Lecture 12

Resistive circuit with operational amplifier

Resistive circuit with operational amplifier

• Circuit model of the operational amplifier
• Analysis of proportional circuit
• Circuit analysis with the ideal operational amplifier

Resistive circuit with operational amplifier

• Operational amplifiers have a wide range of applications, first applied in the 1940s. With the development of integrated circuits, they gradually developed into integrated operational amplifiers, greatly reducing costs.

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• It is a high-gain amplifier that can achieve amplification multiples of several thousand, tens of thousands, or even hundreds of thousands of times, compared to the current controlled current source of a transistor (less than 100 times).

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• It can be used for operations such as addition and subtraction, multiplication, differentiation, integration, etc.

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• It can also be used for signal filters, voltage comparators, signal generators, and so on.

Resistive circuit with operational amplifier

provide power for current/voltage amplification

advantage:

• high gain
• large input resistor, small output resistor

Resistive circuit with operational amplifier

notes:

• 8 pins
• Pin2: inverting input
• Pin3: non-inverting input
• pin4,7, power supply
• pin6: output
• pin1,5: connecting to zero potentiometer (to make sure the output is zero when the input is zero.
• pin8: not used.

Resistive circuit with operational amplifier

Sometimes, we need a simplified diagram and the circuit model.

notes:

• a - inverting input with u^-
• b - non-inverting input with u⁺
• A – open-loop voltage amplification factor (voltage gain), up to ~100K
• u_d = u⁺-u^-

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Resistive circuit with operational amplifier

Note:

• u_sat is lower than the voltage of the power supply (on page 5).
• eps is a very small number, e.g., when u_sat = 13V, eps=0.13mV
• any problem with eps?

Resistive circuit with operational amplifier

• Circuit model of OA

Note:

• When u⁺ = 0; u_o = -Au^-
• When u^- = 0; u_o = Au⁺
• The above explains why u^- is called inverting input and u⁺ is called non-inverting input.

Resistive circuit with operational amplifier

• Circuit model of ideal OA

• In the linear amplification region, A is approximated to be infinity, which means u_d = 0 & u_o = a limited number.
• R_i is infinite.
• R_o = 0.

Proportional Circuit Analysis: Reverse Proportioner

• The open-loop operation of an operational amplifier is extremely unstable, which requires several external components (such as resistors, capacitors, etc.) to form a closed-loop circuit

Resistive circuit with operational amplifier

Equivalent Circuit of Reverse Proportioner

Resistive circuit with operational amplifier

• Applying the node voltage method

The ratio between u_o and u_i only depends on the ratio of R_f to R₁, and is independent of the internal parameters of the operational amplifier. The minus sign indicates that the symbols of uo and ui are opposite (Reverse Proportioner)

Resistive circuit with operational amplifier

• The above approximation is based on the assuming that the OA is an ideal OA.

• In the linear amplification region, A is approximated to be infinity, which means u_d = 0 & u_o = a limited number.

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• R_i is infinite.

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• R_o = 0.

• u⁺ = 0; u^- =0
• i₁ = u_i/R₁, i₂ = -u_o/R_f,

• i₁ = i_2; u_o = -u_i*R_f/R₁

• To ensure that u_o is not saturated, the value of u_i is set to a finite interval.
• Rf is between the output and the inverting input of OA, called negative feedback.

• In the linear amplification region, A is approximated to be infinity, which means u_d = 0 & u_o = a limited number.

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• R_i is infinite. i⁺ = i^- =0

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• R_o = 0.

Applying KCL at the inverting input terminal

u_i1/R₁ + u_i2/R₂ + u_i3/R₃ + u_o/R_f = 0

u_o = -(u_i1/R₁ + u_i2/R₂ + u_i3/R₃)*R_f

= -(k₁*u_i1 + k₂*u_i2 + k₃*u_i3)

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Then u_o can be inverted by applying another reverse proportioner (or inverting amplifier)

• Specifically, we can connect a resistive voltage divider to the op-amp’s output, and then connect the middle terminal of that voltage divider back to the op-amp’s inverting input

• Supplemental reading materials.

Resistive circuit with operational amplifier

• Operational amplifiers are linear devices that have all the properties required for nearly ideal DC amplification and are therefore used extensively in signal conditioning, filtering or to perform mathematical operations such as add, subtract, integration and differentiation.

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• An Operational Amplifier, or op-amp for short, is fundamentally a voltage amplifying device designed to be used with external feedback components such as resistors and capacitors between its output and input terminals. These feedback components determine the resulting function or “operation” of the amplifier and by virtue of the different feedback configurations whether resistive, capacitive or both, the amplifier can perform a variety of different operations, giving rise to its name of “Operational Amplifier”.

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Resistive circuit with operational amplifier

• An Operational Amplifier is basically a three-terminal device which consists of two high-impedance inputs. One of the inputs is called the Inverting Input, marked with a negative or “minus” sign, ( – ). The other input is called the Non-inverting Input, marked with a positive or “plus” sign ( + ).

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• A third terminal represents the operational amplifiers output port.

Equivalent Circuit of an Ideal Operational Amplifier

Resistive circuit with operational amplifier

• In a linear operational amplifier, the output signal is the amplification factor, known as the amplifiers gain ( A ) multiplied by the value of the input signal and depending on the nature of these input and output signals, there can be four different classifications of operational amplifier gain.

Voltage  – Voltage “in” and Voltage “out”

Current  – Current “in” and Current “out”

Transconductance  – Voltage “in” and Current “out”

Transresistance  – Current “in” and Voltage “out”

• Since most of the circuits dealing with operational amplifiers are voltage amplifiers, we focus on voltage amplifiers only, (Vin and Vout).

Resistive circuit with operational amplifier

• An operational amplifier only responds to the difference between the voltages on its two input terminals, known commonly as the “Differential Input Voltage”. Then if the same voltage potential is applied to both terminals the resultant output will be zero. An Operational Amplifiers gain is commonly known as the Open Loop Differential Gain, and is given the symbol (A_o).

Resistive circuit with operational amplifier

• Open Loop Gain

Infinite – The main function of an operational amplifier is to amplify the input signal. The more open loop gain it has, the better the Op-Amp. Open-loop gain is the gain of the op-amp without positive or negative feedback and for such an amplifier the gain will be infinite but typical real values range from about 20,000 to 200,000.

Resistive circuit with operational amplifier

• Input impedance, (ZIN)

Infinite – Input impedance is the ratio of input voltage to input current and is assumed to be infinite to prevent any current flowing from the source supply into the amplifiers input circuitry ( IIN = 0 ). Real op-amps have input leakage currents from a few pico-amps to a few milli-amps.

Resistive circuit with operational amplifier

• Output impedance, (ZOUT)

Zero – The output impedance of the ideal operational amplifier is assumed to be zero acting as a perfect internal voltage source with no internal resistance so that it can supply as much current as necessary to the load. This internal resistance is effectively in series with the load thereby reducing the output voltage available to the load. Real op-amps have output impedances in the 100-20kΩ range.

Resistive circuit with operational amplifier

• Bandwidth, (BW)

Infinite – An ideal operational amplifier has an infinite frequency response and can amplify any frequency signal from DC to the highest AC frequencies so it is therefore assumed to have an infinite bandwidth. With real op-amps, the bandwidth is limited by the Gain-Bandwidth product (GB), which is equal to the frequency where the amplifiers gain becomes unity.

Resistive circuit with operational amplifier

Note:

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From these “idealized” characteristics above, we can see that the input resistance is infinite, so no current flows into either input terminal (the “current rule”) and that the differential input offset voltage is zero (the “voltage rule”). It is important to remember these two properties as they will help us understand the workings of the Operational Amplifier with regards to the analysis and design of op-amp circuits.

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However, real Operational Amplifiers such as the commonly available uA741, for example do not have infinite gain or bandwidth but have a typical “Open Loop Gain” which is defined as the amplifiers output amplification without any external feedback signals connected to it.

Resistive circuit with operational amplifier

• Offset Voltage, (VIO)

Zero – The amplifiers output will be zero when the voltage difference between the inverting and the non-inverting inputs is zero, the same or when both inputs are grounded. Real op-amps have some amount of output offset voltage.

Resistive circuit with operational amplifier

• From these “idealized” characteristics above, we can see that the input resistance is infinite, so no current flows into either input terminal (the “current rule”) and that the differential input offset voltage is zero (the “voltage rule”). It is important to remember these two properties as they will help us understand the workings of the Operational Amplifier with regards to the analysis and design of op-amp circuits.

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• However, real Operational Amplifiers such as the commonly available uA741, for example do not have infinite gain or bandwidth but have a typical “Open Loop Gain” which is defined as the amplifiers output amplification without any external feedback signals connected to it.

Resistive circuit with operational amplifier

• For a typical operational amplifier, this open loop gain can be as high as 100dB at DC (zero Hz). Generally, an op-amps output gain decreases linearly as frequency increases down to “Unity Gain” or 1, at about 1MHz. This effect is shown in the following open loop gain response curve.

Open-loop Frequency Response Curve

• From the graph, the gain of the amplifier at 100kHz is given as 20dB or 10, then the gain bandwidth product is calculated as:

GBP = A x BW = 10 x 100,000Hz = 1,000,000.

• Similarly, the operational amplifiers gain at 1kHz = 60dB or 1000, therefore the GBP is given as:

GBP = A x BW = 1,000 x 1,000Hz = 1,000,000.

• The Voltage Gain (A) of the operational amplifier can be calculated as A = V_out/V_in

• in Decibels or (dB) is given as A=20log(V_out/V_in)

Resistive circuit with operational amplifier

• An Operational Amplifiers Bandwidth

The operational amplifiers bandwidth is the frequency range over which the voltage gain of the amplifier is above 70.7% or -3dB (where 0dB is the maximum) of its maximum output value as shown below.

• Here we used the 40dB line as an example. The -3dB or 70.7% of Vmax down point from the frequency response curve is given as log(70.7)*20 = 37dB.

• 37 = 20 log (A) therefore, A = anti-log (37 ÷ 20) = 70.8

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• GBP ÷ A = Bandwidth, therefore, 1,000,000 ÷ 70.8 = 14,124Hz, or 14kHz

Resistive circuit with operational amplifier

• The Open Loop Gain ( A_o ) of an operational amplifier can be very high, as much as 1,000,000 (120dB) or more.

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• However, this very high gain is of no real use to us as it makes the amplifier both unstable and hard to control as the smallest of input signals, just a few micro-volts, (μV) would be enough to cause the output voltage to saturate and swing towards one or the other of the voltage supply rails losing complete control of the output.

• We can therefore afford to lose some of this high gain by connecting a suitable resistor across the amplifier from the output terminal back to the inverting input terminal to both reduce and control the overall gain of the amplifier. This then produces and effect known commonly as Negative Feedback, and thus produces a very stable Operational Amplifier based system.

Resistive circuit with operational amplifier

• Negative Feedback is the process of “feeding back” a fraction of the output signal back to the input, but to make the feedback negative, we must feed it back to the negative or “inverting input” terminal of the op-amp using an external Feedback Resistor called R_ƒ. This feedback connection between the output and the inverting input terminal forces the differential input voltage towards zero.

• This effect produces a closed loop circuit to the amplifier resulting in the gain of the amplifier now being called its Closed-loop Gain. Then a closed-loop inverting amplifier uses negative feedback to accurately control the overall gain of the amplifier, but at a cost in the reduction of the amplifiers gain.

Circuit theorem

Operational amplifier or Op Amp as they are generally called are linear DC amplifiers. An op amp is a three-terminal device, one called the inverting terminal, one non-inverting terminal and the other is called the output terminal.