simple Alternating Current (AC) sources and the fundamental techniques used for AC circuit analysis.

• Unlike Direct Current (DC) sources, ac voltage sources vary in both magnitude and polarity as a function of time.

• The analysis involves exploring how voltage, current, and power relate within an AC context.

• A critical distinction is made between average values and effective values (often referred to as RMS, though the formal term RMS is introduced later in the series) for voltage and current supplied by an AC source.

Review of Fixed DC Sources and Analysis

• A fixed DC source is defined as one that does not vary in polarity or magnitude as a function of time.

• Example: Consider a fixed $5\,V$ DC source connected to a $100\,\Omega$ resistive load.

  • The voltage drop across the resistor is a constant $5\,V$.

  • The source supplies a constant current: $I = \frac{V}{R} = \frac{5\,V}{100\,\Omega} = 50\,mA$.

  • The power dissipated by the load is constant: $P = V \times I = 5\,V \times 50\,mA = 250\,mW$.

• Characteristics of Fixed DC:

  • Current magnitude remains constant.

  • Current flows in only one direction (unidirectional).

  • Plots of voltage, current, and power against time result in flat, horizontal lines.

Time-Variant DC Sources

• A time-variant DC source varies in magnitude as a function of time but does not change polarity.

• Common examples include pulsed DC, sawtooth waveforms, and triangle waveforms.

• The defining characteristic is that the waveform remains on one side of the horizontal (time) axis when plotted.

• Current may change in value, but it still flows in one direction only.

• Pulsed DC Waveform Example:

  • Magnitude alternates between $5\,V$ and $0\,V$ at regular intervals.

  • A 50% duty cycle is defined as having equal periods of fully "on" ($5\,V$) and fully "off" ($0\,V$).

  • Analysis of a $100\,\Omega$ load with a $1\,s$ on/off cycle:

  • From $0$ to $1\,s$ ($5\,V$ applied): $I = 50\,mA$, $P = 250\,mW$.

  • From $1$ to $2\,s$ ($0\,V$ applied): $I = 0\,mA$, $P = 0\,mW$.

  • Average Power Dissipation: Given equal spans of $250\,mW$ and $0\,mW$, the load dissipates an average of $125\,mW$ over the analysis period.

Introduction to Alternating Current (AC) Sources

• An AC source is defined as one that varies both magnitude and polarity as a function of time.

• Because polarity is variant, both current magnitude and current direction vary.

• Despite these variations, the fundamental circuit analysis techniques established in DC (Ohm's Law and Power equations) remain valid for instantaneous analysis.

• AC Measurement Setup:

  • For analysis purposes, an ammeter is placed in series (left to right, "in" to "out").

  • A voltmeter is placed across the load (positive to negative, top to bottom).

  • These connections remain fixed, even when the source polarity swaps.

• Polarity in AC:

  • Polarity symbols (+ and -) on an AC source schematic typically indicate the direction the source initiates operation during the positive half-cycle. They do not imply a fixed polarity over time.

  • A "positive" half-cycle at $5\,V$ means the top terminal is positive and the bottom is negative.

  • A "negative" half-cycle at $-5\,V$ means the polarity has swapped, with the bottom terminal now positive and the top terminal negative.

Basic AC Circuit Analysis Example

• Scenario: A $12\,V$ AC source that swaps polarity every second connected to a $200\,\Omega$ resistive load.

• Interval 0 to 1 second (Positive Half-Cycle):

  • Source orientation: Positive to negative (top to bottom).

  • Voltage: $+12\,V$.

  • Current: $I = \frac{12\,V}{200\,\Omega} = +60\,mA$.

  • Power: $P = 12\,V \times 60\,mA = +720\,mW$.

• Interval 1 to 2 seconds (Negative Half-Cycle):

  • Source orientation: Positive to negative (bottom to top).

  • The fixed meter connections now read negative values.

  • Voltage: $-12\,V$ (meaning $12\,V$ bottom to top).

  • Current: $-60\,mA$ (meaning $60\,mA$ right to left).

  • Power: $P = (-12\,V) \times (-60\,mA) = +720\,mW$.

• Conclusions:

  • Power dissipation is positive in both half-cycles because the product of two negatives is a positive.

  • From the perspective of the resistor, the direction of current flow is irrelevant to the amount of heat/energy dissipated.

  • The negative signs for voltage and current are simply indicators of direction relative to the fixed measurement equipment.

Average vs. Effective Values

• The Mathematical Trap: In a purely mathematical sense, the average value of AC voltage and current over a full cycle is zero because the positive and negative halves cancel each other out.

• The Physical Reality: Because power is constantly being delivered to the load, a "zero" average is misleading for practical work. Terminology must be precise.

• Effective Values: These are the constant DC-equivalent values that would deliver the same amount of average power to the load.

• Power Calculation Formulas:

  • $P = V \times I$

  • $P = I^2 \times R \rightarrow I = \sqrt{\frac{P}{R}}$

  • $P = \frac{V^2}{R} \rightarrow V = \sqrt{P \times R}$

• Effective Value Calculation:

  • Using the previous $12\,V$, $200\,\Omega$ example where $P_{\text{average}} = 720\,mW$:

  • Effective Current ($I_{\text{eff}}$): $\sqrt{\frac{720\,mW}{200\,\Omega}} = 60\,mA$.

  • Effective Voltage ($V_{\text{eff}}$): $\sqrt{720\,mW \times 200\,\Omega} = 12\,V$.

  • Note: Effective values have no sign; they represent the magnitude of a DC equivalent.

Complex AC Waveform Analysis: Stair-Step Waveform

• Scenario: A cyclic source with a duration of $16\,s$ per full cycle and a $300\,\Omega$ resistive load.

• Voltage Schedule (Cycle 1):

  • $0\,s$ to $1\,s$: $0\,V$

  • $1\,s$ to $3\,s$: $+30\,V$

  • $3\,s$ to $5\,s$: $+60\,V$

  • $5\,s$ to $7\,s$: $+30\,V$

  • $7\,s$ to $9\,s$: $0\,V$

  • $9\,s$ to $11\,s$: $-30\,V$

  • $11\,s$ to $13\,s$: $-60\,V$

  • $13\,s$ to $15\,s$: $-30\,V$

  • $15\,s$ to $16\,s$: $0\,V$

• Power Calculations ($P = \frac{V^2}{R}$):

  • At $0\,V$: $P = 0\,W$.

  • At $30\,V$: $P = \frac{30^2}{300} = 3\,W$.

  • At $60\,V$: $P = \frac{60^2}{300} = 12\,W$. (Doubling voltage quadruples power).

• Average Power ($P_{\text{avg}}$) Calculation:

  • Sum the instantaneous power multiplied by duration: $(0\,W \times 1\,s) + (3\,W \times 2\,s) + (12\,W \times 2\,s) + (3\,W \times 2\,s) + (0\,W \times 2\,s) + (3\,W \times 2\,s) + (12\,W \times 2\,s) + (3\,W \times 2\,s) + (0\,W \times 1\,s) = 72\,Watt-seconds$.

  • Divide by total cycle time: $\frac{72\,Ws}{16\,s} = 4.5\,W$.

• Calculation of Effective Values for Stair-Step:

  • $V_{\text{eff}} = \sqrt{4.5\,W \times 300\,\Omega} \approx 36.7\,V$.

  • $I_{\text{eff}} = \sqrt{\frac{4.5\,W}{300\,\Omega}} \approx 122.5\,mA$.

Millisecond Timescale AC Analysis Example

• Scenario: A pulse wave with a $10\,ms$ cycle time and a $240\,\Omega$ resistive load.

• Voltage Profile (Positive Half: 0-5 ms):

  • $0-1\,ms$: $30\,V$

  • $1-2\,ms$: $80\,V$

  • $2-3\,ms$: $160\,V$

  • $3-4\,ms$: $80\,V$

  • $4-5\,ms$: $30\,V$

• Calculated Values:

  • At $30\,V$: $I = 125\,mA$, $P = 3.75\,W$.

  • At $80\,V$: $I \approx 333.3\,mA$, $P \approx 26.7\,W$.

  • At $160\,V$: $I \approx 666.7\,mA$, $P \approx 106.7\,W$.

• Average Dissipation and Effective Values:

  • Average Power ($P_{\text{avg}}$): Approximately $33.5\,W$.

  • Effective Voltage ($V_{\text{eff}}$): Approximately $89.7\,V$.

  • Effective Current ($I_{\text{eff}}$): Approximately $373.5\,mA$.

Sinusoidal AC Sources

• Sinusoidal sources have a smooth, continuously variant analog nature.

• Example Analysis: A sine wave peaking at approximately $170\,V$ (standard peak for $120\,V$ RMS systems) and a $400\,\Omega$ load.

  • At $84.9\,V$: $I = 212.3\,mA$, $P \approx 18\,W$.

  • At $147\,V$: $I = 367.5\,mA$, $P \approx 54\,W$.

  • At $169.7\,V$ (Peak): $I = 424.3\,mA$, $P \approx 72\,W$.

• Key Terms:

  • In Phase: In a purely resistive load, current and voltage peak and valley at exactly the same time. This synchronous oscillation is called being "in phase."

  • Power Waveform: The product of sinusoidal voltage and current forms a sinusoidal shape shifted entirely to the positive side of the axis.

  • Averaging Sinusoids: The sinusoidal power wave is symmetric around a horizontal line. For a wave peaking at $72\,W$ and bottoming at $0\,W$, the average power is exactly mid-way: $36\,W$.

• Effective Values for this Sine Wave:

  • $P_{\text{avg}} = 36\,W$

  • $V_{\text{eff}} = \sqrt{36\,W \times 400\,\Omega} = 120\,V$.

  • $I_{\text{eff}} = \frac{120\,V}{400\,\Omega} = 300\,mA$.

Conceptualizing AC through DC Equivalents

• The sandbox/sandpile metaphor: Averaging power can be thought of as taking uneven piles of sand and leveling them out to fill the valleys. The resulting horizontal level is the average power.

• AC is delivered in "bursts," whereas fixed DC delivers a constant "push."

• Increasing the oscillation speed of AC (e.g., from $16\,s$ cycles to standard power grid frequencies) allows devices like light bulbs to appear constant in intensity due to thermal inertia and the persistence of human vision.

Advantages of AC over DC

• Transmission: AC is substantially easier to generate and transmit over long distances than DC.

• Magnetic Fields: Changing voltage and current in AC naturally produce changing magnetic fields. These fields can be harnessed for:

  • Electric Motors.

  • Transformers (allowing for voltage level changes).

• DC does not innately carry a changing magnetic field, limiting its use in these specific electromagnetic applications.