Lesson 13

Coordinate Systems

Picking back up with phase angles. Designers need to be able to display and understand the phase angle of a circuit. That circuit would contain resistance, inductive and/or capacitive reactance. The display is done using polar coordinates.

Let’s look at figure E5-1 which could be on your exam.  

This two-dimensional chart shows rectangular coordinates on an X and Y scale. On the horizontal axis or X axis, plot the impedance of a pure resistance. The vertical or Y axis is for reactance. A logarithmic Y-axis scale is most often used for graphs of circuit frequency response.  

Let’s recap what happens on each axis. What rectangular coordinates are used to graph the impedance of a circuit? The X axis represents the resistive component and the Y axis represents the reactive component.

Point 1 in Figure E5-1 best represents the impedance of a series circuit consisting of a 300-ohm resistor and a 19-picofarad capacitor at 21.200 MHz.

Point 3 represents the impedance of a series circuit consisting of a 300-ohm resistor and an 18-microhenry inductor at 3.505 MHz.

Point 4 represents the impedance of a series circuit consisting of a 400-ohm resistor and a 38-picofarad capacitor at 14 MHz.

Phase angle and magnitude are how impedances are described in polar coordinates. For instance, pure inductive reactance in polar coordinates would be shown as a positive 90-degree phase angle.

A phasor diagram is used to show the phase relationship between impedances at a given frequency. Capacitive reactance in rectangular notation is represented as: –jX . That’s important in two questions. 

The impedance 50-j25 represents 50 ohms resistance in series with 25 ohms capacitive reactance. 0 – j100 represents pure capacitive reactance of 100 ohms in rectangular notation. 

Measuring Q

If you want to know how efficient your resonant circuit operates you measure the Q factor. This is also known as the Quality factor. Q generally applies to many components that have resonant circuits or devices. The higher the Q, the less dampening or restriction there is on the circuit resonance. The higher the Q, the higher the Quality.  

You can calculate the Q in different ways. In an RLC parallel resonant circuit Q is: resistance divided by the reactance of either the inductance or capacitance. Series circuits calculate Q differently.

What happens in 2 different circuits when you increase the Q?  

In a series resonant circuit, increasing Q means internal voltages increase. So as you would design a circuit, you would need to manage those voltages.  

In an impedance matching circuit, increasing Q means the matching bandwidth is decreased. That might impact an antenna tuning circuit. 

So we know how to measure Q and calculate the resonant frequency of a circuit. We can put those together to calculate the half-power bandwidth of a circuit. This is a simple calculation. Half power bandwidth equals the resonant frequency of the circuit in kilohertz divided by the Q factor. 

BW = f/Q  

There are two half-power bandwidth calculations needed for the exam. First, it’s a circuit with a resonant frequency of 7.1 MHz and a Q of 150.  Here are the steps:

  • Take the frequency of 7.1 MHz and convert it to kilohertz. That gives you 7,100. 
  • Divide the 7,100 kilohertz by the Q of 150

Your answer is 47.3 kHz. Once you convert the units, it’s a simple division!

Second calculation. The circuit has a resonant frequency of 3.7 MHz and a Q of 118.  

  • Convert 3.7 MHz to 3,700 kilohertz.
  • Divide 3,700 by the Q of 118.

Your answer is 31.4 kHz. Take advantage of the ability to use a calculator on the exam to get these simple answers.  

You remember how capacitors work from earlier studies. They are made of two conductive surfaces, like two plates of metal separated in the middle by an insulator. Capacitors store a charge. We can measure the time it takes to charge a capacitor using the term “one time constant.”

One time constant is the time required for the capacitor in an RC circuit to be charged to 63.2% of the applied voltage. It’s also the time for a cap to discharge to 36.8% of its initial voltage.

T is the symbol for one time constant. T equals Resistance in ohms times Capacitance in farads.

T = RC

Ready to calculate T in a circuit? This circuit has two 220-microfarad capacitors and two 1-megohm resistors, all in parallel.  

  • When resistors are connected in parallel, you get reduced resistance. Two one megohm resistors in parallel equals 0.5 megohms.
  • When capacitors are connected in parallel you get increased capacitance. Two 220-microfarad capacitors in parallel equals 440 microfarads. 
  • Using those in the formula, T = 0.5 megohms times 440 microfarads. That comes out to 220.

That result is in seconds. So, the answer is 220 seconds. Easy enough to remember, because the question and the answer both have the number 220.

Practical Electrical Concepts

When building circuits, there are practical considerations to take into account. That’s in addition to the theoretical ones we have been covering. Here’s a look at some of those. 

The reactive power discussion covered current and voltage. What about wattage? Reactive power is wattless, nonproductive power.  So there are times you will want to minimize it. Long lead lengths in VHF and above can be subject to reactance. Keep lead lengths short for components to minimize inductive reactance.   

There is another benefit of short connections at microwave frequencies. They are used to reduce phase shift along the connection.

What if you used bigger conductors? As a conductor’s diameter increases, it increases electrical length. So, an antenna that uses thicker cable or elements will be electrically longer. 

AC electrical current has a tendency to have the largest density near the surface of a conductor. This is called the skin effect. This impacts frequency too. With conductor skin effect resistance increases as frequency increases because RF current flows closer to the surface. Skin effect is also the primary cause of loss in film capacitors in RF applications.

Selecting the right inductor or capacitor is not a purely theoretical choice. For instance, electrolytic capacitors can be unsuitable for use in RF applications. They have issues with parasitic inductance. It works the other way for inductors. They can parasitically have inter-turn capacitance which creates a self-resonance.

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