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8. Electrical Components

• In mechanics we sum moments and forces, in electrical systems we sum current and voltage.

• There are many analogies between mechanical and electrical systems. Superficially there appears to be major differences. But, the difference between finding forces in a truss, and calculating mesh currents is small.

8.1 Mathematical Properties
• The most fundamental laws for doing circuits calculations are Kirchof’s current and voltage laws

• When doing (linear) circuit analysis there are three main areas,

DC (Direct Current): the simplest form of analysis.

AC (Alternating Current): the voltage and currents are continually changing.

Transient: here we look at the specific changes in a circuit as voltages, etc. are changed.

• Most electrical circuits are assumed to be at rest initially, but this is not essential.

• When electrical circuits reach DC steady state (infinite time), capacitors act as if they are open circuits, and inductors act as if they are short circuits.

• Typical components in electrical circuits include,

resistors

voltage/current sources

capacitors

inductors

op-amps

• Recall that current is the flow of electrons (matter) and hence is conserved.

• We can also consider the power and energy in an electrical system

#### 8.1.1 Resistors

• Resistors are the simplest electrical element, and these are assumed to be perfectly linear.

• Resistors tend to dissipate energy in a system as heat.

• One common application for resistors is as voltage dividers. This allows a voltage to be reduced, as a ratio of two resistance values. This technique is best used when there is no current drawn out.

#### 8.1.2 Voltage and Current Sources

• There are a variety of voltage and current sources. The simplest are chemical batteries.

• Find the output voltage Vo,

• These sources are assumed to generate a constant voltage or current. In truth the supply will weaken with age, or if excessive loads are drawn: this is very common with batteries.

• We will also use dependent sources. These allow us to model some devices that would normally be difficult to work with. The voltage and current values are calculated using other values in the circuit.

Problem 8.1 What if the input current is 1sin(2t)A?

#### 8.1.3 Capacitors

• Capacitors are two isolated metal plates very close together. We can force charge onto the plates, where it is stored. Hence, there is some capacity for storing electrons.

• Because of the electric field, when we change the voltage on a capacitor, it appears to allow a current to flow. But, in actuality when charge collects on one plate, it forces charge out of the other plate.

• When the voltage to a capacitor is not changing it will not allow any current flow. As a result we tend to call these devices DC blocking.

• Some capacitors (especially larger ones >1 uF) have a polarity. The symbols for these indicates the positive direction. If these are connected backwards, they will often explode (literally).

Problem 8.2 Find the current as a function of time.

#### 8.1.4 Inductors

• Inductors are a necessary components in many circuits, but when possible we try to design these out. This is because they are large, prone to trouble (eg, wave a key near one) and costly.

• Inductors are typically made by wrapping a wire coil. This involves many turns (loops) and will often be given an inner core with a higher magnetic permeability.

• Inductors are the opposite of capacitors. The more current we pass through, the higher the voltage. As current flows a magnetic field is built, and when the current is removed the field collapses generating current.

• These devices tend to allow DC current to pass, but resist changing current. Hence these devices are often known as AC blocking.

Problem 8.3 Find the current as a function of time.

#### 8.1.5 Op-Amps

• Op-amps are not simple devices, but they can be made to act that way.

• Most inexpensive op-amps will handle frequencies up to 100KHz, and gains up to 100,000.

• They find good application for special functions such as,

adders/subtractors/multipliers/dividers

amplifiers

impedance isolators

etc.

• The basic calculation for an op-amp assumes that both inputs will be driven to the same voltage.

Problem 8.4 Find the input/output ratio,

• The more complex op-amp model includes a dependent voltage source

8.2 Example Systems
• To develop equations for electrical circuits there are some items to follow,

1. For each component, assume a current flow, then label the current direction, and resulting voltage polarity across each element.

2. Examine the circuit,

Look for sections where components can be combined to have simpler values.

Look for elements such as voltage dividers, current dividers, etc.

Look for nodes where the voltage will be known, apply Kirchhof’s current law.

Look for branches where the current will be known, apply the Voltage law.

If multiple loops with separate sources exist, use the mesh method.

3. Develop equations

4. Eliminate unwanted values.

• Consider the system below,

Problem 8.5 Find the equation relating the output and input voltages,

8.3 Problems
Problem 8.6 Find the combined values for resistors, capacitors and inductors in series and parallel.

Problem 8.7 Given the circuit below, find the transfer function. Simplify the results

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Answer 8.7

Problem 8.8 a) Write the differential equations for the system pictured below.

b) Put the equations in input-output form.

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Answer 8.8

Problem 8.9 Develop the differential equations for the system below, and use them to find the response to the following inputs. Assume that the circuit is off initially.

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Answer 8.9 a) oscillating, amplitude -0.5 to 0.5, f=100 rad/sec

b) oscillating, amplitude -0.25 to 0.25, f=1000000rad/sec

c) first order response, time const=.001 approx., final value=0.5

Problem 8.10 Find the input-output equation for the circuit below, and then find the natural frequency and damping coefficient.

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Answer 8.10

Problem 8.11 Examine the following circuit and then derive the differential equation.

Problem 8.12 Consider the following circuit.

a) Develop a differential equation for the circuit.

b) Put the equation in state variable matrix form.

Problem 8.13 Examine the following circuit and then derive the differential equation.

Problem 8.14 Consider the following circuit.

a) Develop a differential equation for the circuit.

b) Put the equation in state variable matrix form.