Analysis of one port and two port networks

Network functions and two-port parameters let engineers describe a circuit by what it does at its terminals rather than by every internal component. A transfer function relates output to input in the Laplace domain; its poles and zeros reveal stability and the shape of the response. Two-port parameters (Z, Y, h, ABCD) model amplifiers, filters, transformers and transmission lines as "black boxes". NEC MCQs often test the definition of each parameter set, conditions for reciprocity and symmetry, and which parameters add when networks are connected.

Key concepts

Network functions

• Driving-point (one-port) function: ratio of voltage and current at the same port — driving-point impedance $Z(s) = V(s)/I(s)$ or admittance $Y(s) = 1/Z(s)$.
• Transfer function: ratio of output at one port to input at another, with zero initial conditions, e.g. voltage ratio $V_2/V_1$, transfer impedance $V_2/I_1$.
• Impedances in s-domain: R, $sL$, $1/(sC)$.

Poles and zeros

• Write $H(s) = K\frac{(s - z_1)(s-z_2)\cdots}{(s-p_1)(s-p_2)\cdots}$. Zeros make H = 0; poles make H infinite.
• Relationship to response:
  • Pole on negative real axis → decaying exponential $e^{-at}$.
  • Complex-conjugate poles in left half plane → damped oscillation; the further left, the faster the decay.
  • Poles on the jω axis (simple) → sustained oscillation (marginally stable).
  • Any pole in the right half plane, or repeated poles on jω axis → unstable (growing response).
• Zeros shape the amplitude of response terms but do not affect stability.

One-port passive circuits: for a passive RLC network, $Z(s)$ is a positive real function; poles and zeros of LC networks lie on the jω axis and alternate; RC impedance poles/zeros lie on the negative real axis and alternate, with the pole nearest the origin.

Two-port parameters (port 1 input $V_1, I_1$; port 2 output $V_2, I_2$; currents into the ports)

Parameters	Equations	Found by	Reciprocal if	Symmetric if
Z (open-circuit impedance)	$V_1 = z_{11}I_1 + z_{12}I_2$, $V_2 = z_{21}I_1+z_{22}I_2$	opening ports ($I=0$)	$z_{12} = z_{21}$	$z_{11} = z_{22}$
Y (short-circuit admittance)	$I_1 = y_{11}V_1 + y_{12}V_2$, $I_2 = y_{21}V_1 + y_{22}V_2$	shorting ports ($V=0$)	$y_{12}=y_{21}$	$y_{11}=y_{22}$
ABCD (transmission)	$V_1 = AV_2 - BI_2$, $I_1 = CV_2 - DI_2$	—	$AD - BC = 1$	$A = D$
h (hybrid)	$V_1 = h_{11}I_1 + h_{12}V_2$, $I_2 = h_{21}I_1 + h_{22}V_2$	mixed	$h_{12} = -h_{21}$	$h_{11}h_{22}-h_{12}h_{21} = 1$

• Example definitions: $z_{11} = V_1/I_1|_{I_2=0}$ (input impedance, output open); $y_{11} = I_1/V_1|_{V_2=0}$.
• h-parameters: $h_{11}$ = input impedance (Ω), $h_{12}$ = reverse voltage ratio (unitless), $h_{21}$ = forward current gain (unitless), $h_{22}$ = output admittance (S). Widely used for BJT models.
• ABCD: A and D are dimensionless, B in ohms, C in siemens. Used for transmission lines.
• Inverse hybrid (g) and inverse transmission parameters also exist.

Inter-relationships

• $[Y] = [Z]^{-1}$, so $y_{11} = z_{22}/\Delta_z$, $y_{12} = -z_{12}/\Delta_z$, where $\Delta_z = z_{11}z_{22} - z_{12}z_{21}$.
• $A = z_{11}/z_{21}$, $B = \Delta_z/z_{21}$, $C = 1/z_{21}$, $D = z_{22}/z_{21}$.
• Not every set exists for every network: a single series impedance has Y parameters but no Z parameters, while a single shunt impedance has Z parameters but no Y parameters.

Interconnections

Connection	Parameters that add
Series-series	Z matrices add
Parallel-parallel	Y matrices add
Series-parallel	h matrices add
Parallel-series	g matrices add
Cascade (tandem)	ABCD matrices multiply

High-yield points

• Z parameters are open-circuit, Y parameters short-circuit; $[Y] = [Z]^{-1}$.
• Reciprocity: $z_{12} = z_{21}$, $y_{12} = y_{21}$, $AD - BC = 1$, $h_{12} = -h_{21}$.
• Symmetry: $z_{11} = z_{22}$, $y_{11} = y_{22}$, $A = D$.
• Cascade: ABCD matrices multiply; series: Z add; parallel: Y add; series-parallel: h add.
• A right-half-plane pole means instability; simple jω-axis poles give sustained oscillation.

Formulas

$[T]_{total} = [T_1][T_2]\quad\text{(cascade)}$

Input impedance of a two-port with load $Z_L$ (ABCD): $Z_{in} = \frac{AZ_L + B}{CZ_L + D}$.

Worked example

A T-network has series arms $Z_a = 2$ Ω, $Z_b = 2$ Ω and shunt arm $Z_c = 4$ Ω. Find the Z parameters and check reciprocity/symmetry.

1. $z_{11}$ (port 2 open): $Z_a + Z_c = 2 + 4 = 6$ Ω.
2. $z_{22}$ (port 1 open): $Z_b + Z_c = 6$ Ω.
3. $z_{12} = z_{21} = Z_c = 4$ Ω.
4. Reciprocal ($z_{12}=z_{21}$) and symmetric ($z_{11}=z_{22}$). ✓
5. $\Delta_z = 36 - 16 = 20$, so $y_{11} = 6/20 = 0.3$ S.

Exam tips

• "Open-circuit" = Z parameters; "short-circuit" = Y parameters.
• Cascade → ABCD multiply; series → Z add; parallel → Y add; series-parallel → h add.
• Reciprocity: $AD - BC = 1$ and $h_{12} = -h_{21}$ (note the minus sign).
• Right-half-plane pole = unstable; jω-axis simple poles = oscillatory.
• Units of $h_{22}$ are siemens (admittance), $h_{11}$ ohms.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. In the hybrid parameters of a two-port network, $h_{21}$ represents:

• A. Short-circuit input impedance
• B. Open-circuit reverse voltage gain
• C. Open-circuit output admittance
• D. Short-circuit forward current gain

Answer: D. $h_{21} = I_2/I_1$ with $V_2 = 0$, a dimensionless short-circuit forward current gain.

2. When two-port networks are connected in cascade, the overall network is most easily found by multiplying their:

• A. Transmission (ABCD) matrices
• B. z-parameter matrices
• C. h-parameter matrices
• D. y-parameter matrices

Answer: A. In cascade, the output of one network is the input of the next, so $[T] = [T_1][T_2]$. z-matrices add for series connection and y-matrices add for parallel connection.

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