Transformer Circuit Review: Ideal Transformers, Conservation of Energy

In a closed system, energy can be transferred through different forms (heat, kinetic energy, potential energy, etc), but not created nor destroyed. For a passive device such as a transformer, the energy in the system must also follow. This is termed conservation of energy.

A transformer is a passive circuit component which follows these basic formulas, where P is the power and n is the number of turns on the transformer:

Pout=Pin

n2/n1=V2/V1

Consider an electrical transformer with turns ratio N, what is the output voltage?  What is the output current?

The output voltage Vout=N*Vin.

A correct answer to this question must satisfy that power is conserved. This means that the output power must equal the input power. Power = voltage * current.

The impedance of the transformer for N turns ratio:

            Zout = Vout/Iout = (N*V­in)/(Iin/N) = N2 * Zin

Mean Squared Noise Power

What does it mean when people say “mean squared”?

The average value of a noise waveform is zero. The square of the waveform mean is also equal to zero. The square of the noise signal and the mean of the square are non-zero. This is because the negative values associated with the zero-mean noise waveform are made positive by squaring, and the entire waveform is positive. Taking the root of the averaged square of the waveform yields the RMS.

The mean of the squared (“mean square”) noise waveform is the noise power with respect to a 1 Ohm resistor (units: V2/Ω=W, “power” if noise signal is a voltage signal, and units I2/Ω=W, also “power” if noise waveform is current).

The power spectral density is the power of the signal in a unit bandwidth.

What is a current noise power spectral density?

The correct definition of current noise spectral density is the mean of the squared current per hertz, <i2>. The units are A2/Hz.

The square of the mean is equal to zero, because the mean of the noise waveform is zero and squaring that number remains zero. The mean of the square is a non-zero number. Taking the square of a noise current results in a positive valued current waveform. Taking the average of the square is a non-zero number used for the spectral density.

Displacement Current

One of Maxwell’s equations, Ampere’s circuit law, tells us that there are two sources of magnetic fields: conduction currents and displacement currents. Conduction current is very familiar to most people: it is flow of electrons through a conductor due to an applied electric field. The electrons hop from atom to atom within the conductor and rate at which this happens is termed displacement current.

The differential form of the equation also shows another source of magnetic field: displacement current. For a static field (not time varying/DC), there is no displacement current and the Ampere equation is

ampere.PNG

However for time varying fields, the right hand side contains an extra term, which is displacement current density.

dcurrent.PNG

“J” is the conduction current density which is equivalent to conductivity multiplied to E field (also known as Ohm’s Law in point form). Taking the surface integral of the second term on the right hand side yields displacement current.

The important distinction here is that displacement current is not due to the flow of electrons directly, but rather a time varying electric field. A common example is that of a capacitor with an AC voltage source applied to the device. While there is no conduction current flowing through the dielectric which separates the plates, there is still a current through the capacitor (displacement current) .

capacitor

The following image shows two surfaces about a capacitor with an AC voltage applied. If Ampere’s law is applied to surface one, the right hand side is equal to the conduction current flowing in the wire. However, if the law is applied to surface two it demonstrates that no conduction flows through the capacitor. The same closed path is used in the integration (L) and therefore the right hand side cannot be zero. This means a new term for displacement current must be inserted to satisfy the equation. This is demonstrated in the equation below the figure.

displacement.PNG

provingd.PNG

Negative Resistance

RF/Photonics Lab
November 2019
Jared Alves

Negative Resistance

Arguably the most fundamental equation in electrical engineering is Ohm’s Law (V = I*R) which states that voltage is proportional to the product of current and resistance. From this equation, it is apparent that increasing a voltage across an element will increase the current through that element assuming the resistance is fixed. With a resistor, electrical energy is dissipated in the form of thermal energy (heat) due to the voltage drop between the terminals of the device. This is in direct contrast to the concept of negative resistance, which causes electrical power to be produced instead of dissipated.

Generally, negative resistance refers to negative differential resistance, as negative static resistance is not typically used.  Static resistance is the standard V/I ratio while differential resistance takes the derivative dV/dI. The following image shows an I-V curve with several slopes. The inverse of B yields a static resistance, and the inverse of line C is differential resistance (both evaluated at the point A). If the differential curve has a negative slope, this indicates negative differential resistance.

negres1

Even when differential resistance is negative, static resistance remains positive. This is because only the AC component of the current flows in the reverse direction. A device would consume DC power but dissipate AC power. This is because the current decreases as the voltage increases, leading to

negres2

A tunnel diode is a semiconductor device that exhibits negative resistance due to a quantum mechanical effect called “tunneling”.