5.3 symmetry

Circuits Symmetry 1 M H Miller Introduction to Physical Symmetry As a matter of faith, supported by observation, Nature is assumed to be causal, not capricious. Given a particular set of physical circumstances the same physical behavior is expected to occur whenever and wherever these circumstances are repeated. This is usually assumed intuitively; for example the analysis of a given circuit leads to the same conclusions whenever and wherever the analysis is done. From this philosophical presumption intellectual giants have inferred what with hindsight seems almost obvious. Not the least of such inferences follows from what is commonly referred to as ‘symmetry’. Suppose, to use a relatively simple illustration, two identical electrical devices are connected in parallel as illustrated; the nature of the devices other than their being identical is not pertinent.. Suppose further a current I flows into the junction of the devices. Circuit analysts would infer directly that the current in each device is the same, referring in a general way to ‘symmetry’ as the justificati on. Indeed it i s a justifica tion. The argument is that b oth paths being the same Nature treats each in the same way. The currents in the two branches must be equal, each being I/2. The circuit drawn to the left affords a classic second illustration. Suppose each branch of the cube corresponds to a 2 Ω resistor; determine the equivalent resistance ‘seen’ by the voltage V. This is a circuit problem amenable to the more usual methods of analysis, e.g. nodal analysis. But the point here is to suggest how ‘symmetry’ may be used to suggest directly certain relationships. Suppose a current of 1-ampere flows into the node and consider an observer looking into that node. Such an observer would find no difference in moving towards node C whichever of the three branches he follows from node A. Nature must respond therefore by dividing the input current equally among the three branches, as shown. Suppose then, to provide a specific illustration, the path taken brings the observer to node B. Again Nature acting in her her even-handed way treats both options equally; the current entering B divides equally as shown. Similar current division occurs at corresponding nodes. Continue to node C, where two branch currents combine, and then on to node D where three currents recombine. Note that KCL requirements are satisfied at all nodes. The voltage drop V can be calculated following any path between nodes A and D. For the specified 2 Ω branch resistances the voltage is V = 2(1/3) + 2(1/6) + 2(1/3) = 10/6 v. The equivalent resistance of the cube is V/1 = 5/3 Ω.

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8/2/2019 5.3 Symmetry

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Circuits Symmetry 2 M H Miller

Another well-known application of symmetry is the circuit configuration commonly called a ‘bridge’drawn below (ignore the dashed connection between nodes A and B for the moment). We suppose that

the two devices marked ‘1’ are identical, as are the two devices marked‘2’ (not necessarily identical to each other). Invoking ‘symmetry’ is a

sufficient basis to assert that the voltage difference from A to B is zero.After all would nature be so capricious as to behave differently along

identical paths? Moreover is not the current Ix = I/2; why would animpartial observer (as we expect Nature to be) choose to split current

between identical paths differently? Moreover if a connection (dottedline) were to be made between nodes A and B would not as much current

flow from A -> B as in the opposite direction. It would; the current in

such a connection must be zero. (The bridge is said to be ‘balanced’.)And of course is not the current in branch 2 (either side) Ix.

Symmetry finds wide application in integrated circuit technology. For a final illustration consider the

circuit drawn below. This is a simplified circuit equivalent for a ‘differential amplifier’, a ubiquitous

component in many integrated circuits. Two identical devices represented by the current-controlled

sources (actually transistors) produce a voltage difference

va – vb = ßR(ia – ib)in response to signal currents ia and ib. Each current can be written as a

sum of an even (‘common mode’) part and an odd (‘differential mode’part as follows:

ia = (ia + ib)/2 + (ia – ib)/2ib = (ia + ib)/2 - (ia – ib)/2

Note that these equations simply are identities.

Now apply Superposition, i.e., consider the circuit response separately tothe common mode input ia = ib = (ia + ib)/2, and the differential mode input ia = -ib = (ia – ib)/2. The

output voltage difference va – vb is zero; this is a conclusion which follows from the symmetry of the

circuit and the equality of the two input signals.

On the other hand the differential mode output is ßR (ia - ib), because of the anti-symmetry of the input

signals. It happens that spurious noise induced in the circuit, for example by electrical radiation, tends

to be the same for both inputs; the inputs ordinarily are very close together. This common mode signalthen is suppressed by the circuit symmetry. On the other hand by using differential mode for a desired

input signal, i.e., making ia = - ib, the desired input is amplified.

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