EC-1.3 Sinusoidal Steady State

What this is and why it exists

Under a steady sinusoid, every voltage and current in a linear circuit is a sinusoid at the same frequency. Only two things are unknown anywhere: amplitude and phase.

Two unknowns fit neatly into one complex number. Doing that turns the differential equations into algebra, and every method from the earlier topics carries over unchanged.

The vocabulary

• Phasor — a complex number holding the amplitude and phase of a sinusoid.
• Impedance — the complex ratio of voltage phasor to current phasor.
• Admittance — the reciprocal of impedance.
• Reactance — the imaginary part of an impedance. It stores energy rather than dissipating it.
• Real power — the power that does work, measured in watts.
• Reactive power — power that moves back and forth without doing work.
• Apparent power — the product of the measured voltage and current magnitudes.
• Power factor — the cosine of the angle between voltage and current.
• Resonance — the frequency at which inductive and capacitive reactances cancel.
• Quality factor — a measure of how sharp a resonance is.

The mental model

The phasor trick rests on one property. Differentiating a sinusoid gives another sinusoid at the same frequency, shifted a quarter cycle and scaled by the frequency. In complex form that is exactly multiplication by the frequency variable. So a derivative becomes a multiplication, and the calculus disappears.

The rule that keeps this honest is to take the real part at the very end, and never in the middle. Everything between is complex arithmetic.

Impedance is what makes the earlier work reusable. A resistor's impedance is its resistance. An inductor's impedance grows with frequency. A capacitor's impedance falls with frequency, and both are imaginary. Once every element has an impedance, node analysis and mesh analysis and every theorem from the previous topic work exactly as before. The only new thing is that the numbers are complex.

Power is where people come unstuck, because three different quantities are all called power.

Real power is what does work and what a meter charges for. Reactive power is energy moving into a reactance and back out again each cycle, doing nothing useful, but requiring real current in the wires. Apparent power is what you get by multiplying the measured voltage and current magnitudes, ignoring any phase difference.

The angle between voltage and current decides the split, and its cosine is the power factor. A factory full of motors draws current lagging behind the voltage. It pays for wires and transformers sized for apparent power, while doing work equal only to the real power. Adding capacitors shifts the angle back towards zero. That is all a power factor correction capacitor does.

Maximum power transfer returns here with one change. The best load is the complex conjugate of the source impedance. The resistive parts match, and the reactive parts are equal and opposite, so they cancel. Every radio-frequency matching network exists to arrange that.

Resonance is where the reactances cancel by themselves. In a series circuit that leaves only the resistance, so the impedance is at its minimum and the current at its peak. In a parallel circuit the same cancellation gives maximum impedance and minimum current from the source. Knowing which is which decides whether your circuit passes that frequency or rejects it.

Quality factor measures the sharpness. A high value means the circuit stores much more energy each cycle than it loses, so its response falls away quickly either side of resonance. Bandwidth is the resonant frequency divided by that quality factor. Selectivity and bandwidth are therefore the same number seen from two sides, which is why you cannot have both.

What you should now be able to explain or do

Explain why representing a sinusoid as a complex number removes the calculus. Write element impedances and reuse node or mesh analysis without change. Separate real, reactive and apparent power, and say what power factor correction does. State the conjugate matching condition and why the reactance cancels rather than matches. Find a resonant frequency, and relate quality factor to bandwidth.

Check yourself