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5. AC Circuit Analysis

• There are a number of techniques used for analysing non-DC circuits.

• These techniques are,

phasors: for single frequency, steady state systems

Laplace transforms: to find steady state as well as transient responses

etc

5.1 Phasors
• Phasors are used for the analysis of sinusoidal, steady state conditions.

• Sinusoidal means that if we measure the voltage (or current) at any point ‘i’ in the circuit it will have the general form,

• Steady state means that the transients have all stopped. This can be crudely though of as the circuit has ‘charged-up’ or ‘warmed-up’.

• Consider the example below,

• Steady state is another important concept, it means that we are not concerned with the initial effects when we start a circuit (these effects are known as the transients). The typical causes of transient effects are inductors and capacitors.

• We typically deal with these problems using phasor analysis. In the example before we had a voltage represented in the time domain,

• Basically to do this type of analysis we represent all components voltages and currents in complex form, and then do calculations as normal.

• Consider the simple example below,

#### 5.1.1 RMS Values

• When dealing with alternating currents we are faced with the problem of how we represent the signal magnitude. One easy way is to use the peak values for the wave.

• Another common method is to use the effective value. This is also known as the Root Mean Squared value.

#### 5.1.2 LR Circuits

• One common combination of components is an inductor and resistor.

#### 5.1.3 RC Circuits

• Capacitors are often teamed up with resistors to be used as filters,

#### 5.1.4 LRC Circuits

• These circuits tend to weigh off capacitors and inductors to have a preferred frequency.

#### 5.1.5 LC Circuits

• Inductor capacitor combinations can be useful when attempting to filter certain frequencies,

5.2 AC Power
• Consider the power system shown below,

• The generator converts some form of mechanical force into electrical power. This power is then distributed to consumers over wires (and through transformers). Finally at the point of application, each load will draw a certain current, at the supply voltage: operating at a rated power. The voltages supplied this way are almost exclusively AC. Also in an ideal situation the load will be pure resistance, but in reality it will be somewhat reactive.

• Another important example of power delivered is when impedence matching between audio amplifiers and audio speakers. Most consumer systems are 50ohm for maximum power transfer and minimum distortion.

#### 5.2.1 Complex Power

• Consider the basic power equation,

5.2.1.1 - Real Power
• The relationship for real power is shown below where the current and resistance are in phase (although the values are rarely perfectly in phase).

• When the current and voltage are D.C. (not charging) the circuit contains pure resistance, and the power is constantly dissipated as heat or otherwise. Notice that the value of P will always be positive, thus it never returns power to the circuit.

5.2.1.2 - Average Power
• An average power can be a good measure of real power consumption of a resistive component.

5.2.1.3 - Reactive Power
• When we have a circuit component that has current ±90° out of phase with the voltage it uses reactive power. In this case the net power consumption is zero, in actuality the power is stored in and released from magnetic or electric fields.

• Consider the following calculations,

5.2.1.4 - Apparent Power
• In all circuits we have some combination of Real and Reactive power. We can combine these into one quantity called apparent power,

5.2.1.5 - Complex Power
• We can continue the examination of power by assuming each is as below,

5.2.1.6 - Power Factor
• The power factor (p.f.) is a good measure of how well a power source is being used.

• It is common to try to correct power factor values when in industrial settings. For example, if a large motor were connected to a power grid, it would introduce an inductive effect. Capacitors can be added to compensate.

5.2.1.7 - Average Power Calculation
• If we want to find the average power, consider the following,

5.2.1.8 - Maximum Power Transfer
• Consider the Thevenin circuit below. We want to find the maximum power transfered from this circuit to the external resistance.

5.3 3-Phase Circuits
• 3-phase circuits are common in large scale power generators and delivery systems.

• These systems carry 3 phases of voltage, each 120 degrees out of phase, on three separate conductors. If these three wires are connected through a balanced load the sum of currents is zero. Most systems provide a fourth wire as a neutral.

• As a result loads can be connected in a delta configuration with no neutral.