E5C: Coordinate Systems and Phasors
Impedance is a complex quantity — it has both a magnitude and a direction. To work with it mathematically and graphically, engineers use two complementary coordinate systems: rectangular notation and polar notation. Both describe the same impedance from different angles, and knowing how to move between them is essential for circuit analysis.
This lesson covers rectangular coordinates, polar coordinates, phasor diagrams, and logarithmic frequency response graphs — all fundamental tools for analyzing RF circuits at the Extra class level.
Rectangular Coordinates and Impedance Notation
In rectangular (Cartesian) coordinates, impedance is written as Z = R + jX, where R is the resistive component and X is the reactive component. The letter j is the imaginary unit used in electrical engineering to distinguish reactance from resistance on the number plane.
When you plot impedance on a rectangular coordinate graph:
- The horizontal (X) axis represents the resistive component (real part).
- The vertical (Y) axis represents the reactive component (imaginary part).
- Positive Y values indicate inductive reactance (+jX).
- Negative Y values indicate capacitive reactance (−jX).
A pure resistance has no reactance, so it plots directly on the horizontal axis at its ohm value. A pure inductive reactance has no resistance, so it plots on the positive vertical axis. A pure capacitive reactance plots on the negative vertical axis.
A mixed impedance like 50 − j25 represents 50 ohms of resistance in series with 25 ohms of capacitive reactance. The negative sign on j25 indicates capacitive reactance. If it were 50 + j25, that would be 50 ohms resistance in series with 25 ohms of inductive reactance.
- Z = R + jX → inductive, X is positive
- Z = R − jX → capacitive, X is negative
- Pure resistance: plots on horizontal axis, no j component
- Pure capacitance: 0 − j|X|
- Pure inductance: 0 + j|X|
Polar Coordinates and Phase Angle
Polar coordinates describe impedance using two values: magnitude and phase angle. This is written as Z = |Z|∠θ, where |Z| is the total impedance magnitude in ohms and θ is the angle in degrees relative to the positive horizontal axis.
Polar notation is particularly useful when you want to display the phase relationship between voltage and current in a circuit. For circuits containing resistance, inductive reactance, and/or capacitive reactance, polar coordinates make the phase angle immediately visible without calculation.
- Pure inductive reactance → phase angle of +90° (voltage leads current by 90°)
- Pure capacitive reactance → phase angle of −90° (current leads voltage by 90°)
- Pure resistance → phase angle of 0° (voltage and current are in phase)
Phasor Diagrams
A phasor diagram is a graphical tool that shows the phase relationships between impedances at a given frequency. Each impedance or voltage is drawn as an arrow (phasor) whose length represents magnitude and whose angle from the horizontal represents phase.
Phasor diagrams make it easy to see how inductive and capacitive reactances partially cancel each other, and how the net reactance combines with resistance to produce the total impedance. They are commonly used in antenna matching, filter design, and transmission line analysis.
Unlike time-domain waveform diagrams, phasor diagrams represent steady-state sinusoidal signals at a single frequency. The angle between phasors directly shows the phase difference between the quantities they represent.
Logarithmic Frequency Response Graphs
When engineers plot the frequency response of a circuit — how gain or attenuation changes with frequency — they almost always use a logarithmic Y-axis. This is because circuit behavior can span many orders of magnitude, and a linear scale would compress important detail into an unreadable range.
On a logarithmic scale, equal visual spacing represents equal ratios rather than equal differences. This format lets the graph show a wide dynamic range clearly and is the standard for filter response curves, amplifier gain plots, and spectrum analyzer displays. The X-axis (frequency) is often logarithmic as well, producing the familiar Bode plot format.
Impedance Calculations with Figure E5-1
Figure E5-1 shows a rectangular coordinate impedance chart with labeled points distributed across the four quadrants. Three exam questions ask you to identify which labeled point corresponds to a specific series circuit at a given frequency. The approach is to calculate the reactance of the component, form the rectangular impedance Z = R ± jX, and identify the correct quadrant and approximate magnitudes.
XC = 1 / (2π × 14×10⁶ × 38×10⁻¹²) ≈ 300 Ω
Z = 400 − j300 → lower-right quadrant (positive R, negative X) → Point 4
XL = 2π × 3.505×10⁶ × 18×10⁻⁶ ≈ 396 Ω
Z = 300 + j396 → upper-right quadrant (positive R, positive X) → Point 3
XC = 1 / (2π × 21.2×10⁶ × 19×10⁻¹²) ≈ 395 Ω
Z = 300 − j395 → lower-right quadrant (positive R, negative X) → Point 1
The key skill is identifying the quadrant first: inductive circuits produce positive j values (upper half of the chart), and capacitive circuits produce negative j values (lower half). The resistive component sets the horizontal position. Once you know the approximate magnitudes of R and X, you can match the impedance to the correct labeled point on Figure E5-1.
E5C Practice Questions
Check Your Knowledge
E5D: RF Effects →
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