Reactance Impedance and Phasors

Welcome to the Frequency Domain

So we now have a way to express AC waveforms in the time domain (provide a time t and we can compute the value) which is great, however in AC circuits what we are particularly interested in is how a circuit responds to different frequencies.

With AC circuits we now have a frequency component which turns out to have an impact on the effective resistance of capacitors and inductors, however resistors maintain their resistance value independent of frequency.

Before we dive into some formulas for "effective resistance" of capacitors and inductors vs frequency let's build some intuition as to why frequency causes this effect. Let's first look at capacitors and assume we apply a high frequency of say 100 MHz to a circuit containing a resistor and capacitor. Now, in terms of charges flowing in the circuit because the frequency is very high we will only have charges move in a given direction for a very short period of time before switching over to the other direction and then repeating this periodically so what this means is we do not have time to charge the capacitor and an uncharged capacitor looks like a short circuit as it has no internal electric field to oppose the charges flowing onto and off the plates. This then means that as we increase frequency the capacitor looks more and more like a short circuit, i.e. less "effective resistance".

For inductors the opposite holds true, for a large frequency the current will act to change the magnetic flux flowing through the coil very quickly which means by Lenz's Law we will have a significant induced voltage acting to oppose the change which greatly resists the current flow.

Reactance

How do we typically refer to this "effective resistance" well we extend the concept of resistance to reactive components (inductors and capacitors) which react to changes in voltage and current and call it reactance which varies with frequency due to the effects described above. Higher reactance means a lower current flows for the same applied voltage, which also applies to resistance.

Like resistance, reactance is also measured in Ohm's. However, reactance can be positive or negative which denotes whether the circuit acts overall inductive (positive) or capacitive (negative). We denote reactance with the symbol X.

For a capacitor, its reactance is given by:

Which we expect as a capacitor has negative reactance and as we increase the frequency the opposition to current decreases i.e. it is inversely proportional. Likewise as we increase the capacitance it takes longer for charges to build up on the plates which reduces the opposition to current, i.e. it is inversely proportional.

For an inductor, its reactance is given by:

Which we also expect as we increase the frequency the current through the inductor is trying to change more rapidly so we have a stronger opposition to the change via an induced voltage which acts to oppose the current. Likewise as we increase the number of turns the inductor has a larger induced voltage and hence opposition to a change in current.

Impedance

How do we then put it all together so we express the total opposition of a circuit in terms of resistive and reactive components? We define that to be known as the impedance. Impedance is simply the summation of the total resistive and reactive components and has the symbol of Z. Before we express it algebraically we need a graphical method to display impedance and this brings us to the concept of phasors.

Phasors

As we have mentioned we express AC waveforms in the time domain by specifying three components which are the amplitude, frequency and phase. If we only consider amplitude and phase then we can use a complex number representation via polar coordinates to show both amplitude and phase in a 2D plane and this is known as a phasor.

For this the magnitude of the phasor represents the amplitude, A, and the angle the phase, ϕ:

In phasor format, we can express a phasor via the following construct:

Which is shorthand for:

Where j is the imaginary number, we use j instead of i as we use i to refer to AC current.

Phase shift caused by Capacitance

As capacitive reactance is an imaginary number, it introduces a phase shift relative to the input voltage.

Well what exactly does that mean and why in real world terms not mathematics? It means the voltage across the capacitor does not directly track the input voltage direction, instead we have a lag in changing voltage (phase shift mathematically speaking). Intuitively speaking if we assume we send a cosine wave without a phase shift then at time 0 the voltage across the capacitor will be 0V which will slowly ramp up as charges enter the plate while the input voltage is reducing and so the output voltage across the capacitor looks like a sine wave, i.e. a phase shifted cosine wave.

TODO: Add equation and show graphs of waveforms and argand diagram.

Phase shift caused by Inductance

Similarly for inductors we have a phase shift albeit in the other direction. Mathematically this is because the imaginary component sits on the other axis compared to the capacitor. Intuitively if we take the same above example at time 0 assuming switching instantaneously there will be a huge voltage across the inductor due to the induced voltage.

TODO: Add equation and show graphs of waveforms and argand diagram.

Example:

1. Represent a sine wave with amplitude 5 and phase 90 degrees in phasor format:
The amplitude is simply 5 and the phase is pi/2 in radians so we simply have:

Impedance - Generalized opposition to current

Putting this all together, we can express impedance by summing resistance and reactance using complex number notation:

And phasor notation:

Examples:

1. Determine the impedance of the circuit shown below:
Recall that impedance is the summation of the resistive and reactive components within the circuit. In the above the resistive component is simply 1k while the reactive component is capacitive which we can use the reactance formula for capacitors to calculate. The total impedance is then given by:

2. Determine the impedance of the circuit shown below:

As above we can break the circuit down in terms of resistive and reactive elements and sum them together: We can then calculate the reactance of the capacitive and then inductive components:

Questions:

1. Determine the impedance of the circuit shown below:

Z=2000+j0.148

Z=1000-j36.122