Two-ports

A two-port is a network you have stopped caring about the inside of. What is left is four numbers relating the voltage and current at one pair of terminals to the voltage and current at the other. Symbulator will find those four numbers for you, and will also take them as an element in a larger circuit.

13.1Find two-port equivalents

Set Type of analysis to Find equivalent and Type of equivalent to Two-port parameters. It takes the network and the two top nodes of the pair of terminals you want to reduce. The two bottom nodes are always assumed to be ground.

There are six kinds of parameters — z, y, h, g, a and b — and you choose which you want.

AS7's Example 19.1

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Determine the z parameters for the circuit.

Solution

First describe the network, exactly as you would describe any circuit:

Circuit Description

r1,1,2,20
r2,2,0,40
r3,2,3,30

Find equivalent, then Two-port parameters, with Parameters set to z — impedance and 1 and 3 in the two node boxes. In DC.

There is no name to invent: version 9 does not store its answers in calculator variables, so the four come back as z11, z12, z21 and z22.

Results gives z11 = 60 Ω, z12 = 40 Ω, z21 = 40 Ω and z22 = 70 Ω.

Correct.

13.2Use two-ports as elements

A two-port can be an element in a bigger circuit. Its description is three fields: a name whose first letter says which kind it is, then the top left node and the top right node. Both bottom nodes are ground, always — which is why neither top node may be 0, and the two may not be the same node; Symbulator stops with a message if they are.

z,1,2

That is a z-type two-port between nodes 1 and 2. An optional fourth term carries its four parameters — see below.

Giving it its four parameters

Three possibilities.

Give them in the description, as an optional fourth term — a list of four, in the order 11, 12, 21, 22:

z,1,2,[40,20j,30j,50]

This is the way to prefer: the parameters travel with the circuit, so they survive being pasted into a link, saved to a file, or handed to any other tool — and they work in every analysis, the equivalent tools included. The entries can be numbers, SI-prefixed values, or expressions, symbols among them.

Store them in Define, calculator-style, and keep the description bare:

z11 = 40
z12 = 20j
z21 = 30j
z22 = 50

Define any subset and the rest stay symbolic. This is the closest cousin of how versions 7 and 8 stored the values in variables before simulating.

Leave them undefined, in which case they stay symbolic and appear in the answers as z11, z12, z21 and z22. That is often what you want — the answer as a formula in the parameters. (Pinning them afterwards in Expert Mode's Add equations still works too, in a plain solve, and an equation or condition there overrides the description's own values.)

What answers do you get? The current entering each port, named with the two-port and the node it sits on — for a two-port z on nodes 1 and 2, that is iz1 and iz2.

AS7's Example 19.2

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Find I₁ and I₂ in the circuit.

Solution

Circuit Description

e,1,0,100
z,1,2,[40,20j,30j,50]
r,2,0,10

The four parameters ride in the description's fourth term. In AC.

aa(iz1) reads 2∠0° A and aa(iz2) reads 1∠-90° A.

13.3Instructive two-port problems

AS7's Example 19.3

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Obtain the y parameters for the network.

Solution

Circuit Description

r1,1,0,4
r2,1,2,2
r3,2,0,8

Find equivalent, Two-port parameters, y — admittance, nodes 1 and 2, DC.

3/4, -1/2, -1/2, 5/8, all in S.

Correct.

AS7's Example 19.4

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Determine the y parameters for the two-port shown. Note the dependent source.

Solution

Circuit Description

r1,1,2,8
r2,2,0,2
r3,2,3,4
j,2,3,2*ir1

y, nodes 1 and 3, DC. A two-port equivalent works perfectly well around a controlled source.

3/20, -1/20, -1/4, 1/4, all in S.

Correct.

AS7's Example 19.5

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Find the hybrid parameters for the two-port network.

Solution

Circuit Description

r1,1,2,2
r2,2,0,6
r3,2,3,3

h — hybrid, nodes 1 and 3, DC.

4, 2/3, -2/3, 1/9.

Correct.

AS7's Example 19.6

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Determine the Thévenin equivalent at the output port of the circuit, whose h parameters are 1000 Ω, −2, 10 and 2×10⁻⁴ S.

Solution

Circuit Description

e,1,0,60
r,1,2,40
h,2,3,[1000,-2,10,2e-4]

Find equivalent, Thévenin / Norton, nodes 3 and 0, DC. Because the parameters ride in the description, they reach the equivalent tools like any other value, and the answers come straight back as numbers.

Leave the fourth term off — h,2,3 alone — and the same run answers with formulas instead:

\[v_{th} = \dfrac{-60\,h_{21}}{h_{11}h_{22} - h_{12}h_{21} + 40\,h_{22}}\]

which is useful in its own right: put vth in Evaluate with the four parameters in its Conditions box, and the same number falls out.

vth comes back -29.69 V and req 51.46 Ω.

Correct.

AS7's Example 19.7

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Find the g parameters as functions of s for the circuit.

Solution

Circuit Description

l,1,2,1
r,2,0,1
c,2,3,1

g — inverse hybrid, nodes 1 and 3, and this time in FD, since the question asks for functions of s.

g11 = \(1/(s+1)\), g12 = \(-1/(s+1)\), g21 = \(1/(s+1)\) and g22 = \((s^2+s+1)/(s(s+1))\).

AS7's Practice Problem 19.7

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For the ladder network, determine the g parameters in the s domain.

Solution

Circuit Description

l1,1,2,1
r1,2,0,1
l2,2,3,1
r2,3,0,1

g, nodes 1 and 3, FD. The four come back as \((s+2)/(s^2+3s+1)\), \(-1/(s^2+3s+1)\), \(1/(s^2+3s+1)\) and \(s(s+2)/(s^2+3s+1)\).

Correct.

AS7's Example 19.8

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Find the transmission parameters for the two-port network.

Solution

Circuit Description

r1,1,2,10
r2,2,0,20
e,2,3,3*ir1

a — transmission, nodes 1 and 3, DC.

1.765, 15.29, 0.0588, 1.176.

Correct.

13.4The gain tool

Network problems ask for gains: voltage gain, current gain, power gain, and the impedance seen at the input. Symbulator has a tool for that, in the Mini-Tools card: choose gain.

It wants four values — the voltage and current at the input, then the voltage and current at the output — and answers with all four figures at once.

Gain Example 1

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For the circuit, find G_v, G_i, G_p and Z_in. The two-port has y parameters y₁₁ = 0.4 S, y₁₂ = −0.002 S, y₂₁ = −5 S and y₂₂ = 0.04 S.

Solution

The source is 1 V because its value does not matter to a gain — it cancels. A symbolic vs would do just as well.

Circuit Description

es,3,0,1
rs,3,1,2
rl,2,0,20
y,1,2,[0.4,-0.002,-5,0.04]

Solve in DC, then open Mini-Tools, choose gain, and give it the four:

v1

v1

i1

iy1

v2

v2

i2

iy2

The voltage gain Av is 55.56, the current gain Ai -9.615, the power gain Ap 534.2, and the input impedance Zi 3.462 Ω.

These are correct.

es,3,0,1
rs,3,1,5
rl,2,0,2
z,1,2,[4,1.5,10,3]

v1

iz1

v2

iz2