A single-formula approach for designing positive summing amplifiers. By Max Bernhardt, Lange Sales

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1 A singleformula approach for designing positive summing amplifiers This circuittheory approach on opamp design and analysis has two benefits: You can use it on all opamp designs without learning special formulas or cases And it makes possible a rigorous method for designing positive summing amplifiers By Max Bernhardt, Lange Sales After a discussion on the general theory of converting an opamp design into circuit theory with an emphasis on creating one formula, I ll give examples of simple circuits to prove the theory behind this method Finally, I ll present a simple positive summing amplifier, which was developed with conversations with Dieter Knollman (eference ) Because op amps are linear devices in the s domain, circuit theory lends itself well to using these devices By using circuit theory, you can reduce many complex systems to basic blocks that can be easily evaluated using advanced computer software The more masochistic among us can readily use pen and paper To use this theory, start with a generalized opamp circuit: VP ZP VP2 VPm VN ZP2 ZPm ZN V VN2 ZN2 Zf VNi ZNi Where the variables are defined in the following manner: VP m = voltage input to the positive side of the amplifier at any particular m th location VN i = voltage input to the negative side of the amplifier at any particular i th location ZP m = input impedance seen at any particular m th location

2 ZN i = input impedance seen at any particular i th location = Feedback impedance Now, convert the opamp circuit into a circuit system by grounding all the voltage inputs and voltage outputs and applying a voltage source at the input terminals of the op amp Looking exclusively at the negative input to the op amp, the circuit looks like the following: ZN ZN2 ZNi Zf From this circuit, you can write an equation for the parallel input impedance seen by the negative input to the op amp of this form: ZN = ZN f ZN ZN 2 ZN i Where is the parallel input impedance seen by the negative input of the operational amplifier Switching to the positive side of the op amp, you can write the equation for the positive input impedance (): ZP = ZP ZP2 ZPm From basic circuit theory and throwing in offset voltage and input bias current, we can then show that the operational amplifier circuit then becomes the following block diagram:

3 VP ZP Ib VP2 ZP2 VPm ZPm Vos A Ts Vo VN VN2 ZN ZN2 Zf VNi ZNi Ib When designing your system, you can set and to be approximately equal, which allows you to ignore the input bias current Ib, because it will sum to zero at your summing node In addition, assume that the pole (open loop 3 db corner set by Ts) caused by the gain stage of the operational amplifier is much smaller than the pole in the rest of the system and can also be ignored Finally, let Vos be very small in order to have this technique follow classical analysis and you get the following block diagram:

4 VP ZP VP2 ZP2 VPm ZPm A Vo VN VN2 ZN ZN2 Zf VNi ZNi Writing the loop equations starting from VP, you get: A V o ZP = VP Then, take the limit as A gets very large: A V o ZP Lim = A VP Because you ve set equal to, the equation for the input voltage (VP ) comes out to: VP V o= * ZP

5 Because the system is linear, you can combine all the input equations and get the generalized case for the positive inputs to be: VPm* V o = m= ZPm Do the same exercise for the negative inputs: V o ZN = VN V o ZN Lim = A VN VN V o = * ZP VN i* V o = i= ZN i Combining the equations for both sides and realizing that the gain from the negative input is multiplied by negative one gives you the generalized formula for all opamp circuits: VPm* V o = m= ZPm i= VN i* ZNi This formula verifies the derivation Designing with op amps: Single formula technique keeps it simple (Dieter Knollman, EDN, March 2, 998) using a circuit theory approach Some examples that prove this equation too be correct are: Negative input gain:

6 V VN From classical theory, the equation to use is: = 4VN Using our equation, you get: 0* VN * V o = Vo = 4VN The example of a positive amplifier is: VP V From classical theory, the equation that you would use is: = VP((/) = VP Using our equation, you get: VP 0* V o = *4

7 Vo = VP The example of a negative summing amplifier is: 2 3 V VN VN2 From classical theory, the equation that you would use is: = 4*VN VN 2 Using our equation, you get: 0* VN VN V o = *4 2 *4 Vo = 4*VN VN 2 Now, derive the positive summing amplifier using this circuitdesign method To do this circuit, balance the Ib to be as equal as possible, therefore, = When you know that the input impedances are equal, the design of this circuit is straightforward First, pick your gains on ZP, ZP2, etc Then, set Zf to meet those gains Finally, select Zi to the Zf circuit to match the input impedance () seen by the positive side of the amplifier Thus, on the following circuit, should equal /

8 VP VP2 V Using our equation, you get: 0* VP VP V o = *4 2 *4 Vo = VP 2 4*VP To show this circuit in Spice, the following circuit was generated 2 4K V0 2 IVm2 200 V 0 IVm 0 K 0 K XOp_amp 4K IVm 200 If you hold V0 constant at 2V and sweep V from V and V, you get the following graph:

9 (V) PosSumOpAmpDC Transfer0 V m 0000e m e V(IVM) V(IVM2) V(IVM) You can use this design method to solve complex difficult problems that would be hard to solve using other classical methods This method can also be used with tools like Matlab to bridge the gap between physical systems and electrical design eference Dieter Knollman, PhD, is a distinquished member of the technical staff of Lucent Technologies (Denver), where he has worked for 33 years In his current position, he designs PBX port circuits Knollman earned a BSEE from the Virginia Polytechnic Institute and State University (Blacksburg, VA) an MSEE from the University of Illinois Urbana/Champaign, and a PhD from the New York University (New York) Author s biography Max Bernhardt is a field application engineer for Lange Sales He works with key leaders in the industry developing and new technologies for the market Max holds a BS in math education and a BSEE from the University of Wyoming

D is the voltage difference = (V + - V - ).

D is the voltage difference = (V + - V - ). 1 Operational amplifier is one of the most common electronic building blocks used by engineers. It has two input terminals: V + and V -, and one output terminal Y. It provides a gain A, which is usually