Operational Amplifiers

This next topic focuses on the operational amplifier, which is one of the most widely used electronic components in the analog world.  The operational amplifier, also known as an op-amp, is a type of differential amplifier.  The output voltage of an op-amp is:

Op-amps are used in a variety of applications, which perform many different operations, hence the name.  External passive elements – such as resistors, caps, inductors, etc – are used to determine the end function.  Before continuing into any applications, you should consider the following properties of an ideal op-amp.

Ideal OP-AMPs have:

Infinite input impedance

Zero Output Impedance

Infinite Differential Gain

Zero Offset Voltage

Zero Input Bias Current

Zero Common Mode Gain

Recalling our previous equation and substituting Aol=∞ produces the following relationship for the non-inverting and inverting inputs. 

This equation merely demonstrates that for an ideal op-amp, both inputs are the same voltage value, so in a sense both inputs are virtually shorted to each other. 

Using these properties, you can solve any ideal op-amp circuit configuration.  To get you started in op-amp analysis, I’ve provided a simple circuit below, which is an inverting op-amp configuration.

Using KCL, let’s guess the current direction and solve for node 2.

What happens when I substitute a voltage for the V+ input?

I’ll leave this next configuration as homework:

You shall not pass!

Sometimes in electronics you only want certain frequencies of a signal to pass -- an example would be an audio system.  Usually by design, the lower frequency signals are passed to the subwoofer (giving it the deep bass), while the higher frequencies are passed to the tweeter (higher dynamic pitch).  This post will touch base on two different passive filters: low-pass, and high-pass.

Before you start, let's review what a capacitor and inductor looks like at various frequencies.

The impedance of a capacitor is defined as:

Sometimes displayed as:

Right off the bat you should notice that a capacitor looks like a variable resistor across frequency that is inversely proportional to its capacitance and frequency value.  Two important frequencies to note are f = 0 Hz and f = ∞ Hz.  At f = 0 Hz the impedance is infinite, which is the same as an open circuit.  When f = ∞ Hz, the impedance value is 0 Ω, which is a short.

The impedance of an inductor is defined as:

Sometimes displayed as:

For the inductor case, at f = 0Hz, the impedance is 0Ω, which is a short.  When the frequency is infinite, f = ∞ Hz, the impedance is infinite, which is the same as an open circuit.

Lowpass Filter

As the name suggests, this passive filter only allows low frequencies through.  A single pole low-pass filter is illustrated in Figure 1.  

Figure 1.  Low-Pass Filter

Before you derive the transfer function think about this circuit from another point of view.  You can develop an intuitive understanding of the circuit when you analyze it around two frequency values, 0Hz and ∞ Hz.  

At 0Hz the capacitor acts like an open-circuit, therefore Vout = Vin.  At ∞ Hz the capacitor resembles a short, and Vout = GND = 0V. 

The transfer function  of this circuit is shown in Figure 2.

Figure 2. Transfer function of low-pass filter

The next step is to plot the magnitude and phase of this transfer function.

Magnitude:

Phase:

Highpass Filter

This filter only allows higher frequencies through. A single pole high-pass filter is displayed in Figure 3.

At 0Hz the capacitor acts like a short, therefore Vout = GND = 0V.  At ∞ Hz the capacitor resembles an open-circuit, and Vout = Vin.

Figure 3.  High-Pass Filter

The transfer function for Figure 3 is shown below.

Figure 4. Transfer function of high-pass filter

Magnitude:

Phase:

Resistors, Capacitors and Inductors! Oh my!

Resistors:

As you may know from the previous blogs, resistors are two terminal passive components that effectively limit current flow.  We can demonstrate resistance by recalling our previous river analogy.

The analogy suggests that water in a river is represented as 'charge'.  Water flowing in any spatial direction can be represented as current.  If you were to add rocks (resistance) in the path of the river you would effectively reduce the current flow.  This is displayed in Figure 1.

Figure 1.  River analogy for resistance

The current through a resistor equates to the potential across the resistive element divided by the resistance of the element, otherwise known as I=V/R.

The properties of a resistor connected in series or in parallel are illustrated in Figure 2.

Resistors in series

 

Two resistors in parallel

Resistors in parallel

Figure 2. Properties of Resistors

Power across is resistor is calculated as: 

Capacitors:

Capacitors are two terminal passive components that are used to store energy in its electric field.

Recalling our river analogy, capacitors can be represented as a bucket.  If you were to bring a bucket to the river you can collect some water (charge).  The bigger the bucket (capacitance value) the more water (charge) you will be able to collect.

Figure 3.  River analogy for capacitance

In the simplest illustration, a capacitor can be represented as two parallel metal plates, as shown in Figure 4.

Figure 4. Capacitance from parallel plates

The amount of charge is dependent on the voltage across the capacitor and the capacitance value.

Q = C * V

Q [Charge in Coulombs]= C[Capacitance value]*V[Voltage across cap]

The properties of a capacitor connected in series or in parallel are illustrated in Figure 5.

Capacitors in series

Capacitors in parallel

Figure 5. Properties of Capacitors

Instantaneous current through a capacitor:

Inductors:

Inductors, also called "coils", are two terminal passive components that are used to resist changes in electrical current and stores energy in it magnetic field.

Using the river analogy again, we can illustrate the inductor as a water wheel that initially resists any big waves converting it into steady current defined by the resistance of the river path, as shown in Figure 6.

Figure 6.  River analogy for inductance

The properties of an inductor are listed in Figure 7.

Inductors in series

Two inductors in parallel

Inductors in parallel

Figure 7. Properties of Inductors

Instantaneous voltage across an inductor: