User:Alfred Centauri

Alfred Centauri

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I am currently a non-traditional (read 'old') Ph.D. student of Electrical Engineering with a goal of teaching EE at my alma mater.

I became interested in electronics in 1973 when I watched my Dad build his first Heathkit - a Vectorscope / Colorbar generator combo. A year later, I built my first Heathkit - the AD-27, a roll-top compact stereo system - which sparked my interest in audio electronics.

My interests here at Wikipedia are electronics (of course!) and physics.

EE nitpickery
For reasons beyond my comprehension, I find myself thinking about the conventional wisdom in EE and asking: "is this true"? A sampling follows...

Do capacitors block DC?
It is often said that capacitors block DC or equivalently, that a capacitor is an open circuit at DC. Is this true? What is true is that the DC steady state current through a capacitor is identically zero. In DC steady state, all circuit voltages and currents are constant. By the fundamental capacitor equation:


 * $$i_C = C \frac{dv_C}{dt}$$

the capacitor current must be zero if the capacitor voltage is not changing with time. Does this imply that a capacitor blocks DC current? Clearly, the answer is no. According to the equation above, a constant (DC) current through a capacitor can exist if the voltage across the capacitor changes at a constant rate. For example, a DC current of 1mA through a capacitor with capacitance of 1000$$\mu$$F causes the voltage across the capacitor to change at the rate of 1V per second. Theoretically then, a capacitor does not block DC current.

Obviously, for any real current source connected to a real capacitor, the capacitor voltage cannot continue to change forever. Eventually, the capacitor dielectric would break down or the DC current source would reach its maximum working voltage. These real world limitations do not take away from the fact that the capacitor does not block DC current.

But isn't the impedance of a capacitor infinite for DC?

The formula for the impedance of a capacitor is given by:


 * $$Z_C = \frac{1}{j 2 \pi f C}$$

where f is the frequency of the AC voltage and current associated with the capacitor. It is often said that "DC is just AC with zero frequency". In a sense, this is true. Thus, it seems reasonable to believe that setting the frequency to zero in the equation above would give the impedance of the capacitor 'at DC'. Mathematically, this is problematic because division by zero is undefined, however it is clear that as the frequency approaches zero, the impedance increases without bound so one can properly say that the impedance of a capacitor 'goes to infinity' as the frequency goes to zero. What exactly does this mean?

To answer this, we need to understand how the formula for the impedance of a capacitor is derived. The first step in any deriviation of the impedance is to assume that the voltage across the capacitor is a sinusoidal function of time with constant amplitude, frequency, and phase. This step is crucial to the derivation of impedance and is equivalent to requiring that the circuit has settled into AC steady state. Now consider the following: if the frequency of this sinusoid is set to zero, the voltage across the capacitor becomes constant. In other words, setting the frequency to zero is equivalent to requiring that the circuit has settled into DC steady state. We already know that the DC steady state current through a capacitor is identically zero so it is reassuring to find that the impedance formula gives the same result. The question then becomes: do all circuits have a DC steady state solution?

Consider the case of a sinusoidal current source connected to a capacitor. Using the impedance form of Ohm's law:


 * $$V_c = I_c Z_C \,$$

we find that the voltage across the capacitor increases without bound as the frequency decreases to zero. How is this to be interpreted?

The voltage given by the formula above is a phasor voltage. This phasor gives the peak amplitude and phase of the sinusoidal voltage across the capacitor. However, the phasor representation assumes that the circuit is in AC steady state or, if the frequency is zero, in DC steady state.

For a circuit to be in DC steady state, the circuit must have a DC steady state solution in which to exist. The circuit composed of a non-zero constant current source and a capacitor has no DC steady state solution. This was made clear earlier when we found that if the current through a capacitor is constant and non-zero, the voltage must be changing at a constant rate. Thus, for a circuit that does not have a DC steady state solution, we should expect a meaningless answer from a method that requires that the circuit is in DC steady state. A infinite voltage amplitude is such a meaningless answer.

Is the DC voltage across an inductor zero?
It is commonly said that an inductor is a 'short circuit at DC'. Is this true? What is true is that the DC steady state voltage across an inductor is identically zero. In DC steady state, all circuit voltages and currents are constant. By the fundamental inductor equation:


 * $$v_L = L \frac{di_L}{dt}$$

the inductor voltage must be zero if the inductor current is not changing with time. Does this imply that the DC voltage across an inductor zero? Clearly, the answer is no. According to the equation above, a constant (DC) voltage across an inductor can exist if the current through the inductor changes at a constant rate. For example, a DC voltage of 1V across an inductor with an inductance of 1H causes the current through the inductor to change at the rate of 1A per second. Theoretically then, the DC voltage across an inductor is not necessarily zero.

Obviously, for any real voltage source connected to a real inductor, the inductor current cannot continue to change forever. Eventually, the inductor would overheat and melt or the DC voltage source source would reach its maximum current rating. These real world limitations do not take away from the fact that the DC voltage across an inductor is not necessarily zero.

But isn't the impedance of an inductor zero for DC?

The formula for the impedance of an inductor is given by:


 * $$Z_L = j 2 \pi f L \,$$

where f is the frequency of the AC voltage and current associated with the inductor. It is often said that "DC is just AC with zero frequency". In a sense, this is true. Thus, it seems reasonable to believe that setting the frequency to zero in the equation above would give the impedance of the inductor 'at DC'. When this is done, the impedance becomes zero. What exactly does this mean?

To answer this, we need to understand how the formula for the impedance of an inductor is derived. The first step in any deriviation of the impedance is to assume that the current through the inductor is a sinusoidal function of time with constant amplitude, frequency, and phase. This step is crucial to the derivation of impedance and is equivalent to requiring that the circuit has settled into AC steady state. Now consider the following: if the frequency of this sinusoid is set to zero, the current through the inductor becomes constant. In other words, setting the frequency to zero is equivalent to requiring that the circuit has settled into DC steady state. We already know that the DC steady state voltage across an inductor is identically zero so it is reassuring to find that the impedance formula gives the same result. The question then becomes: do all circuits have a DC steady state solution?

Consider the case of a sinusoidal voltage source connected to an inductor. Using the impedance form of Ohm's law:


 * $$I_l = \frac{V_l}{Z_L} \,$$

we find that the current through the inductor increases without bound as the frequency decreases to zero. How is this to be interpreted?

The current given by the formula above is a phasor current. This phasor gives the peak amplitude and phase of the sinusoidal current through the inductor. However, the phasor representation assumes that the circuit is in AC steady state or, if the frequency is zero, in DC steady state.

For a circuit to be in DC steady state, the circuit must have a DC steady state solution in which to exist. The circuit composed of a non-zero constant voltage source and an inductor has no DC steady state solution. This was made clear earlier when we found that if the voltage across an inductor is constant and non-zero, the current must be changing at a constant rate. Thus, for a circuit that does not have a DC steady state solution, we should expect a meaningless answer from a method that requires that the circuit is in DC steady state. A infinite current amplitude is such a meaningless answer.

Does current flow?
This subject has been covered quite well by User:Wjbeaty but the short answer is no. An air current is a flow of air. A water current is a flow of water. An electric current is a flow of electric charge. Charge flows, not current.

In virtually any textbook on electrical engineering, you will find statements such as: "The current that flows through R1...". Why not just say: "The current through R1..."? Why write something like: "The applied voltage causes a current to flow...". Instead, write: "the applied voltage causes a current through...". It's easy - give it a try

Is the voltage across a short circuit always zero?
The answer is - it depends! More specifically, the answer depends on how one defines a short circuit. In some texts, one may find that a short circuit is defined as a zero resistance path for electric current. In others, one may find that a short circuit always has zero volts across it. These may seem like equivalent definitions but it turns out that they aren't!

If we accept the definition of a short circuit as a zero resistance path for electric current, then we can approximate a short circuit with a resistor that has a very small resistance. Place such a resistor across a voltage source. According to KVL, the voltage across the resistor must equal the voltage across the voltage source. According to Ohm's law, the voltage across the resistor is equal to the product of the current and the resistance.

Now, let the resistance of this resistor approach zero. In the limit as the resistance goes to zero, we see that the current through the resistor goes to infinity. But the product of infinity and zero is mathematically an indeterminate form. That is, the product may be zero or it may be any other number or it may even by infinite.

To determine what the product is, we need more information. By KVL, we already know that the product must equal the voltage of the source. Thus, we conclude that there can be a non-zero voltage across a zero ohm resistor if the current through the resistor is infinite.

Conversely, if we accept the idea that a short circuit always has zero volts across it, a short circuit is not at all like a zero ohm resistor but is instead, an ideal voltage source where the voltage is exactly zero. Personally, I choose to think of a short circuit as a resistor with a zero resistance.

Do frequencies resonate?
Is it 'resonant' frequency or 'resonance' frequency? 

Ground or common?
The so-called 'ground' symbol is ubiquitous in virtually any schematic diagram. Yet, rarely is this node actually connected to 'ground'. Further, in many circuits, there is more than one 'ground', i.e., analog ground, digital ground, rf ground, etc. But technically, the term 'grounded' means connected to the Earth. If this node is not actually connected to ground, a more appropriate name might be 'common' or 'return', i.e., +5V return, analog common, etc.

Is it phase or phase angle?
There seems to be no clear definition of phase or phase angle. In fact, it appears that the terms are used interchangeably. That is, 'phase' is short for 'phase angle'. For example, in many textbooks, one will find something like this:


 * Let $$f(t) = A \cos(2 \pi f t + \phi) \,$$


 * Where A is the amplitude, f is the frequency, and $$\phi$$ is the phase angle.

Then, in the next few paragraphs and pages, $$\phi$$ is referred to as the phase.

Assuming that the word phase is short for phase angle, the question is: does the term phase by itself have a separate meaning?

Consider the term phase plot. In the study of differential equations, a phase plot is introduced to plot the derivative of a variable against the variable, e.g., velocity against position. The mathematical space formed by these two quantities is called a phase space of the system and a point in this space is a phase of system.. This use of the word phase hints that phase is not a simple number.

{to be continued}

User:Alfred Centauri/sandbox