Resistance, Capacitance, and Inductance (R-C and R-L Circuit)

Our aim here is to discuss the very important concepts and give intuitive explanations of the subjects. We'll start with the resistor which is the most straightforward, then with the less understood and often misunderstood concepts of capacitance and inductance. We will discuss the R-C and R-L circuits as well. These along with Ohm's Law, Kirchoff's Voltage Law (KVL) and Kirchoff's Current Law (KCL) are the very foundations of understanding the complete behaviors of practically any circuit one might encounter.

Resistance
It is a general term that refers to the opposition to the free flow of current in a circuit. More accurately, it refers or applies to DC (direct current). Resistance doesn't apply only to resistors but also to a host of other components in the circuit. For example, a copper trace has a resistance according to its dimensions, as well as the leads of he components.When a current flows through a resistor a potential difference is generated across it, called voltage. That voltage has a polarity depending on the direction of current flow.

To a resistor, applying a voltage, e.g. a battery, produces a corresponding amount of current, which according to Ohm's Law is I = V/R, where I is the current produced in A (amperes), V the voltage applied in  V (volts), and R the resistance (in ohms), of the resistor.

Figure 1

Plot below shows a linear relationship between voltage and current, where slope is positive. This is pretty obvious from the Ohm's law equation. A practical conclusion we can make off the plot is, changing the voltage by certain amount (we will call it delta V), produces a corresponding change in current exactly predicted by Ohm's Law. If we can measure the change in current (delta i) by applying a change in voltage (delta v), we can calculate for the resistance value of the component where we applied the source.

Figure 2

In real world we'd like to talk more about impedance, a term that applies to a changing source or input signal. To a resistor that automatically equates to resistance, as we have shown above to be equal to the delta v over delta i. We call this response to the change in voltage over the change in current, the impedance. This is almost synonymous to resistance, but we need to keep in mind that this applies to a changing input signal.

Our battery in the first figure can be replaced by an alternating current (AC) source, which will exactly have the same effect in the response of the resistor. An AC is a 'complex' signal, that has an amplitude and phase (against a reference signal). Its complex behavior, its phase in particular, ideally doesn't influence the response of the resistor. Figure below shows the current flowing in the resistor in phase with the AC voltage applied to it.

Figure3

Capacitance
This is generally defined as the ability of something to store an electrical charge. In electrical and electronics, a component we call capacitor is made up of two conductor plates, with an insulator (dielectric) in between that prevents the charge to cross from one plate to another. Thus an application of voltage between two plates creates accumulation of electrical charges (positive on one side, and negative on the other) on each of the plate. The quality and physical dimensions of the plates and the dielectric determines the capacity  of the device to store charge. The capacitance is directly proportional to the area of the plates, and indirectly proportional to the distance between the plates.The standard (SI) unit of capacitance is Farads (F), after Michael Faraday. One (1) Farad of capacitance C means a charge of one (1) coulomb, q, will have a potential difference across a capacitor of one (1) volt V, whereby,

C = q/V

Figure 4

Consider a current source sourcing a constant current to the capacitor. The voltage across the capacitor will increase such that,

I = C x (dV/dt)

Figure 5

The change in voltage with respect to time, from the equation above, is equal to I/C.  To put it in another way, sourcing a constant current to the capacitor will produce a constant change in voltage across the capacitor proportional to its capacitance C.

Capacitors come in different types (material used), shapes, and sizes according to its value and applications. Due to its distinct behaviors in DC and AC, it has find many uses in electronics circuits such as in filters, bypassing, coupling, and decoupling, among others. Consider the circuit below, where a capacitor in series with a resistor are applied with a voltage (DC) upon closing the switch.

Figure 6 R-C Series circuit, capacitor charging

The closing of the switch provides a transient stimulus to the circuit. At exactly when the switch is closed (t = 0), with the charge in the capacitor at its initial value from its previous state (in this case 0v), the voltage across the resistor is at maximum, which means the current flow is also at maximum at the start. From the figure above, with 0 volts as the the capacitor's initial voltage, one can think of the capacitor being shorted the moment the switch was closed, which also means that initial voltage drop across the resistor is Vs. The current then starts to decay towards zero, while the voltage at the capacitor charges up to the supply voltage. This is a very important concept that needs to be understood accurately. As the capacitor charges, it follows that the voltage across the resistor decays accordingly. One may think of a capacitor as a water tank and the resistor that regulates the flow of water, which is the current. Only, the tank gets filled in a negative exponential manner, where the voltage across the capacitor is given by:

where RC is equivalent to one time constant tau, in seconds.  At one time constant, the capacitor has charged up to ~63% of the supply voltage. The greater the value of resistance or capacitance, the longer the time it takes for the capacitor to charge up. We often hear the words larger time constant which basically means the time it takes to charge or discharge the capacitor. The capacitor can be considered fully charged 5 time constants or greater.

As it becomes fully charged, the current flow decreases until it becomes zero, thereby rendering the circuit open and the IR drop across the resistor also becomes zero. Therefore in the steady state, or we say as time goes to "infinity", the capacitor appears open to the DC input voltage. To a steady DC voltage, its resistance is very large or "infinite".

The capacitor voltage equation above is the solution to the differential equation from the R-C circuit we have in figure 6, with a step input voltage represented by the closing of the switch.

A few things we need to clarify. Let's assume that the capacitor has an initial stored voltage from previous state when the switch was closed. The capacitor then will try to charge up to the supply voltage Vs from its initial value in the same manner when the capacitor has no initial charge, having the same time constant.

Figure 7

Plot above shows the voltage across the capacitor, where Vc1 is the initial voltage of the capacitor when the switch was closed. From the plot we can write the equation of the capacitor voltage, modifying our first equation,

One can also think of the capacitor as a memory device, because it remembers or retains the voltage when we open the switch. It will ideally hold its voltage as long as there is no path for it to discharge.

Next, imagine we have another switch to provide a discharge path for the capacitor after fully charging to Vs.

Figure 8

Intuitively, this will be the opposite of what happened during the charging, with the same time constant it will take to fully discharge.

Applying an AC voltage as an input to our circuit creates an interesting behavior in the capacitor. With a sinusoidal input, the capacitor acts like a resistor with an impedance, Xc, a function of the frequency of the signal frequency,

As the frequency increases, the capacitive reactance, Xc, decreases.  The signal frequency and the capacitance is inversely proportional to the capacitive reactance, or its impedance.

The capacitor is not however as straightforward as a resistor in the circuit in the presence of an ac signal, because the voltage across the capacitor does not occur simultaneously with its current, that is, they are not in the same phase. In fact the voltage in the capacitor lags its current by 90 degrees. This creates just a little bit of complication in the circuit, because we're going to have to deal with vectors in the analysis due to the phase differences.

Figure 9

                    

Figure 10 R-C Vector Analysis

Because the resistor and the capacitor are in series, Kirchoff's Voltage Law dictates that their vector sum must be equal to the input voltage VIN. It is important to realize that because they are in series, one and the same current flow in each. In resistor the current and the voltage are in phase, it follows that since the capacitor current leads its voltage by 90 degrees, the voltage across the resistor leads the capacitor voltage by 90 degrees as well. If the VR and VC are of the same magnitude, VIN lies exactly 45 degrees between VR and VR. We say that the output, VC, lags the input voltage by 45 degrees.

Figure 11 Input leads output by 45 degrees.

Going back to figure 9, as the frequency of the input signal increases, we've already said that the capacitive reactance decreases, and therefore the voltage across it decreases as well. It also follows that the voltage across the resistor increases. As the frequency tries to increase much further, VR gets even larger and the phase lag of the output VC becomes closer to 90 degrees. These phenomenon has significant implications in the applications and use of the circuit.  The series R-C circuit as configured in figure 9 acts as filter to input signals of different frequencies. The output, which is across the capacitor, is at maximum equal to the input voltage at DC (0 Hertz), and becomes smaller as the frequency increases. The circuit is basically called a low-pass filter, because it allows signals at lower frequencies but rejects those at higher frequencies. The frequency at which the signal starts to roll-off or go down in level ( this is -20dB per decade increase in frequency) is called the break frequency, or cut-off frequency. The -20dB per decade means a ten-fold decrease in the output voltage per ten-fold increase in frequency. The cut-off frequency is given by:

You may have noticed the similarity with our previous equation on XC, this is because at the break frequency, the R resistance is equal to the capacitive reactance (impedance) XC. Also at this frequency, the output voltage is shifted 45 degrees from the input VIN. 

Inductance

Inductor is in many ways a device that behaves opposite to that of a capacitor. An inductor usually consists of a wire that is wound to form a coil. Inductance is the ability to store electrical energy in its magnetic field, while capacitance stores the energy in its electric field. It is when the magnetic flux formed by the current flowing through the conductor changes, as the current changes and produces an expanding or contracting magnetic field that cuts through the conductor, an electromagnetic force (emf) is induced (Faraday's Law) that opposes the change in current that originally produced it (Lenz' Law).

The voltage across the inductor is formed as a function of its inductance L, given in Henry (H), and the change in current (derivative) with respect to time.

Figure 12

One can think of inductance as the opposite of capacitance, and has a behavior that is complementary opposite to the capacitance.While capacitance opposes the change in voltage, inductance opposes the change in current in it. Imagine when the capacitor  charges up and the charging voltage is withdrawn, the capacitor holds the voltage with same polarity as the charging source. With the inductor, when the source is withdrawn after the current settled, there will be a large change in current (to zero!), and a voltage opposite to the charging source is developed in order to sustain the current flowing in the same direction. This voltage is called back-emf, or counter emf. The inductor stores charge in a similar way as the capacitor does, but in a different mechanism.

Figure 13.a

Figure 13.b

So taking from our analysis of the capacitance above, we can formulate its transient (when we apply our signal via a switch) and AC behavior.The current flow in the circuit is given by:

where L is the inductance. At t = 0, or the moment the Vin is applied, the current is zero and the voltage across the inductor is the same as Vin, as shown in the picture above, Figure 13.a. As the current goes up towards steady state, the voltage across the inductor goes down and approaches zero. Where for the R-C circuit the time constant is RC, the time constant is L/R for the inductor-resistor circuit, also given in seconds. In frequency domain, where the input is sinusoidal, the impedance of the inductor is given by:

in ohms.

The voltage across the inductor leads its current by 90 degrees. Intuitively,  the current is zero at the maximum inductor voltage, and goes up towards maximum when the inductor voltage becomes close to zero. Ideally at steady state, there is no voltage across the inductor, because it has ideally zero resistance.

At the frequency where the series resistance is equal to the inductive reactance, the voltage across the inductor leads the input voltage by 45 degrees. At this frequency since they are in series and the current is the same, the resistor voltage is equal in magnitude to the inductor voltage. The same vector analysis as with our R-C circuit can be used, except that the inductor voltage now leads the resistor voltage in phase by 90 degrees.