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All conductors always have resistance, inductance and capacitance. As a circuit designer it is important to know how large these so-called parasitics are and when they are significant.
In general parasitics become more important with increasing frequency.
All parasitics end up being proportional to length therefore as a general rule conductors should be kept as short as possible. This typically means that wires and cables are the first to show parasitics.
Capacitors and inductors begin to have influence on circuit behaviour as soon as there is non-zero frequency, which includes switched DC.
Using the Laplace transform and complex math (i.e. imaginary numbers) one can combine resistors, capacitors and inductors into a single quantity called impedance.
Impedance is made up of resistance (R) and reactance (X), where capacitors and inductors are called reactive components. Note that resistance is mathematically real while reactance is imaginary.
\(Z(f) = R(f) + jX(f)\)
Alternatively, this can be written in exponential form:
\(Z(s) = |Z|e^{j\theta}\)
The magnitude of Z is given as:
\(|Z| = \sqrt{R^2 + X^2}\)
And the angle as:
\(\theta = \arctan{\frac{X}{R}}\)
For a capacitor the reactance is :
\(X_C(f) = -\frac{1}{2\pi fC}\)
And for an inductor:
\(X_L(f) = 2\pi fL\)
When in exponential form, capacitors have an angle of -90° and inductors +90°.
Two element series circuits are very straightforward, just adding the two terms together.
RC circuit:
\(Z_{RC}(f) = R – j\frac{1}{2\pi fC}\)
RL circuit:
\(Z_{RL}(f) = R + j2\pi fL\)
An RLC circuit is a bit more complicated:
\(Z_{RLC} = R + j2\pi f – j\frac{1}{2\pi fC} = R -j\frac{1 + (2\pi f)^2LC}{2\pi fC}\)
Analogous to voltage dividers is the current divider.
When there are parallel branches in a circuit they all share the same voltage by definition. The resulting current depends on the ratio of the resistances.
The common saying that current takes the path of least resistance is a misnomer.
Current actually takes all paths simultaneously, with inverse proportion to the branch resistance. It’s definitely not as easy to say!
A saying that would be true would be to say that current takes the path of zero resistance, if zero resistance existed.
Parallel branches share a common voltage. In a resistive circuit therefore each branch will carry current:
$$I_n = \frac{V}{R_n}$$
The total current will then be:
$$I = I_1 + … + I_n = \frac{V}{R_1} + … + \frac{V}{R_n}$$
$$I = \frac{V}{R_{parallel}} + \frac{V}{R_k}$$
Each branch can then be written as:
$$I_k = I \frac{1}{1 + \frac{R_k}{R_{parallel}}}$$
The majority of electronic systems are too complex for convenient hand analysis and of the the most common tools for simplification is the small-signal approximation.
In general, electronic components are non-linear and very quickly become tedious to work with.
In fact, many common terms such as gain or output impedance rely on the small-signal approximation.
From calculus we have Taylor’s theorem that lets us approximate almost any function as a sum of terms, near a specific point.
f(x) near x= a can be approximated as:
$$f(x) = f(a) + \frac{df(a)}{dx}(x – a) + \frac{1}{2!}\frac{d^2f(a)}{dx^2}(x – a)^2 + \frac{1}{3!}\frac{d^3f(a)}{3! dx^3}(x – a)^3 + …$$
The individual terms are called first order, second order etc. based on the power of x.
This is a very powerful approximation that is used very frequently but rarely explicitly mentioned.
The first step to the approximation is to define the operating point. In a typical transistor circuit design, this would be the DC bias point.
If f(x) is a transfer function then ‘a’ is the input and f(a) is the output.
This is also frequently called the bias point, or steady state solution.
The gain, A, of a circuit is the linear multiplication term relating the output to input:
$$v_o(t) = Av_i(t)$$
In general gain varies with the operating point and intuitively it is equal to:
$$A(a) = \frac{df(a)}{dx}$$
This is of course the first-order term from Taylor’s theorem.
The typical amplifier or receiver is designed to be linear, therefore the gain is the only desirable term.
The second and third order terms then represent unwanted non-linear distortion.
Typically the second and third order terms are the dominant distortion products and rarely are higher order terms discussed.
A large amount of effort is spent in analog design to minimize these distortion terms.
Each term in the Taylor expansion has an 1/k!(x-a)^k term. When x is near a, each higher order term is less and less significant. For example:
$$0.9 > 0.9^2 > 0.9^3 …$$
When this assumption is true, we can say it is a small-signal and can calculate the transfer function using just the linear terms.
Typically, electronic circuits are analyzed using the small-signal assumption, with the higher-order effects added in afterwards if required.
Differential equations are the heart of a vast array of engineering theory, and as such, one can create models of entirely different branches of physics by using circuit theory.
The usefulness of this concept here is that electronics engineers can gain insight into mechanical systems by using their circuit knowledge. They can also use standard electronics tools like SPICE to incorporate mechanical outputs, such as motors or temperature rise.
In the recent past, this was useful for designing analog computers to solve mechanical problems.
The fundamental electrical components are resistors, capacitors and inductors.
Their voltage-current relationships are:
$$v(t) = i(t)R$$
$$i(t) = C \frac{dv}{dt}$$
$$v(t) = L \frac{di}{dt}$$
The key equations of motion are Newton’s laws, the main one being:
$$F(t) = ma(t) = m\frac{dv}{dt}$$
In a rotational system, we have:
$$M(t) = I\frac{dw}{dt}$$
The steady state thermal equation is:
$$T = P\theta + T_{env}$$
Dynamically there is Newton’s law of cooling:
$$\frac{dT}{dT} = \frac{T_{env} – T}{t_0}$$
The steady state temperature rise equation is identical to the solution of a DC resistor circuit.
The equivalents are:
Voltage (V) = temperature (°C)
Current (A) = power (W)
Resistance ($$\Omega$$) = thermal resistance (°C/W)
The solution is then very intuitive:
Temperature rise is caused by heat flow through a non-zero thermal resistance, exactly how voltage difference is caused by current flow through non-zero electrical resistance.
The power is a current source, the resistance a resistor, and the ambient temperature is the reference voltage. The resulting voltage at the current source is the new temperature of the device.
Typically a component datasheet will provide a $$\theta_{j-a}$$ which is the thermal resistance from thr component junction (j) to the ambient (a) environment. This can easily be added to a SPICE simulation.
Having established thr equivalence above, we can choose an electrical element to represent the thermal time constant.
Since temperature is similar to voltage and the thermal time constant slows it down, a capacitor is the appropriate electrical element (inductors slow down current).
A capacitor added in parallel to the current source results in an equation identical to Newton’s law of cooling:
$$i_c(t) = C \frac{dv}{dt}$$
$$i_c(t)R = RC \frac{dv}{dt}$$
$$v(t) = RC \frac{dv}{dt}$$
$$\frac{dv}{dt} = \frac{v_c(t) – v_{ref}}{RC}$$
$$\frac{dT}{dt} = \frac{T(t) – T_{env}}{t_0}$$
Therefore, the thermal time constant is equivalent to RC, which is the familiar electrical time constant.
Common-mode voltage is a term that seems to come from the world of differential amplifiers but finds itself being used generically in the EMC domain.
An ideal differential amplifier only sees the difference between its inputs:
$$v_o = A (v_h – v_l)$$
Each of the two inputs can be defined as two common components, a differential and a common mode.
$$v_h = v_{cm} + \frac{v_{id}}{2}$$
$$v_l = v_{cm} – \frac{v_{id}}{2}$$
The differential amplifier then ideally outputs:
$$v_o = Av_{id}$$
In reality, there is a finite common-mode rejection ratio, CMRR, such that:
$$v_o = Av_{id} + CMRRv_{cm}$$
Waveforms need to be quantitatively defined in many different ways.
Peak value: maximum instantaneous value
Peak to peak value: difference between maximum and minimum instantaneous values throughout the entire range
RMS value: room-mean-square. Provides power equivalent value as if the waveform was DC.
\(RMS = \sqrt{\frac{1}{T}\int^T_0 x^2(t)dt}\)
Frequency: repetition rate per second of a periodic waveform
Period: duration in seconds of a repetitive waveform
$$x(t + T) = x(t)$$
Instantaneous power is always given by:
$$p(t) =v(t) i(t)$$
This result is simple but when discussing power consumption it is the average power that is needed.
To calculate average we first obtain the total energy used:
$$E = \int^T_0 v(t) i(t)dt$$
Then we simply divide by the time interval to obtain average power:
$$P = \frac{E}{T}$$
For DC voltage and current the integration simplifies into a multiplication, with the familiar:
$$E = VIT, P = VI$$
For a resisitive device one can use Ohm’s Law:
$$P = \frac{1}{T}\int^T_0 \frac{v^2(t)}{R}dt$$
This is the origin of the RMS value, such that:
$$P = \frac{V^2_{RMS}}{R}$$
Ohm’s law is one of the first equations taught to electronics students, showing the voltage-current relationship in a resistor.
It can be derived from Maxwell’s equations and holds true for linear homogenous materials. This typically means pure metals.
Any device designed to meet Ohm’s law is a called a resistor.
Really, Ohm’s law doesn’t need any basis in physics because it’s just a defined as a ratio of voltage to current.
\(R = V/I\)Nothing is 100% Ohmic because the equation is limited by the construction of devices and material properties.
This includes:
-materials heat up as they dissipate power, which changes their resistance
-at some temperature a material will melt or otherwise change, destroying the device
-at high enough voltages materials will break down and will spontaneously conduct
-dissimilar metals generate small voltages based on the thermoelectric effect
-chemical residues can generate voltages from galvanic action
-mechanical strain creates voltage in piezoelectric materials
-at extremely low voltage the intrinsic noise of a device will dominate the intended signal
-certain current mechanisms operate with fundamentally different physics, including ionic fluids or charged particle streams (electron beams, particle accelerators)
The ratio interpretation allows the intuition of Ohm’s law to be used as a conceptual tool with many more applications that just resistors.
All devices have an intrinsic voltage-current relationship which can be linear, non-linear or even be a multi-valued function.
At any point one can define a differential or small-signal resistance:
This idea is used with surprising frequency even if rarely discussed. Incandescent lightbulbs have such a strong temperature dependency in normal use that one needs to discuss “cold” and “hot” resistances.
Constant power loads (like switching DC-DC regulators) have a negative differential input resistance which is important to their application.
Perhaps most importantly, the majority of active devices are characterized for input and output resistance despite the behaviour being created by complex, non-linear device physics.
It is this non-linear that forces any component datasheet to specify in detail the operating point, including temperature and voltage ranges.
Understanding that a resistance is very often more than just Ohm’s law is a key point in understanding electronic devices.
Resistor Divider:
\(V_o = V_i \times \frac{1}{1 + \frac{R_{src}}{R_{load}}}\)
The resistor divider is a common circuit but it is much more valuable as a concept.
The voltage divider represents fundamental circuit behaviour across countless applications.
Battery:
A 9V battery has 100 ohms internal resistance.
What is the output voltage when powering a 900 ohm load?
Power Wiring:
A 100ft long extension cord of 16 AWG copper wire is used with a 1200 W space heater. What is the voltage at the heater?
Appliances are rated assuming nominal voltage (120 V), so a 1200 W heater is really \(R = 120^2/1200 = 12 \Omega\)
16 AWG is 0.004 ohm/ft, so with 100 ft of line and return there is 0.8 ohms.
Therefore the heater will only get:
\(V_o = 120 \times \frac{1}{1 + 0.8/12} = 112.5 V\)