# Time Varying Signals

There are several ways to represent signals within the time domain, that is, with respect to time; the simplest way is to say that:

- is the voltage at a current time
- eg: would be the voltage when time is equal to zero or when the signal crosses the y-axis
- is the amplitude of the signal
- is the angular frequency of the signal (measured in rad s
^{-1})- is the frequency of the signal

- is the current time

Remember that

We can also describe a signal based on its gradient at t=0; if the gradient of the signal is positive before t=0, the signal is described as leading, if it is negative then it is described as lagging.

If the sine wave does not begin at t=0 then our equation is slightly different:

- is the offset in radians
- It is positive for leading signals
- and negative for lagging signals

To put this into perspective, we can say that a cosine wave has an offset of so our equation would be:

# Complex Numbers

Before we continue onto frequency varying signals, we will quickly run through complex numbers to give your memory a little refresh if it needs it, but feel free to skip to the next section if you want.

Complex numbers are numbers that are represented in the form

*z*is a complex number- a is the real part
- b
*j*is the imaginary part*j*is equal to- Note that it is more commonly referred to as
*i*but in the engineering world we use*j*

This is known as Cartesian form.

## Modulus & Argument

A complex number can be plotted on an *Argand Diagram* where the horizontal represents the real part and the vertical axis represents the imaginary part:

Using this diagram we can then calculate the argument, which is the size of the angle from the positive real axis. In the diagram above this is easy to calculate:

In the example above this would be:

## Polar Form

Complex numbers can be represented in Polar Form:

*r*is the modulus of the complex number- is the argument (or phase) of the complex number

## Trigonometric Form

We can also represent complex numbers in trigonometric form:

*r*is the modulus of the complex number- is the argument (or phase) of the complex number
- is the real part
- is the imaginary part

Using the Euler Relationship we can convert between Polar and Trigonometric form:

# Frequency Varying Signals

As well as signals in the time domain, in analogue electronics we often have to deal with signals in the frequency domain, that is, with respect to frequency; we do this using complex numbers and *Phasors*.

Let’s take our original equation:

(Trigonometric Form)

(Polar Form)

(Angular Form or as a *Phasor*)

We will be using complex numbers when dealing with multiple voltages that have different phases.

# Reactive Circuits

A reactive circuit is one which contains components that store energy, such as a capacitor or inductor. These components will have a phase shift between voltage and current.

**Impedance **is the resistance to the complex ratio of voltage and current within AC circuits. Take a look at the following circuit:

is the impedance, note the rightwards harpoon on top of Z, V and *I*, this signifies that this quantity is complex. We can convert ohm’s law into Phasor form so that we can easily deal with complex circuits:

Let’s imagine that is equal to 5 Volts and that is equal to 2+*j* Amps:

In this example, the current is leading the voltage by 26.6°, this means that the phase angle of the impedance will be negative; let’s now calculate :

## Capacitors

For an ideal capacitor, the voltage and current are related by the equation:

- C is the capacitance
- is the change in voltage with respect to the change in time

We can re-write the above equation to take complex voltage into account

Where *I* is the amplitude of the current. This is a very important equation to remember and we can re-write it in Phasor form:

If we want to calculate the impedance of the capacitor:

## Inductors

For an ideal inductor, the voltage and current are related by the equation:

- L is the inductance
- is the change in current with respect to the change in time

We can re-write the above equation to take complex voltage into account

Where is the amplitude of the voltage. This is a very important equation to remember and we can re-write it in Phasor form:

If we want to calculate the impedance of the inductor:

# Resistive Circuits

A resistive circuit is one which contains components that do not store energy, such as a resistor. These components will only dissipate energy and will have voltage and current in phase. Let’s first imagine a resistor in an AC circuit:

We can calculate the voltage drop between A and B as:

- is power with respect to time
- is voltage with respect to time
- is current with respect to time

Since we are dealing with an AC circuit, the voltage and current will not always be constant and so we can easily calculate the average power:

# Complex Power

## Root Mean Square

The Root Mean Square (RMS) is a measure of the magnitude of a varying quantity. This is especially useful when we have a signal such as a sine wave (which alternates between positive and negative) and we want to find an ‘average’ value of it.

The root mean square of a voltage can be calculated with

## Power

The equation to calculate power (measured in Watts) in electronics is:

- is the complex conjugate of

The RMS value of current and voltage multiplied together will give the average power:

- P is the Active Power: this is power dissipated and is measured in Watts (W).
- Q is the Reactive Power: this is power that is stored and is measured in Volt Amps (VA).

### Example

Therefore, if we have and :

Let’s now look at applying these equations to our capacitor, inductor and resistor.

## Capacitor

For a capacitor, since current leads voltage; therefore to calculate the average power:

Therefore

## Inductor

For an inductor, since voltage leads current; therefore to calculate the average power:

Therefore

## Resistor

For a resistor, since both voltage and current are in phase; we can calculate average power with:

There is no imaginary part so all the power is dissipated:

## Power Factor

is known as the *Power Factor* and it can be used to determine the load is resistive or reactive. Since this is the real part of our complex power into the load, it is equal to the power dissipated, therefore:

- If then and so the load is
**Resistive** - If then and so the load is
**Reactive**

# Examples

We will now take a look at several examples to put this all into practice.

## Resistor and Inductor in Series

Similarly, a resistor and capacitor in series would have the equation:

## Resistor and Capacitor in Parallel

*Remember that impedances follow the same circuit laws as resistances.*

# Summary

We have gone into a lot of detail in this tutorial and there is a lot to remember. To finish up we will provide a summary of what we have been through:

Capacitor | ||||

Inductor | ||||

Resistor |

- If phase angle is
**negative**: current leads voltage - If phase angle is
**positive**: voltage leads current

Capacitor | ||

Inductor | ||

Resistor |