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z Parameters And h Parameters Of Two Port Networks

**(a)** What is a *one-port network*?
**(b)** What is a *two-port network*?
**(c)** What are z parameters?
**(d)** What are h parameters?
**(e)** Figure 1.2 is a two-port network. Find:

(i) the voltage-gain ratio, V_{2}/V_{1}, in terms of the z parameters.

(ii) the current-gain ratio, I_{2}/I_{1} in terms of the h parameters.

(iii) the current-gain ratio, I_{2}/I_{1}, in terms of the z parameters.

**The strings**:
S_{7}P_{3}A_{32} (force - push).
**The math**:

Pj Problem of Interest is of type *force* (push).

**(a)** A *pair* of terminals constitute a *port*: if there is an identifiable voltage across the terminals and the current into one terminal is the same as the current out of the other terminal.

Figure 1.1(a) consists of two *one-port* networks (A and B). Network, A is a *one-port* network when viewed from from the left of terminals 1, 2. Network B is a *one-port* network when viewed from the right side of terminals 1,2.
**(b)** Figure 1.1(b) is a *two-port* network if currents I_{1in} and I_{1out} are equal; and currents I_{2in} and I_{2out} are equal. The four variables V_{1}, V_{2}, I_{1} and I_{2} (only two of which can be independent) characterize the network.
**(c)** z parameters are the shortened name of *open circuit impedance parameters*. They are the coefficients of the dependent currents of a two-port network given that both voltages V_{1} and V_{2} are independent and the linear network contains no independent sources. In other words, the coeffiecients in the following equations:

V_{1} = z_{11}I_{1} + z_{12}I_{2} -------(1)

V_{2} = z_{21}I_{1} + z_{22}I_{2}.-------(2)

Where z_{11} = V_{1}/I_{1} (I_{2} is assumed = 0)

z_{12} = V_{1}/I_{2} (I_{1} is assumed = 0)

z_{21} = V_{2}/I_{1} (I_{2} is assumed = 0)

z_{22} = V_{2}/I_{2} (I_{1} is assumed = 0)
**(d)** h parameters are the shortened name of *hybrid parameters*. As the name suggests, they are the coefficients of a dependent voltage and a dependent current. In other words, a voltage source and a current source are independent while the other voltage and current are dependent. So, h parameters are the coefficient in the following equations given that V_{1} and I_{2} are independent and V_{2} and I_{1} are dependent:

V_{1} = h_{11}I_{1} + h_{12}V_{2} -------(3)

I_{2} = h_{21}I_{1} + h_{22}V_{2}.-------(4)

Where h_{11} = V_{1}/I_{1} (V_{2} is assumed = 0)

h_{12} = V_{1}/V_{2} (I_{1} is assumed = 0)

h_{21} = I_{2}/I_{1} (V_{2} is assumed = 0)

h_{22} = I_{2}/V_{2} (I_{1} is assumed = 0)

**(ei)** Consider V_{2} and R_{L} of figure 1.2:

By Ohms Law, -V_{2}/R_{L} = I_{2}

From equation (2), I_{2} = (V_{2} - z_{21}I_{1})/z_{22}-------(5)

From equation (1) I_{1} = (V_{1} - z_{12}I_{2})/z_{11} = (V_{1} - z_{12})(-V_{2}/R_{L}))/z_{11}-------(6)

Replacing I_{1} in (5) by its value in (6), we have:

-V_{2}/R_{L} = [V_{2} - z_{21}((V_{1} - z_{12})(-V_{2}/R_{L}))/z_{11})]/z_{22}-------(7)

So, (z_{21}z_{12}/R_{L}))/z_{11})]/z_{22})V_{2} - V_{2}/z_{22} - V_{2}/R_{L} = -(z_{21}/z_{11}z_{22})V_{1}

So, (z_{21}z_{12}/R_{L}z_{11}z_{22})V_{2} - V_{2}/z_{22} - V_{2}/R_{L} = -(z_{21}/z_{11}z_{22})V_{1}

So,V_{2}/V_{1} = z_{21}R_{L}/(z_{11}R_{L} + z_{11}z_{22} - z_{12}z_{21}).
**(eii)** Consider V_{2} and R_{L} of figure 1.2:

By Ohms Law, -V_{2}/R_{L} = I_{2}

Let V_{1} and I_{2} be the independent variables, then, equation (4) holds:

I_{2} = h_{21}I_{1} + h_{22}V_{2}

Since V_{2} = -I_{2}R_{L}, we have

I_{2} = h_{21}I_{1} - h_{22}I_{2}R_{L}

So, I_{2}/I_{1} = h_{21}/(1 + h_{22}R_{L}).
**(eiii)** Consider V_{2} and R_{L} of figure 1.2:

By Ohms Law, -V_{2}/R_{L} = I_{2}

From equation (2), I_{2} = (V_{2} - z_{21}I_{1})/z_{22} -------(5)

Substitute V_{2} = -I_{2}R_{L} in (5), we have:

I_{2} = (-I_{2}R_{L} - z_{21}I_{1})/z_{22}

So, I_{2}(z_{22} + R_{L}) = - z_{21}I_{1})

So, I_{2}/I_{1} = - z_{21}/(z_{22} + R_{L}).

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

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