This is the second blog post in the series of fundamental concepts of electric circuits and signals with the Tek 1202B oscilloscope. In this post I will show an intuitive and analytical view of capacitance and capacitors, then I will analyze charge storage mechanisms and I will compare in two experiments how capacitors charge from a constant voltage source through a resistor versus from a constant current. Following that I will show how capacitors are used in RC low-pass and high-pass filters. I will be using the new to illustrate these concepts.
Two conductors separated by air or any dielectric material form a capacitor. Capacitors store electric charge, and capacitance measures the ratio between the amount of charge stored and the voltage between the two conductors. Capacitance uses the symbol C, and is measured in Farads (F).
C = capacitance (F)
Q = charge stored (C)
V = voltage between the two conductors (V)
Besides storing energy, capacitors have the property of conducting transient currents. The conduction process does not occur through charge transfer from one conductor to the other since there is an insulator in between, but rather by charge displacement in one conductor due to charge variation in the other conductor. This current is equal to:
I = transient current through the capacitor (A)
C = capacitance (F)
dV/dt = how fast the voltage changes (V/s)
A Simple Capacitor: Parallel Plate Capacitor
A well-known and very simple to analyze capacitor is made of two parallel conductive plates separated by a dielectric, as shown in the following figure.
The charge +Q equals the charge –Q and is called charge stored in the capacitor. The electric field is generated between the two conductive plates, and consists of uniformly distributed field lines inside the capacitor and fringe field lines at the edges. The typical analysis neglects the fringe field and simplifies the calculation of capacitance as:
C = capacitance (F)
ε0 = permitivity of free space (ε0 = 8.854 x 10-12 F/m)
εr = relative permitivity of the dielectric material
S = area of the conductive plate (m2)
t = separation between the two plates (m)
Notice that the capacitance increases with area and decreases with separation. The capacitance also depends on the dielectric material. The lowest capacitance occurs for air dielectric, and it increases for different materials depending on their relative permitivity.
In integrated circuits, packages and printed circuit boards, the capacitance of power and ground planes can often be approximated as parallel plates capacitance. However, for thin wires, the fringe field starts to play a significant role, and the parallel plate approximation is no longer valid.
If we manufacture a parallel plate capacitor, we get a similar device as shown in the figure below diagram (a).
The parallel plates have their own resistance and inductance, and the terminals are dominantly inductive. The equivalent circuit of a real capacitor, shown in diagram (b), consists of the capacitor C in series with an “equivalent series inductance” (ESL) and an “equivalent series resistance” (ESR). A high value leakage resistor RL is placed in parallel with the capacitor. Because of the parasitic elements, the frequency dependence of impedance does no longer look like an ideal capacitor, but more like a resonant circuit as shown in the figure below (logarithmic scale).
The impedance decreases with frequency up to the “self resonance frequency” f0, after which it starts to increase. The impedance increase with frequency after f0 makes the capacitor behave like an inductor. The minimum impedance equals the ESR value.
Capacitors charge and discharge following the formula: I=CdV/dt. Let’s look at two examples: charging from a constant current source and charging from a constant voltage source through a resistor.
Charging from constant current
When charging a capacitor from a constant current source “I” is constant in I=CdV/dt formula. Solving for V(t) we find out that V=(1/C)*I*t, which is a linear variation in time ( a straight line). To visualize this I have setup an experiment in which I charged a 100nF capacitor from a constant current source and I probed the voltage on the capacitor and the charging current with a Tektronix TBS1202B-EDU oscilloscope. The figure below shows the measured waveforms of voltage and current during capacitor charging.
The yellow trace represents the voltage on the capacitor and the blue trace represents the charging current. The charging current is measured as voltage drop on a 1.5kOhm resistor connected in series with the capacitor. At the beginning the current is equal to zero, then it starts flowing at constant intensity for about 7.5ms, after which it stops flowing. The value of the current can be calculated using Ohm’s law applied on the 1.5kOhm series resistor as V=IR => I=V/R, where we can read V from the blue waveform as about 1.8mV (2mV/division), so I=1.8mV/1.5kOhms=1.2uA.
Going now back to the capacitor charging formula, V=(1/C)*I*t, we can insert the values for C and I so the charging formula becomes: V=(1/100e-9)*1.2e-6*t=12*t. Based on this, we expect the voltage on the capacitor to increase linearly with time following the formula: V(t) = 12*t. Let’s see if this formula correlates with the experimental measurement. The charging voltage is shown by the yellow trace, and indeed it varies linearly with time starting from 0V and reaching about 100mV (2 vertical divisions multiplied by 50mV/division) in about 7.5ms (3 horizontal divisions multiplied by 2.50ms/division). And our calculated formula predicts that the voltage on the capacitor will charge in the 7.5ms to a value equal to V(7.5ms)=12*7.5=90mV. This value is close to the measured 100mV considering all the variations involved here like +/-10% tolerance of resistor value from the nominal 1.5kOhm, and +/-10% tolerance in capacitor value from 100nF. Below I have inserted a video that shows this measurement:
Charging from constant voltage through a resistor
When charging a capacitor from a constant voltage source through a resistor the same formula I=CdV/dt applies, but the voltage variation on the capacitor is no longer a straight line. As I have annotated on the figure below, solving the integral we get an exponential type variation for the voltage on the capacitor.
The square type trace (blue trace) represents the voltage applied to the capacitor through a 100Ohm resistor and the yellow trace represents the voltage variation on the capacitor. We can notice the exponential increase of voltage on the capacitor, which approaches asymptotically the voltage level of the blue trace.
The picture below shows how the current varies during capacitor charging:
So the current has a step right at the beginning after which it gradually decreases until it reaches asymptotically the zero value.
Capacitors Used in RC Filters
We can construct low-pass and high-pass filters using a capacitor and a resistor. Let’s take a look next at each of these types of filters.
Low Pass Filter
A low pass filter will let pass low frequency spectral components and will block high frequencies. To illustrate this I have setup an experiment in which I built a low pass RC filter using a 100nF capacitor and a 100Ohms resistor. I then connected a sinusoidal signal generator to the input of the filter and I used the Tektronix TBS1202B-EDU oscilloscope to probe the input and the output signals. The following figure shows this experiment setup.
I have annotated this picture to show the capacitor, resistor, and the measured amplitude and frequency on the TBS1202B-EDU oscilloscope. The schematic of the low-pass RC filter is shown in the bottom-right corner of the picture (input is on the left and output on the right). Notice that at 2kHz frequency the amplitude of the input signal, shown by the blue trace (channel 2 of the oscilloscope), is equal to 1V and the amplitude of the output signal (yellow trace; channel 1 of the oscilloscope) is also 1V. So at 2KHz the filter lets pass the entire signal without attenuation.
Next let’s increase the frequency to see what happens. The following figure shows the measured input and output amplitudes at 16.6kHz.
Notice that the amplitude of the output signal has decreased to 0.7V. So as we expected, when we increased the frequency the amplitude of the output signal decreased. In this case the output amplitude is 0.7V, which can be expressed also as Vout=0.7 x Vin (since Vin in this case is 1V). I chose to show this particular case because it is very close to the cutoff frequency of this filter. The cutoff frequency is the frequency at which the output amplitude is reduced to 0.707 of the input amplitude. For this RC filter the cutoff frequency is calculated as
So based on the resistor and capacitor values the cutoff frequency should be 1/(2*3.14*100*100e-9)= 15.9kHz, which is close to 16.6kHz measured in this experiment (considering the +/-10% variation of the resistor and capacitor values from the nominal value marked on them).
Let’s see what happens when we increase even more the frequency. The following picture shows the waveforms at 49.4kHz, which is three times the cutoff frequency.
Notice that the amplitude at the output has decreased to 0.4V, or expressed as ratio of the input, has decreased to 0.4 of the 1V input value.
The following picture shows the RC filter operation at 82kHz, which is about five times the cutoff frequency.
The amplitude of the output signal has been reduced to 0.3V.
Next let’s increase even more the frequency. The following figure shows what happens at 165kHz, which is about ten times the cutoff frequency.
Notice that the amplitude at the output has been reduced to 0.18V.
So as we increase the frequency the low-pass RC filter attenuates the amplitude of the output signal, towards the zero value.
High Pass Filters
High pass filters would let pass high frequency spectral components and attenuate low frequency components. The following figure shows an experiment in which I built a high-pass RC filter using a 100nF capacitor and a 100Ohms resistor. Same as in the previous experiment, I connected a sinusoidal signal generator to the input of the filter and I used the Tektronix TBS1202B-EDU oscilloscope to probe the input and the output signals. The following figure shows this experiment setup.
The schematic of the high-pass RC filter is shown in the bottom-right corner of the picture (input is on the left and output on the right). The frequency value chosen here, 16.6kHz, is close to the cutoff frequency of this filter. The cutoff frequency of the RC high-pass filter is defined the same way as for the RC low-pass filter, and the same formula is used to calculate it.
Let’s see now what happens if we increase the frequency above the cutoff value. The following picture shows the RC filter operation at 82kHz, which is about five times the cutoff frequency.
Notice that both the output and input amplitudes are equal to 1V, so the RC filter lets the entire signal pass without significant attenuation.
If we move towards the other direction, lowering the operating frequency from the cutoff value, the amplitude at the output starts decreasing. The figure below shows the high-pass filter operation at 0.93kHz.
In this example at 0.93kHz the amplitude at the output has been reduced from 1V to 0.12V. So as we lower the frequency the high-pass filter attenuates the signal towards the zero value.
To conclude, in this blog post I have covered fundamental concepts of capacitors, charging of capacitors, and application of capacitors in RC low-pass and high-pass filters. In the following blog post I will talk about inductors.